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NUMBER PATTERNS ANSWERS - Browse Patterns

NUMBER PATTERNS ANSWERS edHelper.com – Number Patterns

RANDOM.ORG – True Random Number Service. RANDOM.ORG offers true random numbers to anyone on the Internet. The randomness comes from atmospheric noise, which for Welcome to Math Homework Answers – Math Homework Answers is a math help site where students, teachers, and math lovers can ask and answer math homework questions to 6/30/2010 · Best Answer: 4a) that is arithmetic series with a = 0.5 and d = 1- 0.5 = 0.5 a(n) = a + d.(n – 1) 20 = 0.5 + 0.5(n-1) 40 = 1 + (n-1) 39 = n – 1 n = 40 S(n

JERC Multiplication Facts Tool – Jonathan Bass – TeachersPayTeachers

Number patterns? – Yahoo!7 Answers

6/16/2013 · Best Answer: The first pattern is cubes, i.e. n^3. The second pattern is an arithmetic sequence with a common difference of 5; i.e. just add 5 each time. 6/4/2013 · 1. an=2(n-1) 2. an=58-9n 3. an=3*(n^n) 4. an=a(n-1)+2n+1 -2,0,2,4, –> 2n – 4 58,49,40,31 –> -9n + 67 because the "nth term" (un) in an 13. This sequence has a difference of 3 between each number. The pattern is + 3 each time.

Comparing Numbers to Thousands – Dawn Harrell – TeachersPayTeachers

Number Pattern for 10*4n-1 – Math Homework Answers

TenMarks teaches you how to be able to understand number patterns. 3/8/2008 · Best Answer: 1) – 1— 3—–7—11 —–4 —–4—-4 —–difference of 4 T1 = –1, T2 = –1+4, T3 = – 1 + 2×4 T20 = – 1 + 19 × 4 = 75 Tn To solve number patterns, you must be able to find the number patterns in data. Once the pattern is found, then you can solve the problem. T

Grade 7 math worksheets, pre algebra word problems, Page 8

complete the number pattern. – Math Homework Answers

how to figure out number patterns?:Number patterns can be defined as. how to figure out number patterns? number patterns 2. Answers. Was this 6/4/2013 · What are the next 3 numbers of each pattern? And what is the nth term? -2, 0, 2, 4 58,49,40,31 3,12,27,48 2,5,10,17 0,2,6,12,20 0,5,12,21 Contains 10 hidden number pattern problems. The answers can be found below. Standard: MATH 1 Grades: (K-2) View worksheet. Number Pattern Quiz

turkey number patterns 1 find the missing numbers in each turkey count

What’s this number pattern? (8, 15, 23)? – Yahoo! Answers NZ

5/20/2010 · Best Answer: A "pattern" can be anything. You could say the list 1234, 2341, 3412 has a pattern because each term moves the first digit of the previous http://answers.ask.com/Computers/Other/what_number In mathematics, a number pattern is a list of numbers that follow a defined series or pattern.

patterns level 1 these one page worksheets introduce patterns students

Learn About Number Patterns video from Answers.com Videos

3/29/2007 · Best Answer: 1, 2, 5, 10, 17, 26, 37. The sequence is 0^2 + 1, 1^2 + 1, 2^2 + 1, 3^2 + 1, 4^2 + 1, 5^2 + 1 and so on 9/21/2011 · Best Answer: They sound like times tables squares. Your son needs to colour multiples of 2 on the 2x square and multiples of 10 on 10x square. This should 2/8/2008 · Best Answer: 1/(1^2) = 1 1/(2^2) = 1/4 1/(3^2) = 1/9 1/(4^2) = 1/16 1/(5^2) = 1/25 1/(6^2) = 1/36 1/(7^2) = 1/49 1^-2 2^-2 3^-2 4^-2 5^-2 = 1/25 6^-2

Simple Geometric Patterns Worksheets

3/8/2008 · Best Answer: 1) – 1— 3—–7—11 —–4 —–4—-4 —–difference of 4 T1 = –1, T2 = –1+4, T3 = – 1 + 2×4 T20 = – 1 + 19 × 4 = 75 Tn To solve the number pattern, read the pattern from beginning to end. Start with the first few numbers and try to figure out what the relationship between them 6/3/2008 · Best Answer: Any of them, plus any other number you care to think about. What comes next in these patterns? 3, 9, 2, 8, 32, 3, 15, 3, 9, 2, 8, 32, 3, 9

February 2013. Best Brain Teasers

How To Solve Number Patterns? – Ask.com

NUMBER PATTERNS (Answers) Look at the numbers then try and work out what the missing numbers are. 1. +2 What is the pattern for the following numbers and what will come next? This is making me crazy. Perhaps I’m just too dense to figure it out. 4, 12, 6, 12, 36, 18, 36 edHelper subscribers – Create a new printable Answer key also includes questions Answer key only gives the answers No answer key

Yahoo! Answers – Number pattern problems? please help?

3258,3308,3358,_,_, Welcome to Math Homework Answers – Math Homework Answers is a math help site where students, teachers, and math lovers can ask and answer math Just as it is written, a number pattern is a pattern of numbers. This can be either sequential or non-sequential and have a regular or irregular spaces between each 6/2/2011 · Best Answer: A series which has some relation. Its simple. For example :- 2,3,4,7,11. all these are prime numbers. 7 ,8,10,13,17. +1 ,+2, +3

display the Calculators and Hundred Boards: Displaying Number Patterns

Help with a number pattern on AOL Answers.

9/19/2007 · Best Answer: okay, got #4 – count the number of open ends when writing the letter. B has no open ends, C has 2, D has none, E has 3, F has 3, so G has 2 Microsoft Word – Year 4 Number Patterns Worksheet – ANSWERS Author: emt2 Created Date: 10/24/2012 12:17:09 PM 6/1/2013 · Best Answer: many patterns is there: e.g. i) 8, 15, 23 = 8, 8+(7), 8+(7+8). 8+ (7+8+9), 8+ (7+8+9+10). = 8, 15, 23, 32, 42. ii) 8, 15, 23

Number Patterns Worksheets for Elementary – David Filipek

Yahoo! Answers – Maths number patterns?

What is the number pattern of the numbers 781013? There is no number pattern in 781013. But there is a number pattern in the numbers 7, 8, 10, 13 4/13/2010 · Best Answer: Question A is powers of 5. Each number is 5 times the one before it. 5 times 1 = 5 5 times 5 = 25 5 times 25 = 125 5 times 125 = 625 The next 2/21/2010 · Best Answer: The progression is: Multiply by 2, add three, multiply by 2, add 3, multiply by 2, add 3 Start = 1 1×2 = 2 2+3 = 5 5×2 = 10 10+3 = 13 13×2

number patterns search for modart modulo art pattern kaleidoscopic

What is number pattern? Explain with example? – Yahoo! Answers India

Making Number Patterns (A) Answers Instructions: Make a number pattern for each of the rules. Start at 63 and subtract 4 each time. 63, 59, 55, 51, 47 3/3/2013 · Best Answer: The ‘relationship’ works both ways. And that is because algebra defines a set of rules, based on the foundations of arithmetic, by which Answers to questions are provided for entertainment purposes only. This page is for providing answers to the question "Does anyone ever see the same number patterns?"

counting game over 100? | SpanishDict Answers

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Binomial Expansion

Binomial Expansion

The binomial theorem is used to expand the binomials to any given power without direct multiplication. The coefficients appeared in the binomial expansion are called Binomial Coefficient. They are the same as their entries of Pascal’s triangle and can be determined by a simple formula using factorials.

Definition of Binomial Coefficient The binomial coefficient $\begin
n\\
k
\end$ is the coefficient of the x k term in the polynomial expansion of the binomial power (1 + x) n .

In combinations, \begin
n\\
k
\end is interpreted as the element -k subsets (the k-combinations) of an n-element set, that is the number of ways that k things can be "chosen" from a set of n things. Hence, \begin
n\\
k
\end is often read as "n choose k" and is called the choose function of n and k. The coefficient of any term in the expansion of (x + a) is the number of combinations of a particular size that can be drawn from a given set.

Binomial Expansion Formula

According to the binomial coefficient properties, the binomial theorem states that the following binomial formula is valid for all positive values of n integers:

Basic Binomial expansion:

If n is a rational number and x is a real number s.t. |x| < 1, then


n. Which represents the product of the first n positive integers are as follows,

Binomial Expansion for Negative Powers

Below you could see expansion for negative powers:

Binomial expansion for (1 + x) n has the form

Changing n to -n and simplifying, we get

Binomial Expansion Examples

Given below are some examples on Binomial Expansion.

Solved Examples

Question 1: Solve (x + 20) 3
Solution:

Step 2: y = 20, n = 3

(x + 20) 3 = x 3 + 3x 3 - 1 20 + $\frac<3 \times (3 - l)><2!>$ x 3 - 2 20 2 + $\frac<3(3 - 1)(3 - 2)><3!>$ x 3 - 3 20 3 +. + 20 3

= x 3 + 60 x 2 + 1200 x + 8000

Question 2: Find binomial coefficient (x + 2) 6 by using the binomial theorem.
Solution:

By comparing (x + 2) 6 with (a + b) n. we havea = x. b = 2 and n = 6.

Substituting in the formula, we may get

(x + 2) 6 = x 6 + 12x 5 + 60x 4 + 160x 3 + 240x 2 + 192x + 64

Co efficient of x 6 = 1

Co efficient of x 5 = 12

Co efficient of x 4 = 60

Co efficient of x 3 = 160

Co efficient of x 2 = 240

Co efficient of x = 192

Factorising help! (maths, read description)?

Factorising help. (maths, read description)?

I have some maths homework to factorise some quadratic equations and i understood them all except these ones: x² -7x +8 =0 x² -4x +21 =0 My teacher said that if the order it a - and then a + then they should both turn out to be minus's. This worked for all the others except these. Please help? Thanks so much if you answer :)

Asked on Jun 14 2014 by SnowdropBee

Your observation is correct. Please recheck the problems to be absolutely sure the final terms are positive. If they are not, the problems factor into (x-8)(x+1) and (x-7)(x+3)
If the problems are stated correctly, there is no factorization.

Answered on Jun 14 2014 by Francis K

okay so i am currently going through the book "Pure Core Maths 1&2 AQA AS-Level. i have exams coming up and am currently going through the topic of factorising polynomials, and this question popped up: b) Factorise f(x)=x^3-5x^2-33x-27, knowing that (x+3) is a factor. i got the right answers (x+3),(x+1) and (x-9). then part two says, "Hence factorise y^6-5y^4-33y^2-27". the answer is (x^2+3),(x^2+1), (x+3) and (x-3). i am not sure why this is the case? can someone please explain it to me. thank you :(

Starting with y^6-5y^4-33y^2-27:
Make the simplifying substitution x = y^2, x^2 = y^4, x^3 = y^6:
x^3 - 5x^2 - 33x - 27
This is the same polynomial you already factored in part one, so:
(x + 3) (x + 1) (x - 9)
Undo the substitution:
(y^2 + 3) (y^2 + 1) (y^2 - 9)
The first two quadratic factors don't factor any farther in real numbers, but the third one is a difference of squares, so:
(y^2 + 3) (y^2 + 1) (y + 3) (y - 3)

A monkey had 75 peaches Each day, he kept a fraction of his peaches, gave the rest away, and then ate one. These are the fractions he decided to keep: 1/2. 1/4. 3/4. 3/5. 5/6. 11/15 In which order did he use the fractions so that he was left with just on peach at the end? Please could you give the answer and explain how you got there. Thanks so much :)

Okay, I don't want to give you the answer, but I'll give you a pretty good place to start.

When you do the problem, try to use the fraction that would fit the least number of situations. For example, you start with 75 peaches. Because 75 is not an even number, you know that you can only use the fractions 3/5 or 11/15, right? Looking at the denominator, which fraction would apply to the least number of situations? 11/15. Right? You can use the fraction 3/5 with any number of peaches divisible by 5, but 11/15 only with peaches divisible by 15. Does that make sense? The reason you want to use the fraction that applies to the fewest number of situations is because there will be fewer chances in the rest of the problem to use that fraction.

So, the monkey starts with 75 peaches. The monkey keeps 11/15, or 55 peaches, and then eats one. He is then left with 54 peaches. Now, which fractions can you use with 54 peaches? Which one of those fractions is able to be used in the FEWEST number of situations (hint: the fraction with the largest denominator).

1: Figure out which fractions you can use
2: Find the fraction that can be used in the fewest number of situations
3: If two fractions have the same denominator, you need to pick one and see if it works. If it doesn't work, try the other one.
4: There's one step where rule #2 doesn't work toward the end, but you'll realize that in the very next step.

I hope that makes sense!

School of Mathematics Help Desk

School of Mathematics Help Desk

(Moved to bottom of this page.) StuRat 16:07, 28 May 2007 (UTC)

How can N dots be placed on the surface of a sphere, equally distanced or evenly spaced. What kind of math is needed for this kind of problems ?

What does evenly spaced mean? The simplest example - put them all on the equator. Are you talking about the w:Platonic solids. Find area of the sphere and divide it by N. If an area element can be devised that has this area with a dot in the middle then you can divide and place dots without pesky off by one errors. Square root of the area element might give an approximatation of the size of the area element. This type of approach is related to limits and calculus. Mirwin 06:23, 20 September 2007 (UTC) This Q was discussed in depth here: w:Reference_desk/Archives/Mathematics/2007_August_29#Distributing_points_on_a_sphere. StuRat 14:32, 27 November 2009 (UTC)

What is a complex plane?

  • It is the set of complex numbers represented as a plane (plane in the usual sense).--Hillgentleman |Talk
Expanding a bit. A complex number has two components which are different types of numbers. A common representation is a real number plus an imaginary number (a real number multiplied by the square root of minus one or i ). The set of complex numbers x + y(i) represented as a plane can be plotted two dimensionally by treating the x as an x coordinate and the y(i) as a y-coordinate. Mirwin 11:41, 25 January 2007 (UTC) "The" complex plane, rather than "a" complex plane, I'd say better. --Jorge 02:22, 6 February 2007 (UTC) Check out complex numbers I will explain all there Graemeb1967 11:12, 27 November 2009 (UTC)

How do you find the capacity of a cylinder in litres?

The formula for the volume of a right cylinder is V =Ah. That is, the volume equals the area of one of the bases times the cylinder's height. In the case of a circular cylinder, the area of a base is A = πr 2. So, the volume of a right circular cylinder is V = πr 2 h. Now, if you start with radius and height in centimeters, this will give you cubic centimeters. If you start with inches, this will give you cubic inches. The final step is to convert these volume units into liters. This site lists the conversion factors needed: [1]. StuRat 21:13, 17 April 2007 (UTC)

If there are twenty-five people in a room and everyone shakes hands with everyone else once, how many handshakes are there? (I got three hundred but my textbook says three hundred and twenty-five which I think is wrong. If you also got any of these two answers please tell me and explain how.) (The preceding unsigned comment was added by Ricky Di Don (talk • contribs ) 16 October 2016‎)

Petroleum Tank Measuring Edit

How do you know how many gallons are in a petroleum tank that is horizontal. What is the math equation. Note, I am not a math wizard so please use examples that a lay person can understand.

What shape is the tank? Also, do you have any more details about the tank (for example, do you know the height of the petroleum as measured from the bottom of the tank)? Deltinu 05:04, 19 January 2007 (UTC)

Interested in Brushing Up, Where Do I Start Edit

I'm interested in brushing up on my math and have "Calculus by Discovery" on loan from a friend. I'm looking for something here that would complement that. I found the Intro to Calculus page very confusing. My question is where do I start, and is there someone to whom I can pose questions or talk to to make sure I'm understanding everything correctly.

Also, on an only tangentially related note, has there been any discussion of a mathematics-through-application course? One that would pose (word) problems and then explain the math in them. I ask because I'm 27 and it took until now for numbers to mean more to me than simply a grade on a report card.

Feel free to pose any question at my Talk Page or just right here. I'll try to answer them to my best. About the course you mention, I don't know if anything has been talked. --Jorge 07:36, 9 February 2007 (UTC)

A section on "Word problems" would be great! Feel to start it or continue asking word problems here. Eventually someone will probably edit the best of this archive into a page such as you suggest. Great idea! Thanks for contributing here. Mirwin 06:27, 20 September 2007 (UTC)

I think that I'm in much the same boat as the one who started this entry. I, too, would like to brush up on my math skills, but I'm pretty sure that I'm further behind than he is (calculus comes after trig, right? ) The trouble is that I'm not just uncertain about how and where to begin; I'm also uncertain of where, exactly, I'm at. I was hoping someone could direct me to some kind of online self-assesment site, test, or application to figure out where I fell behind. I've been out of school sometime now, and it seems like everything after 8th grade is just a blur. I'm primarily interested in algebra, trigonometry and geometry, but I think it makes sense to assess where I'm at before I start looking for learning materials. Any advice?

Khanacademy.org has lots of content including, most notably, a collection of video lectures covering different topics. It might be helpful. 173.154.158.197 06:51, 22 March 2011 (UTC)

Equal Temperament Question Edit

Hi, can you outline the steps to get the answer 261.626 by the below equation on a scientific calculator?

You can do this, as is, using parens, but it's far easier if you do part of it in your head and/or use the calculator memory (or write down) parts of it. Here are some ways to do it:

Here's the long way:

Here's a slightly shorter way (doing the subtraction in your head):

Here's an even shorter way (doing the division in your head, too), which eliminates the need for parens:

Note that the ± key is labeled +/- on the calculator at that site, and is located right under the + sign. It toggles the current displayed value between positive and negative. That calculator doesn't appear to do rounding, so you will need to round off to 3 decimal places yourself. StuRat 16:22, 28 May 2007 (UTC)

Why are the above equations in two different fonts? How can I change the top equation to the font size of the bottom eqation? Calgea 18:17, 28 May 2007 (UTC)

If the equation is simple enough, it will be rendered as HTML. There's an option in your preferences page to change this. You can also force PNG rendering by inserting and removing a space (\,\;) -- take a look: g = A r = R ω 2 = L ( 2 π / T ) 2 <\displaystyle g=Ar=R\omega ^<2>=L(2\pi /T)^<2>\,\;> . See also m:Help:Displaying a formula. HTH. --HappyCamper 18:38, 28 May 2007 (UTC) I agree with the above except you only need the "\;" part. So, this formula:

<math>g = Ar = R\omega^2 = L(2\pi/T)^2 \;</math>

StuRat 02:16, 29 May 2007 (UTC)

Wikiversity Talk: Naming Conventions # Removing Course Numbering Scheme Edit

At "Removing Course Numbering Scheme" Talk #1 & "Removing Course Numbering Scheme" Talk #2 are Discussions in which the Undersigned would wish to have the benefit of the thinking of our Mathematicians. Could some Mathematicians please weigh in on these Discussions. Thank you in advance for your assistance.

(posted here as I'd prefer an algorithmic explanation than program code)

For a set of small rectangles whose dimensions are random, place these rectangles within the minimum number of fixed size larger rectangles? Any thoughts? If it helps the context is image printing. Sfan00 IMG 23:47, 21 March 2011 (UTC)

Just to try to understand better, I have some questions: 1) Are each of the small rectangles the same size as each other. 2) Are each of the large rectangles the same size as each other. For now, I'm assuming that the small rectangles vary in size, while the large rectangles do not. Some general thoughts: A) This is going to require lots of trial-and-error, so a computer program probably is the best way to go, if you have lots of rectangles. B) I think the edges of all the rectangles should always be parallel, for ideal packing. Thus, the only possible rotation is 90 degree (180 or 270 just give you the same thing as 0 and 90). Only rotate the small rectangles. C) Start by placing the largest of the small rectangles (if they are different sizes). Then place the next largest small rectangle, etc. until all have been placed, trying the 90 degree rotation at each step. D) Calculate the "packing factor" (percent of space used) at each step, then try the next arrangement, until you find the best arrangement. StuRat 00:13, 26 March 2011 (UTC) For context,
  • The smaller rectangles are not all the same size, and the sizes are random. (representing bitmap graphics areas). However it is a finite set of rectangles.
  • the larger rectangles are a fixed size ( representing a sheet of printer paper. for example A4)

I don't need a 100% optimal soloution ( which would be rather hard to compute for any given set of random rectangles I'm told) You are correct about the simplification you note in (B), that the only possible orientations are landscape or portrait..

Starting with the largest rectangle is where I would have started as well.

In fact, I would probably go further and arrange the incoming rectangles so that all have the same orientation, either portrait or landscape. with the longest side of the smaller rectangles parllel to the longest side of the larger one.

In reading around I've come across some demonstrations of shelf-packing algorithms here - http://users.cs.cf.ac.uk/C.L.Mumford/

Sfan00 IMG 11:53, 26 March 2011 (UTC)

I don't think there's any reason why arranging all the rectangles in the same orientation is likely to produce optimal packing. Here's a counter-example: StuRat 20:52, 26 March 2011 (UTC)

Physics Homework Help

1. In SI units, velocity is measured in units of meters per second (m/s). Which of the following combinations of units can also be used to measure velocity?
(a) cm/s (b) cm/s 2 (c) m 3 /(mm 2 s 2 ) (d) km/s (e) miles per hour (f) km/hour (g) miles/cm

2. A typical airplane can fly at a speed of 400 miles per hour. What is its speed in meters per second?

3. In SI units, acceleration is measured in units of meters per second squared (m/s 2 ). Which of the following combinations of units can also be used to measure acceleration?
(a) cm/s (b) cm/s 2 (c) m 3 /(mm 2 s 2 ) (d) km/s 2 (e) miles per hour (f) miles/(minutes) 2

4. A falling baseball has an acceleration of magnitude 9.8 m/s 2. What is its acceleration in feet per second squared?

5. An elite runner can run 100 m in 10 s. What is his average speed?

6. Consider the motion of a sprinter running a 100-m dash. When it is run outdoors, this race is run along a straight line portion of a track, so this is an example of motion in one dimension. Draw qualitative plots of the position, velocity, and acceleration as functions of time for the sprinter. Start your plots just before the race starts and end them when the runner comes to a complete stop.

7. A jogger maintains a speed of 3.0 m/s for 200 m until he encounters a stoplight, and he abruptly stops and waits 30 s for the light to change. He then resumes his exercise and maintains a speed of 3.5 m/s for the remaining 50 m to his home. (a) What was his average velocity for this entire time interval? (b) What were his maximum and minimum velocities, and how do they compare with this average?

8. A hockey puck that is sliding on an icy surface will eventually come to rest. The (horizontal) force that makes it stop is due to friction between the puck and the ice. Draw qualitative plots of the position, velocity, and acceleration as functions of time for the puck. Pay special attention to the sign of the acceleration.

9. Consider a marble falling through a very thick fluid, such as molasses. Draw qualitative plots of the position, velocity, and acceleration as functions of time for the marble, starting from the moment it is released in the molasses.

10. A person riding on a skateboard is initially coasting on level ground. He then uses his feet to push on the ground so that he speeds up for a few seconds, and then he coasts again. Draw qualitative plots of his position, velocity, and acceleration as functions of time.

11. In the drop zone. Consider a skydiver who jumps from an airplane. In a typical case, the skydiver will wait for a (short) time before opening her parachute. Draw qualitative plots of the position, velocity, and acceleration as functions of time for the skydiver, starting from the time she jumps from the plane, assuming (somewhat unrealistically) that she falls straight down. Here, the position, y, is her vertical height above the ground. Be sure to indicate the time at which she opens the parachute. Hint: Consider how the velocity will vary with time after the chute is open for a long time.

12. Consider a skier who coasts up to the top of a hill and then continues down the other side. Draw a qualitative plot of what the skier’s speed might look like.

13. Figure P2.13 shows three motion diagrams, where each dot represents the location of an object between equal time intervals. Assume left-to-right motion. For each motion diagram, sketch the appropriate position–time, velocity–time, and acceleration–time graphs.

14. A bicycle is moving initially with a constant velocity along a level road. The bicyclist then decides to slow down, so she applies her brakes over a period of several seconds. Thereafter, she again travels with a constant velocity. Draw qualitative sketches of her position, velocity, and acceleration as functions of time.

15. Figure P2.15 shows several hypothetical position–time graphs. For each graph, sketch qualitatively the corresponding velocity–time graph.

16. The position–time graphs in Figure P2.15 each describe the possible motion of a particular object. Give at least one example of what the object and motion might be in each case.

17. Figure P2.17 shows several hypothetical velocity–time graphs. For each case, sketch qualitatively the corresponding acceleration–time graph.

18. Figure P2.17 shows several hypothetical velocity–time graphs. For each case, sketch qualitatively the corresponding position–time graph.

19. Give examples of objects whose motion might be described by the graphs in Figure P2.17.

20. Figure P2.20 shows several hypothetical acceleration–time graphs. For each case, sketch qualitatively the corresponding velocity–time graph.

21. Give examples of objects whose motion is described by the plots in Figure P2.20.

22. Match each of the following examples of motion to one of the position–time graphs in Figure P2.22. (a) A person at the beginning of a race, starting from rest (b) A runner near the end of a race, just after crossing the finish line (c) A ball dropped from a window, hitting the ground below and then bouncing several times (d) A bowling ball as it rolls down a lane, just after it leaves the bowler’s hand.

23. Match each of the examples of motion in Problem 22 to one of the velocity–time graphs shown in Figure P2.23.

24. Consider the position–time graph shown in Figure P2.24. Make a careful graphical estimate of the velocity as a function of time by measuring the slopes of tangent lines. What is an approximate value of the maximum velocity of the object?

25. For the object described by Figure P2.24, estimate the average velocity (a) over the interval from t = 2.0 s to t = 4.0 s and (b) over the interval from t = 1.0 s to t = 5.0 s.

27. Figure P2.27 shows the velocity–time curve of a falling brick. Make a careful estimate of the slope to find the acceleration of the brick at t = 3.0 s.

28. Using a graphical approach (i.e. by estimating the slope at various points), find the qualitative behavior of the acceleration as a function of time for the object described by the velocity–time graph in Figure P2.28.

29. For the object described by the velocity–time graph in Figure P2.29, estimate the average acceleration over the interval from t = 0 s to t = 50 s and over the interval from t = 100 s to t = 200 s.

30. Draw a position–time graph for an object whose velocity as a function of time is described by (a) Figure P2.28 and (b) Figure P2.29.

31. Draw a graph showing the position as a function of time for an object whose acceleration is (a) constant and positive, (b) constant and negative, and (c) positive and increasing with time.

32. A car travels along a straight, level road. The car begins a distance x = 25 m from the origin at t = 0.0 s. At t = 5.0 s, the car is at x = 100 m; at t = 8.0 s, it is at x = 300 m. Find the average velocity of the car during the interval from t = 0.0 s to t = 5.0 s and during the interval from t = 5.0 s to t = 8.0 s.

33. A squirrel falls from a very tall tree. Initially (at t = 0), the squirrel is at the top of the tree, a distance y = 50 m above the ground. At t = 1.0 s, the squirrel is at y = 45 m, and at t = 2.0 s, it is at y = 30 m. Estimate the average velocity of the squirrel during the intervals from t = 0.0 s to t = 1.0 s and from t = 1.0 s to t = 2.0 s. Use these results to estimate the average acceleration of the squirrel during this time.

34. The space shuttle takes off from Cape Canaveral in Florida and orbits Earth 18 times before landing back at Cape Canaveral 24 hours and 15 minutes later. During this time, it moves in a circular orbit with radius 6.7 × 10 6 m. (a) What is the average speed of the space shuttle during its journey? (b) What is the average velocity?

35. Figure P2.35 shows the velocity as a function of time for an object. (a) What is the average acceleration during the interval from t = 0 s to t = 20 s? (b) Estimate the instantaneous acceleration at t = 5 s, (c) at t = 10 s, and (d) at t = 20 s.

36. Figure P2.36 shows the acceleration as a function of time for an object. (a) If the object starts from rest at t = 0, what is the velocity of the object as a function of time? (b) If the object instead has a velocity of +40 m/s at t = 0, how does your result for part (a) change?

37. A rabbit runs in a straight line with a velocity of +1.5 m/s for a period of time, rests for 10 s, and then runs again along the same line at +0.60 m/s for an unknown amount of time. The rabbit travels a total distance of 1200 m, and its average speed is 0.80 m/s. (a) What is the total time the rabbit spends running at 1.5 m/s? (b) How long does it spend running at 0.60 m/s?

38. Figure 2.22 shows one of Galileo’s experiments in which a ball rolls up an incline. A ball that is initially rolling up the incline will roll up to some maximum height and then roll back down the incline. Draw qualitative plots of the position and velocity as functions of time for the ball. Take x = 0 at the bottom of the ramp.

39. Consider the inclined plane (i.e. ramp) in Figure 2.23A, and assume it is covered with a layer of ice so that it is extremely slippery (frictionless). The horizontal surface of the bottom is not covered with ice, however. A box is released from the same height on the ramp for a number of trials and slides down the ramp. For each trial, the box encounters a different horizontal surface at the bottom of the smooth ramp. Draw qualitative plots to compare the velocity as a function of time for each case: (a) a grass field, (b) an asphalt street, (c) a hardwood floor, (d) an ice rink, and (e) an ideal, frictionless surface.

40. An unsafe way to transport your penguin. A flatbed truck hauls a block of ice on top of which stands a penguin (see Fig. P2.40). Assume the block of ice is frozen solid to the bed of the truck, but the top surface of the ice is extremely slippery. The truck skids to a stop from an initial velocity of 20 m/s. Describe what happens to the penguin in terms of Newton’s first law. How does it differ from what happens to the driver of the truck, who is wearing a seat belt? Draw and compare the velocity–time graphs of the penguin and truck driver.

41. A flatbed railcar is moving at a slow but constant velocity. A man stands in the railcar, facing sideways (perpendicular) to the motion of the railcar. The man holds a baseball at arm’s length and drops it onto the railcar bed. Where does the principle of inertia predict that the ball will land, (a) directly below the drop point on the railcar, (b) slightly in front of the drop point on the railcar, or (c) slightly behind the drop point on the railcar? Explain.

42. The person shown in Figure 2.1 is pushing on a refrigerator that sits on a level floor. Assume the floor is very slippery so that the only horizontal force on the refrigerator is due to the person. If the force exerted by the person has a magnitude of 120 N and the refrigerator has a mass of 180 kg, what is the magnitude of the refrigerator’s acceleration?

43. Action and reaction. Use Newton’s third law along with his second law to find the acceleration of the man in Problem 42 as he pushes the refrigerator. Assume the man has a mass of 60 kg and the floor is very slippery so that there is no frictional force between the floor and the man.

44. An object is found to move with an acceleration of magnitude 12 m/s 2 when it is subjected to a force of magnitude 200 N. Find the mass of the object.

45. In SI units, force is measured in newtons, with 1 N = 1 kg m/s 2. Which of the following combinations of units can also be used to measure force?
(a) g·m/s (b) g ·cm 2 /s (c) kg · m 4 /(s 2 ·cm 3 ) (d) g·cm/s 2 (e) kg·miles/(minutes) 2

46. In the U.S. customary system of units, mass is measured in units called slugs. Suppose an object has a mass of 15 kg. Use the conversion factors inside the front cover of this book to express the mass of the object in slugs.

47. In the U.S. customary system of units, force is measured in units of pounds (abbreviated lb). Suppose the force on an object is 150 lb. Using the conversion factors inside the front cover of this book, express this force in units of newtons.

48. A force is found to be 240 g·cm/s 2. Convert this value into units of newtons.

49. A cannon is fired horizontally from a platform (Fig. P2.49). The platform rests on a flat, icy, frictionless surface. Just after the shell is fired and while it is moving through the barrel of the gun, the shell (mass 3.2 kg) has an acceleration of +2500 m/s 2. At the same time, the cannon has an acceleration of -0.76 m/s 2. What is the mass of the cannon?

50. According to Newton’s third law, for every force there is always a reaction force of equal magnitude and opposite direction. In each of the examples below, identify an action–reaction pair of forces. (a) A tennis racket hits a tennis ball, exerting a force on the ball. (b) Two ice skaters are initially at rest and in contact. One of the skaters then pushes on the other skater’s back, exerting a force on that skater. (c) A car is moving at high speed and runs into a tree, exerting a force on the tree. (d) Two cars are moving in opposite directions and collide head-on. (e) A person leans on a wall, exerting a force on the wall. (f) A hammer hits a nail, exerting a force on the nail. (g) A mass hangs by a string tied to a ceiling, with the string exerting a force on the mass. (h) A bird sits on a telephone pole, exerting a force on the pole.

51. Tom has two ways he could drive home from work. He could take Highway 99 for 45 miles with a speed limit of 65 mi/h, or he could take Interstate 5, which would take him a bit out of the way at 57 miles, but with a speed limit of 75 mi/h. (a) Assuming Tom misses the rush hour and drives at an average speed equal to the maximum speed limit, which route gets him home the fastest? How much time does he save? (b) If Tom breaks the law and travels at 75 mi/h on Highway 99, how much time would he save (if he is not stopped to get a speeding ticket) compared to driving at the legal speed limit on Highway 99?

52. Throwing heat. In professional baseball, pitchers can throw a fastball at a speed of 90 mi/h. (a) Given that the regulation distance from the pitcher’s mound to home plate is 60.5 ft, how long does it take the ball to reach home plate after the ball leaves the pitcher’s hand? (b) It takes (on average) 0.20 s for the batter to get the tip of his bat over home plate. How much time does that give him to react? (c) The average fast pitch in professional women’s softball is about 60 mi/h, where the regulation distance from the pitcher’s mound to home plate is 40.0 ft. How do the travel time of a pitched ball and reaction time of a batter in softball compare with those in baseball? For simplicity, assume the pitcher releases the ball just above the center of the pitcher’s mound. Keep three significant figures in your calculation.

53. Figure P2.53 shows the position as a function of time for an object. (a) What is the average velocity during the period from t = 0.0 s to t = 10.0 s? (b) What is the average velocity between t = 0.0 s and t = 5.0 s? (c) Between t = 5.0 s and t = 10.0 s? (d) Explain why and how your answers to parts (a), (b), and (c) are related.

54. On your vacation, you fly from Atlanta to San Francisco (a total distance of 3400 km) in 4.0 h. (a) Draw a qualitative sketch of how the speed of your airplane varies with time. (b) What is the average speed during your trip? (c) Estimate the top speed during your trip. Hint: You reach your top speed about 10 minutes after taking off. (d) What is your average acceleration during the first 10 minutes of your trip? (e) What is the average acceleration during the central hour of your trip?

55. Figure P2.55 shows the position as a function of time for an apple that falls from a very tall tree. (a) At what time does the apple hit the ground? (b) Use a graphical approach to estimate and plot the acceleration as a function of time. (c) Make a sketch of the velocity of the apple as a function of its height above the ground.

56. A cheetah runs a distance of 100 m at a speed of 25 m/s and then runs another 100 m in the same direction at a speed of 35 m/s. What is the cheetah’s average speed?

57. Astronaut frequent-flyer miles. The average speed of the space shuttle while in orbit is about 8900 m/s. How far does the space shuttle travel during a mission that lasts 7.5 days?

58. A cat is being chased by a dog. Both are running in a straight line at constant speeds. The cat has a head start of 3.5 m. The dog is running with a speed of 8.5 m/s and catches the cat after 6.5 s. How fast did the cat run?

59. Predator and prey. The author’s cat enjoys chasing chipmunks in the front yard. In this game, the cat sits at one edge of a yard that is w = 30 m across (Fig. P2.59), watching as chipmunks move toward the center. The cat can run faster than a chipmunk, and when a chipmunk moves more than a certain distance L from the far end of the yard, the cat knows that it can catch the chipmunk before the chipmunk disappears into the nearby woods. If the cat’s top speed is 7.5 m/s and a chipmunk’s top speed is 4.5 m/s, find L. Assume the chipmunk and cat move along the same straight-line path.

60. A thief is trying to escape from a parking garage after completing a robbery, and the thief’s car is speeding (v = 12 m/s) toward the door of the parking garage (Fig. P2.60). When the thief is L = 30 m from the door, a police officer flips a switch to close the garage door. The door starts at a height of 2.0 m and moves downward at 0.20 m/s. If the thief’s car is 1.4 m tall, will the thief escape?


1. A spacecraft that is initially at rest turns on its engine at t = 0. If its mass is m = 3000 kg and the force from the engine is 45 N, what is the acceleration of the spacecraft?

2. A hockey puck moves on an icy surface that is frictionless, with a constant speed of 30 m/s. How long does it take the puck to travel the length of the hockey rink (60 m)?

3. An ice skater moves without friction on a frozen pond. While traveling at 8.0 m/s, she finds that it takes 17 s to travel the length of the pond. How long is the pond?

4. An object moves with a constant acceleration of 4.0 m/s2. If it starts with an initial speed of 30 m/s, how long does it take to reach a velocity of 250 m/s?

5. A car has a velocity of 10 m/s at t = 7.0 s. It then accelerates uniformly and reaches a velocity of 42 m/s at t = 12.0 s. What is its acceleration during this period?

6. A constant force of 400 N acts on a spacecraft of mass 8000 kg that has an initial velocity of 30 m/s. How far has the spacecraft traveled when it reaches a velocity of 5000 m/s?

7. A rocket-powered sled of mass 3500 kg travels on a level snow-covered surface with an acceleration of +3.5 m/s 2 (Fig. P3.7). What are the magnitude and direction of the force on the sled?

8. Your car has a dead battery. It is initially at rest, and you push it along a level road with a force of 120 N and find that it reaches a velocity of 2.0 m/s in 50 s. What is the mass of the car? Ignore friction.

9. You are a newly graduated astronaut preparing for your first trip into space. Plans call for your spacecraft to reach a velocity of 500 m/s after 2.4 minutes. If your mass is 75 kg, what force will be exerted on your body? Assume the acceleration is constant.

10. Pulling g’s. Suppose again you are the astronaut in Problem 9. When most people are subjected to an acceleration greater than about 5 × g, they will usually become unconscious (“black out”). Will you be in danger of blacking out?

11. An object with an initial velocity of 12 m/s accelerates uniformly for 25 s. (a) If the final velocity is 45 m/s, what is the acceleration? (b) How far does the object travel during this time?

12. A hockey puck has an initial velocity of 50 m/s and a final velocity of 35 m/s. (a) If it travels 35 m during this time, what is the acceleration? (b) If the mass of the puck is 0.11 kg, what is the horizontal force on the puck?

13. An elevator is moving at 1.2 m/s as it approaches its destination floor from below. When the elevator is a distance h from its destination, it accelerates with a = -0.50 m/s2. where the negative sign indicates a downward vertical direction. (“Up” is positive.) Find h.

14. An airplane must reach a speed of 200 mi/h to take off. If the runway is 500 m long, what is the minimum value of the acceleration that will allow the airplane to take off successfully?

15. A drag racer is able to complete the 0.25-mi course in 6.1 s. (a) If her acceleration is constant, what is a? (b) What is her speed when she is halfway to the finish line?

16. Consider a sprinter who starts at rest, accelerates to a maximum speed vmax, and then slows to a stop after crossing the finish line. Draw qualitative plots of the acceleration, velocity, and position as functions of time. Indicate the location of the finish line on the x–t plot.

17. Draw a qualitative plot of the total force acting on the ball in Figure 3.14 as a function of time. Begin your plot while the ball is still in the thrower’s hand and end it after the ball comes to rest back on the ground.

18. A barge on a still lake is moving toward a bridge at 10.0 m/s. When the bridge is 40.0 m away, the pilot slows the boat with a constant acceleration of -0.73 m/s 2. (a) Use Equation 3.3 to find the time it takes the barge to reach the bridge. Note that you will obtain two answers! Do both calculated times correspond to possible physical situations? (b) Find the final velocity for each time using Equation 3.1. (c) For each set of answers for t and v, sketch plots of velocity versus time and position versus time, and use them to describe the two situations.

19. To measure the height of a tree, you throw a rock directly upward, with a speed just fast enough that the rock brushes against the uppermost leaves and then falls back to the ground. If the rock is in the air for 3.6 s, how tall is the tree?

20. Your car is initially traveling at a speed of 25 m/s. As you approach an intersection, you spot a dog in the road 30 m ahead and immediately apply your brakes. (a) If you stop the instant before you reach the dog, what was the magnitude of your acceleration? (b) If your velocity is positive while you are slowing to a stop, is your acceleration positive or negative?

21. A car is traveling at 25 m/s when the driver spots a large pothole in the road a distance 30 m ahead. She immediately applies her brakes. If her acceleration is -27 m/s 2. does she manage to stop before reaching the pothole?

22. A bullet is fired upward with a speed v0 from the edge of a cliff of height h (Fig. P3.22). (a) What is the speed of the bullet when it passes by the cliff on its way down? (b) What is the speed of the bullet just before it strikes the ground? (c) If the bullet is instead fired downward with the same initial speed v 0. what is its speed just before it strikes the ground? Express your answers in terms of v 0. h, and g. Ignore air drag. Assume the bullet is fired straight up in (a) and (b) and straight down in (c).

23. A ball is thrown upward with a speed of 35 m/s from the edge of a cliff of height h = 15 m (Fig. P3.22). (a) What is the speed of the ball when it passes by the cliff on its way down to the ground? (b) What is the speed of the ball when it hits the ground? Ignore air drag. Assume the ball is thrown straight up. A rock is dropped from a tree of height 25 m into a lake (depth 5.0 m) below. After entering the water, the rock then floats gently down through the water at a speed of 1.5 m/s to the bottom of the lake. What is the total elapsed time?

24. A rock is dropped from a tree of height 25 m into a lake (depth 5.0 m) below. After entering the water, the rock then floats gently down through the water at a speed of 1.5 m/s to the bottom of the lake. What is the total elapsed time?

25. You are a passenger (m 110 kg) in an airplane that is accelerating on the runway, beginning to take off. The force between your back and your seat is 400 N. Starting from rest, how far does the plane travel as it accelerates to a takeoff speed of 130 m/s?

26. A motivated mule can accelerate an empty cart of mass m = 180 kg from rest to 5.0 m/s in 10 s. If the cart is loaded with 540 kg of wood, how long will it take the mule to get the cart to 5.0 m/s? Assume a constant acceleration and that the mule exerts the same force as when the cart is empty.

27. A person has a weight of 500 N. What is her mass?

28. A book of mass 3.0 kg sits at rest on a horizontal table. What is the normal force exerted by the table on the book?

29. Under siege. Twenty soldiers hold a long beam of heavy wood against a fortified castle door and attempt to push it open. If each soldier is able to supply a forward force on the beam of 80 N, what normal force does the castle door exert on the beam if the door does not open?

30. Two books of mass m 1 = 8.0 kg and m 2 = 5.5 kg are stacked on a table as shown in Figure Q3.16. Find the normal force acting between the table and the bottom book.

31. The table in Figure Q3.16 is now sitting in an elevator, with m 1 = 9.5 kg and m 2 = 2.5 kg. The normal force between the floor and the bottom book is 70 N. Find the magnitude and direction of the elevator’s acceleration.

32. A grand piano with three legs has a mass of 350 kg and is at rest on a level floor. (a) Draw a free-body diagram for the piano. Show the force of the floor on each leg as a separate force in your diagram. (b) What is the total force of the piano on the floor?

33. Your friend’s car is broken and you volunteer to push it to the nearest repair shop which is 2.0 km away. You carefully move your car so that the bumpers of the two cars are in contact and then slowly accelerate to a speed of 2.5 m/s over the course of 1 min. (a) If the mass of your friend’s car is 1200 kg, what is the normal force between the two bumpers? (b) If you then maintain a speed of 2.5 m/s, how long does it take you to reach the repair shop?

34. A tall strongman of mass m = 95 kg stands upon a scale while at the same time pushing on the ceiling in a small room. Draw a free-body diagram of the strongman (Fig. P3.34) and indicate all normal forces acting on him. If the scale reads 1100 N (about 240 lb), what is the magnitude of the normal force that the ceiling exerts on the strongman?

35. A bodybuilder configures a leg press apparatus (Fig. P3.35) to a resistance of 230 lb (approximately 1000 N). She pushes the weight to her full extension and comes to rest. (a) What is the normal force on each of her feet? (b) Assume she is able to move the push-plate of the leg-press machine at an acceleration of 0.50 m/s2 for the first half of the displacement. What is the normal force on each foot during this period of acceleration? Assume the resistive force is due to a weight of 230 lb hanging from the end of the cable.

36. Three blocks rest on a frictionless, horizontal table (Fig. P3.36), with m 1 = 10 kg and m 3 = 15 kg. A horizontal force F = 110 N is applied to block 1, and the acceleration of all three blocks is found to be 3.3 m/s 2. (a) Find m 2. (b) What is the normal force between blocks 2 and 3?

37. A car is moving with a velocity of 20 m/s when the brakes are applied and the wheels lock (stop spinning). The car then slides to a stop in 40 m. Find the coefficient of kinetic friction between the tires and the road.

38. A hockey puck slides with an initial speed of 50 m/s on a large frozen lake. If the coefficient of kinetic friction between the puck and the ice is 0.030, what is the speed of the puck after 10 s?

39. Your moving company runs out of rope and hand trucks, so you are forced to push two crates along the floor as shown in Figure P3.39. The crates are moving at constant velocity, their masses are m 1 = 45 kg and m 2 = 22 kg, and the coefficients of kinetic friction between both crates and the floor are 0.35.

40. You are given the job of moving a refrigerator of mass 100 kg across a horizontal floor. The coefficient of static friction between the refrigerator and the floor is 0.25. What is the minimum force that is required to just set the refrigerator into motion?

41. The coefficient of kinetic friction between a refrigerator (mass 100 kg) and the floor is 0.20, and the coefficient of static friction is 0.25. If you apply the minimum force needed to get the refrigerator to move, what will the acceleration then be?

42. After struggling to move the refrigerator in the Problem 41, you finally apply enough force to get it moving. What is the minimum force required to keep it moving with a constant velocity? Assume μ k = 0.20.

43. You are trying to slide a refrigerator across a horizontal floor. The mass of the refrigerator is 200 kg, and you need to exert a force of 350 N to make it just begin to move. (a) What is the coefficient of static friction between the floor and the refrigerator? (b) After it starts moving, the refrigerator reaches a speed of 2.0 m/s after 5.0 s. What is the coefficient of kinetic friction between the refrigerator and the floor?

44. A driver makes an emergency stop and inadvertently locks up the brakes of the car, which skids to a stop on dry concrete. Consider the effect of rain on this scenario. Using the values in Table 3.2, how much farther would the car skid (expressed in percentage of the dry-weather skid) if the concrete were instead wet?

45. Jeff Gordon (a race-car driver) discovers that he can accelerate at 4.0 m/s 2 without spinning his tires, but if he tries to accelerate more rapidly, he always “burns rubber.” (a) Find the coefficient of friction between his tires and the road. Assume the force from the engine is applied to only the two rear tires. (b) Have you calculated the coefficient of static friction or kinetic friction?

46. Antilock brakes. A car travels at 65 mi/h when the brakes are suddenly applied. Consider how the tires of a moving car come in contact with the road. When the car goes into a skid (with wheels locked up), the rubber of the tire is moving with respect to the road; otherwise, when the tires roll, normally the point where the tire contacts the road is stationary. Compare the distance required to bring the car to a full stop when (a) the car is skidding and (b) when the wheels are not locked up. Use the coefficients of kinetic and static friction from Table 3.2 and assume the tires are rubber and the road is dry concrete. (c) How much farther does the car go if the wheels lock into a skidding stop? Give your answer as a distance in meters and as a percent of the nonskid stopping distance. (d) Can antilock brakes make a big difference in emergency stops? Explain.

47. A hockey puck slides along a rough, icy surface. It has an initial velocity of 35 m/s and slides to a stop after traveling a distance of 95 m. Find the coefficient of kinetic friction between the puck and the ice.

48. A rock is dropped from a very tall tower. If it takes 4.5 s for the rock to reach the ground, what is the height of the tower?

49. A baseball is hit directly upward with an initial speed of 45 m/s. Find the velocity of the ball when it is at a height of 40 m. Is there one correct answer for v or two? Explain why.

50. A squirrel is resting in a tall tree when it slips from a branch that is 50 m above the ground. It is a very agile squirrel and manages to land safely on another branch after only 0.50 s. What is the height of the branch it lands on?

51. Basketball on the Moon. If LeBron James can jump 1.5 m high on Earth, how high could he jump on the Moon (assume an indoor court), where g = 1.6 m/s 2 ?

52. An apple falls from a branch near the top of a tall tree. If the branch is 12 m above the ground, what is the apple’s speed just before it hits the ground?

53. A ball is thrown directly upward with an initial velocity of 15 m/s. If the ball starts at an initial height of 3.5 m, how long is the ball in the air? Ignore air drag.

54. Two children are playing on a 150-m-tall bridge. One child drops a rock (initial velocity zero) at t = 0. The other waits 1.0 s and then throws a rock downward with an initial speed v 0. If the two rocks hit the ground at the same time, what is v 0 ?

55. A rock is dropped from a tall bridge into the water below. If the rock begins with a speed of zero and has a speed of 12 m/s just before it hits the water, what is the height of the bridge?

56. A roofing tile falls from rest off the roof of a building. An observer from across the street notices that it takes 0.43 s for the tile to pass between two windowsills that are 2.5 m apart. How far is the sill of the upper window from the roof of the building?

57. You are standing at the top of a deep, vertical cave and want to determine the depth of the cave. Unfortunately, all you have is a rock and a stopwatch. You drop the rock into the cave and measure the time that passes until you hear the rock hitting the floor of the cave far below. If the elapsed time is 8.0 s, how deep is the cave? Hints: (1) Sound travels at a constant speed of 340 m/s. (2) Consider two separate time periods. During the first period, the rock undergoes free fall and lands at the bottom of the cave. During the second period, sound travels at a constant velocity back up the cave.

58. Your friend is an environmentalist who is living in a tree for the summer. You are helping provide her with food, and you do so by throwing small packages up to her tree house. If her tree house is 30 m above the ground, what is the minimum (initial) speed you must use when throwing packages up to her?

59. You are standing across the street from a tall building when the top of the building (h = 80 m) is hit by lightning and a brick is knocked loose. You see the lightning strike and immediately see that the brick will fall to hit a person standing at the base of the building. You then run toward the person and push him out of the path of the brick, the instant before the brick reaches him. If you are initially 25 m from the building, how fast do you have to run?

60. A cable attached to a block of mass 12 kg pulls the block along a horizontal floor at a constant velocity. If the tension in the cable is 5.0 N, what is the coefficient of kinetic friction between the block and the floor?

61. A crate of mass 55 kg is attached to one end of a string, and the other end of the string runs over a pulley and is held by a person as in Figure 3.21. If the person pulls with a force of 85 N, what is the crate’s acceleration?

62. A car of mass 1200 kg is being moved by a large crane in preparation for lowering it onto a junk pile. If the car is accelerating downward at 0.20 m/s 2. what is the tension in the cable?

63. You work for a moving company and are given the job of pulling two large boxes of mass m 1 = 120 kg and m 2 = 290 kg using ropes as shown in Figure P3.63. You pull very hard, and the boxes are accelerating with a = 0.22 m/s 2. What is the tension in each rope? Assume there is no friction between the boxes and the floor.

65. In traction. When a large bone such as the femur is broken, the two pieces are often pulled out of alignment by the complicated combination of tension and compression forces that arise from the muscles and tendons in the leg (see the X-ray image in Figure P3.64A). To realign the bones and allow proper healing, these forces must be compensated for. A method called traction is often employed. If a total tension force of 400 N is applied to the leg as depicted in Figure P3.64B to realign the parts of the femur, how much mass m must be attached to the bottom pulley?

66. What is the approximate speed at which the force of air drag on a car is equal to the weight of the car? Hint: Start by estimating the mass and the frontal area of the car.

67. A bullet of mass 10 g leaves the barrel of a rifle at 300 m/s. Assuming the force on the bullet is constant while it is in the barrel, find the magnitude of this force. Hint: Start by estimating the length of the barrel.

68. When an airplane takes off, it accelerates on the runway starting from rest. What is the magnitude of this runway acceleration for a passenger jet? Hint: You will need to find (or estimate) one or more quantities such as the takeoff velocity, the distance traveled on the runway, and the time spent on the runway.

69. Calculate the terminal velocity for a pollen grain falling through the air using the drag force Equation 3.20. Assume the pollen grain has a diameter of 1 mm and a density of 0.2 g/cm 3. If this grain is released from the top of a tree (height 10 m), estimate the time it will take to fall to the ground. Hint: The pollen grain will reach its terminal velocity very quickly and will have this velocity for essentially the entire motion. Your answer will explain why pollen stays in the air for a very long time.

70. Hang time. LeBron James decides to jump high enough to dunk a basketball. What is the approximate force between the floor and his feet while he is jumping from the floor? Hint: Start by estimating the distance he moves while in contact with the floor and the height he must jump for his hands to reach just above the rim.

71. Consider a skydiver who lands on the ground with a speed of 3 m/s. What is the approximate force on the skydiver’s legs when she lands?

72. Calculate the terminal velocity for a baseball. A baseball’s diameter is approximately d = 0.070 m, and its mass is m = 0.14 kg. Express your answer in meters per second and miles per hour.

73. A skydiver opens her parachute immediately after jumping from an airplane. Approximately how long does it take to reach her terminal velocity? Hint: Use half the maximum acceleration as an estimate of the average acceleration during this time period.

74. Calculate the force of air drag on a hockey puck moving at 30 m/s. A hockey puck is approximately 3.0 cm tall and 8.0 cm in diameter.

75. The following items are dropped from an airplane. Rank them in order from lowest terminal velocity to highest and justify your ranking. (a) Bowling ball (d) Watermelon (b) Beach ball (e) Cantaloupe (c) Spear or javelin (f) Apple (pointing downward)

76. A Styrofoam ball of radius 28 cm falls with a terminal velocity of 5.0 m/s. What is the mass of the ball?

77. You are a secret agent and find that you have been pushed out of an airplane without a parachute. Fortunately, you are wearing a large overcoat (as secret agents often do). Thinking quickly, you are able to spread out and hold the overcoat so that you increase your overall area by a factor of four. If your terminal velocity would be 43 m/s without the coat, what is your new terminal velocity with the coat? Could you survive impact with the ground? How about over a lake?

78. Calculate the drag force on a bullet that is 5.0 mm in diameter and moving at a speed of 600 m/s. If the mass of the bullet is 10 g, compare this force to the weight of the bullet and to your own weight. Assume the drag force on the bullet is given by Equation 3.20.

79. Hail forms high in the atmosphere and can be accelerated to a high speed before it reaches the ground. Estimate the terminal velocity of a spherical hailstone that has a diameter of 2.0 cm

80. Calculate the terminal velocity for a bacterium in water. Use the expression for the drag force in Equation 3.23. The values of other quantities you need are given in Example 3.10. How long does it take a bacterium to fall from the top of a lake of depth 5.0 m to the bottom?

81. The force exerted on a bacterium by its flagellum is 4 × 10 -13 N. Find the velocity of the bacterium in water. Assume a size r = 1 μm.

82. The kick experienced when firing a rifle can be explained by Newton’s third law. A .22-caliber rifle has a mass of M = 5.2 kg, and a bullet with a mass m = 3.0 g leaves the barrel of the gun at a velocity of 320 m/s. (a) If the bullet starts from rest and leaves the gun barrel after t = 0.010 s, what was the acceleration of the bullet? (b) What was the force on the bullet? (c) What was the magnitude of the force exerted on the gun? (d) What acceleration did the gun experience? (e) Compare the ratio of M and m to the ratio of the acceleration of each object.

83. What is your reaction time? The following simple method can be employed to determine reaction time. A partner holds a meter stick by pinching it at the top and letting it hang vertically. To measure your reaction time, place your thumb and forefinger just below the base of the meter stick, ready to pinch it when it falls. Without signaling, your partner releases the meter stick; it accelerates due to gravity at a rate of 9.8 m/s 2. and you grab it as fast as possible. (a) If your thumb pinches the meter stick at the 45-cm mark, what was your reaction time? Using a similar calculation, one can calibrate a “Grab-it Gauge” such as that shown in Figure P3.83. (b) Calculate the distance to draw each line from the bottom starting point for reactions times of 0.14, 0.16, 0.18, 0.20, and 0.22 s.

84. Deer in the headlights. There are two important time intervals to consider when coming to an emergency stop while driving. The first is the driver’s reaction time to get a foot on the brake pedal, and the second is the time it takes to decelerate the car to rest. Consider a car moving at 30 m/s (about 65 mi/h) when the driver sees a deer in the road ahead and applies the brakes. (a) If the driver’s reaction time is 1.1 s, how far the does the car travel before the brakes are applied? (b) If the deer is 100 m away when the driver sees it, what acceleration is needed to stop the car without hitting the deer? (c) If the concrete streets are wet, will the car be able to stop without hitting the deer? (d) If the car cannot stop in time, how fast will it be going when it strikes the deer? If the car can stop in time, how far away from the deer will it come to rest?

85. A boy pushes a 3.1-kg book against a vertical wall with a horizontal force of 40 N. What is the minimum coefficient of friction that will keep the book in place without sliding?

86. An impish young lad stands on a bridge 10 m above a lake and drops a water balloon on a boat of unsuspecting tourists. Although the boat is traveling at a speed of 7.5 m/s, the boy manages to land the balloon right on the deck of the boat. How far away from the base of the bridge was the boat when the boy released the balloon? Assume he just lets the balloon go without throwing it (i.e. he simply drops it).

87. Two mischievous children drop water balloons from a bridge as depicted in Figure P3.87. If each water balloon is approximately 30 cm in diameter, what is the time interval between when the first balloon was let go and the second balloon was dropped? Assume both balloons were let go exactly above the bridge railing. Take measurements directly from the figure and scale appropriately.

88. A subway train is designed with a maximum acceleration of 0.20 m/s 2. which allows for both passenger safety and comfort. (a) If subway stations are 1.2 km apart, what is the maximum velocity that can be obtained between stations? (b) How long does it take to travel between two stations? (c) The train stops for a total of 45 s at each station. What is the overall average velocity of the train from station to station?

89. A spring scale indicates that a helium balloon tied to it produces a tension of 0.20 N in the string. The string is then cut, and the balloon rises until it comes to rest on the ceiling. (a) Draw a free-body diagram of the balloon on the ceiling. (b) What is the normal force exerted by the ceiling on the balloon?

90. (a) Draw free-body diagrams for the two blocks and for the person in Figure P3.63. (b) For each horizontal force in part (a), identify the corresponding reaction force.

91. A car is outfitted with a flat piece of plywood mounted vertically on its front bumper. As seen in Figure P3.91, a block of wood is simply placed in front of the car just as the car begins to accelerate. (a) If the coefficient of static friction between the block and the plywood is μ s = 0.90, what acceleration is needed to keep the block from falling? (b) Safety concerns limit the maximum speed of the car to 50 m/s (about 110 mi/h). How long can the car keep the block from falling this way? (c) If the mass of the wooden block is doubled, how does the answer to part (a) change?

92. Two spheres, one of wood and one of steel, have the same mass. (a) How many times greater is the terminal velocity of the steel sphere of diameter 10 cm, than that of the wood sphere of diameter 40 cm? (b) Another two spheres, again one of wood and one of steel, are this time exactly the same size. Find the ratio of the terminal velocity of the steel sphere of mass 40 kg to that of the wood sphere of mass 10 kg.

93. A block of mass M 1 = 3.0 kg rests on top of a second block of mass M 2 = 5.0 kg, and the second block sits on a surface that is so slippery that the friction can be assumed to be zero (see Fig. P3.93). (a) If the coefficient of static friction between the blocks is μ s = 0.21, how much force can be applied to the top block without the blocks slipping apart? (b) How much force can be applied to the bottom block for the same result?

94. Draw free-body diagrams for the two crates in Figure P3.39. For each force, identify the corresponding reaction force.

95. A bullet of mass m = 10 g and velocity 300 m/s is shot into a block of wood that is firmly attached to the ground. The bullet comes to rest in the wood at a depth of 8.1 cm. (a) What was the acceleration of the bullet? (b) How long did it take for the bullet to come to rest once it entered the wood? (c) What was the frictional force exerted by the wood on the bullet?

96. Consider a small sailboat with a triangular sail of height 10 m and width at the base of 5.0 m. (a) Assuming a wind speed of 15 mi/h relative to the boat, estimate the force exerted by the wind on the sail. (b) If the sailboat is moving with a constant speed, what is the drag force due to the water? (c) Suppose the speed of the wind relative to the boat is doubled to 30 mi/h. By what factor does the speed of the boat relative to the water increase? Hint: Assume as in Stokes’s expression (Eq. 3.23) that the drag force due to the water is proportional to the speed of the boat relative to the water.

97. High dive. The cliff-divers of Acapulco are famous for diving from steep cliffs that overlook the ocean into places where the water is very shallow. (a) Suppose a cliff-diver jumps from a cliff that is 25 m above the water. What is the speed of the diver just before he enters the water? (b) If the water is 4.0 m deep, what is the acceleration of the diver after he enters the water? Assume this acceleration is constant and it begins at the moment his hands enter the water.

98. The pedestrian walkway on the Golden Gate Bridge is about 75 m above the water below. This bridge is (unfortunately) a popular spot for some unhappy people, who attempt to jump off. Ignore air drag and calculate (a) the time it takes for an object to fall from the bridge to the water and (b) the speed of the object just before it hits the water. Express your answer to (b) in meters per second and miles per hour. (c) Use Equation 3.22 to calculate the terminal velocity of a person. Will air drag be important for a person who jumps off the Golden Gate Bridge?

99. The surfaces where bones meet are lubricated by a fluid like substance, which makes the coefficient of friction between bones very small (see Table 3.2). What is the approximate lateral (i.e. horizontal) force required to make the bones in a typical knee joint slide across each other? For simplicity, assume the surfaces in the knee are fl at and horizontal, and consider an adult of average mass.

1. Two ropes are attached to a skater as sketched in Figure P4.1 and exert forces on her as shown. Find the magnitude and direction of the total force exerted by the ropes on the skater.

2. The three forces shown in Figure P4.2 act on a particle. If the particle is in translational equilibrium, find F 3 (the magnitude of force 3) and the angle θ 3 .

3. Several forces act on a particle as shown in Figure P4.3. If the particle is in translational equilibrium, what are the values of F 3 (the magnitude of force 3) and θ 3 (the angle that force 3 makes with the x axis)?

4. A man is lazily floating on an air mattress in a swimming pool (Fig. P4.4). (a) Draw a free-body diagram for the man and for the mattress. (b) Identify the reaction forces for all the forces in your free-body diagrams in part (a). (c) If the mass of the man is 110 kg and the mass of the mattress is 2.5 kg, what is the upward force of the water on the mattress? What is the force that the man exerts on the mattress? Note: Keep three significant figures in your answer.

5. A person leans against a wall (Fig. P4.5). Draw a free-body diagram for the person.

6. The sled in Figure 4.2 is stuck in the snow. A child pulls on the rope and finds that the sled just barely begins to move when he pulls with a force of 25 N, with the rope at an angle of 30° with respect to the horizontal. (a) Sketch the sled and all the forces acting on it. Also choose a coordinate system. (b) Determine the components of all the forces on the sled along the coordinate axes. (c) Write the conditions for static equilibrium along your two coordinate directions. (d) If the sled has a mass of 12 kg, what is the coefficient of friction between the sled and the snow? (e) Is this the coefficient of static friction or the coefficient of kinetic friction?

7. A system of cables is used to support a crate of mass m = 45 kg as shown in Figure P4.7. Find the tensions in all three cables.

8. Balancing act. The tightrope walker in Figure P4.8 gets tired and decides to stop for a rest. During this rest period, she is in translational equilibrium. She stops at middle of the rope and finds that both sides of the rope make an angle of θ = 15° with the horizontal. (a) Sketch the walker and the rope. Show all the forces acting on the point of the rope on which the walker stands (call this point P). Also include a coordinate system. (b) Determine the components of all the forces on point P along the coordinate axes. (c) Write the conditions for static equilibrium along your two coordinate directions. (d) If the mass of the tightrope walker is 60 kg, what is the tension in the rope?

9. A flag of mass 2.5 kg is supported by a single rope as shown in Figure P4.9. A strong horizontal wind exerts a force of 12 N on the flag. Find the tension in the rope and the angle θ the rope makes with the horizontal.

10. A car of mass 1400 kg is parked on a very slippery hillside (Fig. P4.10). To keep it from sliding down the hill, the owner attaches a cable. (a) Sketch all the forces on the car. Include coordinate axes in your sketch. (b) Determine the components of all the forces on the car along the coordinate axes. (c) Write the conditions for static equilibrium along your two coordinate directions. (d) If there is no frictional force between the road and the tires, what is the tension in the cable?

11. A crate is placed on an inclined board as shown in Figure P4.11. One end of the board is hinged so that the angle θ is adjustable. If the coefficient of static friction between the crate and the board is μs = 0.30, what is the value of θ at which the crate just begins to slip?

12. Two blocks of mass m 1 = 45 kg and m 2 = 12 kg are connected by a massless string that passes over a pulley as shown in Figure P4.12. The coefficient of static friction between m1 and the table is μ s = 0.45. (a) Will this system be in static equilibrium? Assume the pulley is frictionless. (b) Find the tension in the string.

13. For the system in Problem 12 and Figure P4.12, how large can m 2 be made without the system starting into motion?

14. Stemming a chimney. A rock climber of mass 60 kg wants to make her way up the crack between two rocks as shown in Figure P4.14. The coefficient of friction between her shoes and the rock surface is μ s = 0.90. What is the smallest normal force she can apply to both surfaces without slipping? Assume the rock walls are vertical. Hint: Why is the normal force between the climber and the rock on the left equal to the normal force between her and the rock on the right?

15. A rock is thrown horizontally with a speed of 20 m/s from a vertical cliff of height 25 m. (a) How long does it take the rock to reach the horizontal ground below? (b) How far will it land from the base of the cliff? (c) What is the velocity (magnitude and direction) of the rock just before it hits the ground?

16. A hockey puck is given an initial velocity such that v x = 12 m/s and v y = 18 m/s, where the x–y plane is horizontal. (a) What is the initial speed of the puck? (b) What angle does the initial velocity make with the x axis? (c) What angle does the initial velocity make with the y axis?

17. A quarterback is asked to throw a football to a receiver who is 35 m away. What is the minimum speed the football must have when it leaves the quarterback’s hand? Assume the ball is caught at the same height as it is thrown.

18. Which of the graphs in Figure P4.18 might be a plot of the vertical component of the velocity of a projectile that is thrown from the top of a building?

19. A soccer ball is kicked with an initial speed of 30 m/s at an angle of 25° with respect to the horizontal. Find (a) the maximum height reached by the ball and (b) the speed of the ball when it is at the highest point on its trajectory. (c) Where does the ball land? That is, what is the range of the ball? Assume level ground.

20. Consider a rock thrown off a bridge of height 75 m at an angle θ = 25° with respect to the horizontal as shown in Figure P4.20. The initial speed of the rock is 15 m/s. Find the following quantities: (a) the maximum height reached by the rock, (b) the time it takes the rock to reach its maximum height, (c) the place where the rock lands, (d) the time at which the rock lands, and (e) the velocity of the rock (magnitude and direction) just before it lands.

21. A football player wants to kick a ball through the uprights as shown in Figure P4.21. The ball is kicked from a distance of 30 m (he’s playing metric football) with a velocity of magnitude 25 m/s at an angle of 30°. Will the ball make it over the crossbar (height 3.1 m)?

22. An airplane is flying horizontally with a constant velocity of 200 m/s at an altitude of 5000 m when it releases a package. (a) How long does it take the package to reach the ground? (b) What is the distance between the airplane and the package when the package hits the ground? (c) How far ahead of the target along the x direction should the airplane be when it releases the package?

23. A horizontal rifle is fired at a target 50 m away. The bullet falls vertically a distance of 12 cm on the way to the target. (a) What is the speed of the bullet just after it leaves the rifle? (b) What is the speed of the bullet just before it hits the target?

24. A bullet of mass 0.024 kg is fired horizontally with a speed of 75 m/s from a tall bridge. If the bullet is in the air for 2.3 s, how far from the base of the bridge does it land?

25. Consider the game of baseball. A pitcher throws a ball to the catcher at a speed of 100 mi/h (45 m/s). If the velocity of the ball is horizontal when it leaves the pitcher’s hand, how far (vertically) will it fall on the way to the catcher? The distance from the pitcher to the catcher is 60.5 ft. Express your answer in meters and in feet.

26. A batted baseball is hit with a speed of 45 m/s starting from an initial height of 1 m. Find how high the ball travels in two cases: (a) a ball hit directly upward and (b) a ball hit at an angle of 70° with respect to the horizontal. Also find how long the ball stays in the air in each case.

27. A juvenile delinquent wants to break a window near the top of a tall building using a rock. The window is 30 m above ground level, and the base of the building is 20 m from his hiding spot behind a bush. Find the minimum speed the rock must have when it leaves his hand. Also find the corresponding launch angle of the rock. Hint: Graph how the initial velocity changes as a function of launch angle to find the minimum value needed and the corresponding launch angle.

28. You are a serious basketball player and want to use physics to improve your free throw shooting. Do an approximate calculation of the minimum speed the ball must have in order to travel from your hand to the basket in a successful free throw. You will have to estimate or find several quantities, including the distance from your hand to the basket, the height of the ball when it leaves your hand, and the height of the basket. Graph how the initial velocity changes as a function of launch angle to find the minimum initial speed of the ball and particular launch angle.

29. Consider the problem of kicking a soccer ball past a goalkeeper into the goal (Fig. P4.29). You are 25 m away from the goal and kick the ball at an angle of 30° with respect to the horizontal, and the ball just passes over the goalkeeper’s hands. Find the initial speed of the ball. Also, calculate how long it takes the ball to get from you to the goal. Hint: You will have to estimate the height of the goalkeeper.

30. A ball is thrown straight up and rises to a maximum height of 24 m. At what height is the speed of the ball equal to half its initial value? Assume the ball starts at a height of 2.0 m above the ground.

31. Two rocks are thrown off a cliff. One rock (1) is thrown horizontally with a speed of 20 m/s. The other rock (2) is thrown at an angle u relative to the horizontal with a speed of 30 m/s. While the two rocks are in the air, rock 2 is always directly above 1. Find θ.

32. A bullet is fi red from a rifle with a speed v 0 at an angle θ with respect to the horizontal axis (Fig. P4.32) from a cliff that is a height h above the ground below. (a) Calculate the speed of the bullet when it strikes the ground. Express your answer in terms of v 0. h, g, and θ. (b) Explain why your result is independent of the value of θ.

33. A high jumper can run horizontally with a top speed of 10.0 m/s. (a) If he can “convert” this velocity to the vertical direction when he leaves the ground, what is the theoretical limit on the height of his jump? (b) How does your result to part (a) compare with the current high-jump record of approximately 2.5 m? Can you explain the difference?

34. A baseball is thrown with an initial velocity of magnitude v0 at an angle of 60° with respect to the horizontal (x) direction. At the same time, a second ball is thrown with the same initial speed at an angle θ with respect to x. If the two balls land at the same spot, what is θ?

35. A golf ball is hit with an initial velocity of magnitude 60 m/s at an angle of 65° with respect to the horizontal (x) direction. At the same time, a second golf ball is hit with an initial speed v 0 at an angle 35° with respect to x. If the two balls land at the same time, what is v 0 ?

36. Consider a swimmer who wants to swim directly across a river as in Example 4.7. If the speed of the current is 0.30 m/s and the swimmer’s speed relative to the water is 0.60 m/s, how long will it take her to cross a river that is 15 m wide?

37. Suppose the swimmer in Figure 4.20 has a swimming speed relative to the water of 0.45 m/s and the current’s speed is 2.5 m/s. If it takes the swimmer 200 s to cross the river, how wide is the river?

38. An airplane has a velocity relative to the air of 200 m/s in a westerly direction. If the wind has a speed relative to the ground of 60 m/s directed to the north, what is the speed of the airplane relative to the ground?

39. Consider again the airplane in Problem 38, but now suppose the wind is directed in an easterly direction. How long does it take the airplane to travel a distance between two cities that are 300 km apart?

40. An airplane flies from Boston to San Francisco (a distance of 5000 km) in the morning and then immediately returns to Boston. The airplane’s speed relative to the air is 250 m/s. The wind is blowing at 50 m/s from west to east, so it is “in the face” of the airplane on the way to San Francisco and it is a tailwind on the way back. (a) What is the average speed of the airplane relative to the ground on the way to San Francisco? (b) What is the average speed relative to the ground on the way back to Boston? (c) What is the average speed for the entire trip? (d) Why is the average of the average speeds for the two legs of the trip not equal to the average speed for the entire trip?

41. Round trip. An airplane flies from Chicago to New Orleans (a distance of 1500 km) in the morning and then immediately returns to Chicago. The airplane’s speed relative to the air is 250 m/s. The wind is blowing at 50 m/s from west to east, so it is perpendicular to the line that connects the two cities. (a) What is the average speed of the airplane relative to the ground on the way to New Orleans? (b) What is the average speed relative to the ground on the way back to Chicago? (c) What is the average speed for the entire trip?

42. A 35-m-wide river flows in a straight line (the x direction) with a speed of 0.25 m/s. A boat is rowed such that it travels directly across the river (along y). If the boat takes exactly 4 minutes to cross the river, what is the speed of the boat relative to the water?

43. Two children pull a sled of mass 15 kg along a frictionless surface as shown in Figure P4.43. (a) Find the magnitude and direction of the sled’s acceleration. (b) How long does it take the sled to reach a speed of 10 m/s?

44. Consider the crates in Problem 12 (Fig. P4.12). Assume there is no friction between m1 and the table. (a) Sketch all the forces on the crates. Include a coordinate system. (b) Determine the components of the forces along the coordinate axes. (c) Write Newton’s second law for motion along both coordinate directions. (d) Solve the equations from part (c) to find the acceleration of the crates and the tension in the string.

45. When a car is traveling at 25 m/s on level ground, the stopping distance is found to be 22 m. This distance is measured by pushing hard on the brakes so that the wheels skid on the pavement. The same car is driving at the same speed down an incline that makes an angle of 8.0° with the horizontal. What is the stopping distance now, as measured along the incline?

46. A block slides down an inclined plane that makes an angle of 40° with the horizontal. There is friction between the block and the plane. (a) Sketch all the forces on the block. Include a coordinate system. (b) Determine the components of the forces along the coordinate axes. (c) Write Newton’s second law for motion along both coordinate directions. (d) If the acceleration of the block is 2.4 m/s 2. what is the coefficient of kinetic friction between the block and the plane?

45. A sled is pulled with a horizontal force of 20 N along a level trail, and the acceleration is found to be 0.40 m/s 2. An extra mass m = 4.5 kg is placed on the sled. If the same force is just barely able to keep the sled moving, what is the coefficient of kinetic friction between the sled and the trail?

48. Two crates are connected by a massless rope that passes over a pulley as shown in Figure 4.25. If the crates have mass 35 kg and 85 kg, what is their acceleration? If the system begins at rest, with the more massive crate a distance 12 m above the floor, how long does it take the more massive crate to reach the floor? Assume the pulley is massless and frictionless.

49. A hockey puck is sliding down an inclined plane with angle θ = 15° as shown in Figure P4.49. If the puck is moving with a constant speed, what is the coefficient of kinetic friction between the puck and the plane?

50. Consider a baseball player who wants to catch the fly ball in Problem 26(b). Estimate the force exerted by the ball on the player’s glove as the ball is coming to rest. Hint: You will have to estimate the acceleration of the ball, and to do so it is useful to estimate the distance the ball travels while coming to rest in the glove. The mass of a baseball is 0.14 kg.

51. Two forces are acting on an object. One force has a magnitude of 45 N and is directed along the +x axis. The other force has a magnitude of 30 N and is directed along the +y axis. What is the direction of the object’s acceleration? Express your answer by giving the angle between a and the x axis.

52. A car is traveling down a hill that makes an angle of 20° with the horizontal. The driver applies her brakes, and the wheels lock so that the car begins to skid. The coefficient of kinetic friction between the tires and the road is μ k = 0.45. (a) Find the acceleration of the car. (b) How long does the car take to skid to a stop if its initial speed is 30 mi/h (14 m/s)?

53. A skier is traveling at a speed of 40 m/s when she reaches the base of a frictionless ski hill. This hill makes an angle of 10° with the horizontal. She then coasts up the hill as far as possible. What height (measured vertically above the base of the hill) does she reach?

54. Consider again the skier in Problem 53, but now assume there is friction with μ k = 0.10 between the skis and the snow. What height does she reach now?

55. An airplane of mass 25,000 kg (approximately the size of a Boeing 737) is coming in for a landing at a speed of 70 m/s. Estimate the normal force on the landing gear when the airplane lands. Hint: You will have to estimate the angle that the landing velocity makes with the horizontal and also the compression (vertical displacement) of the landing gear shock absorbers when the plane contacts the ground.

56. Consider the rock-on-a-string accelerometer in Figure 4.29. What is the accelerometer angle if the airplane has a horizontal acceleration of 1.5 m/s 2 ?

57. Consider the accelerometer design shown in Figure P4.57, in which a small sphere sits in a hemispherical dish. Suppose the dish has an acceleration of a x along the horizontal and a small amount of friction causes the sphere to reach a stable position relative to the dish as shown in the figure. Find the position of the sphere (i.e. find the angle θ) as a function of a x .

58. Consider a commercial passenger jet as it accelerates on the runway during takeoff. This jet has a rock-on-a-string accelerometer installed (as in Fig. 4.29). Estimate the angle of the string during takeoff.

59. Estimate the terminal velocity for a golf ball.

60. Make qualitative sketches of the trajectory of a batted baseball (v 0 = 50 m/s) with and without air drag.

61. A skier travels down a steep, frictionless slope of angle 20° with the horizontal. Assuming she has reached her terminal velocity, estimate her speed.

62. Consider the motion of a bicycle with air drag included. We saw how to deal with the motion on a hill in connection with Figure 4.32. Now assume the bicycle is coasting on level ground and is being pushed along by a tailwind of 10 mi/h (4.5 m/s). If the bicycle starts from rest at t = 0, what is the acceleration of the bicycle at that moment? Assume that there are no frictional forces (e.g. from the bicycle’s tires or bearings, or any other source) opposing the motion of the bicycle.

63. What is the approximate magnitude of the drag force on an airplane (such as a commercial jet) traveling at 250 m/s?

64. Consider the system of blocks in Figure P4.64, with m 2 = 5.0 kg and θ = 35°. If the coefficient of static friction between block 1 and the inclined plane is μ s = 0.25, what is the largest mass m 1 for which the blocks will remain at rest?

65. A water skier (Fig. P4.65) of mass m = 65 kg is pulled behind a boat at a constant speed of 25 mi/h. If the tension in the horizontal rope is 1000 N (approximately 225 lbs), what are the magnitude and direction of the force the water exerts on the skier’s ski?

66. An abstract sculpture is constructed by suspending a sphere of polished rock along the vertical with four cables connected to a square wooden frame 1.2 m on a side as shown in Figure P4.66. If the stone has mass m = 8.0 kg and the bottom two cables are under a tension of 25 N each, what is the tension in each top cable?

67. An airplane pulls a 25-kg banner on a cable as depicted in Figure P4.67. When the airplane has a constant cruising velocity, the cable makes an angle of θ = 20°. (a) What is the drag force exerted on the banner by the air? (b) The weather changes abruptly and the pilot finds herself in a 20-minute rainstorm. After the storm, she finds that the cable now makes an angle of θ water = 30°. Assuming the velocity and drag force are the same after the storm, find the mass of the water absorbed by the banner’s fabric during the storm.

68. A piece of wood with mass m = 2.4 kg is held in a vise sandwiched between two wooden jaws as shown in Figure P4.68. A blow from a hammer drives a nail that exerts a force of 450 N on the wood. If the coefficient of static friction between the wood surfaces is 0.67, what minimum normal force must each jaw of the vise exert on the wood block to hold the block in place?

69. Stomp rocket. A “stomp rocket” is a toy projectile launcher illustrated in Figure P4.69i, which uses a blast of air to propel a dart made of plastic. The air blast is produced by jumping on a plastic bladder as shown in Figure P4.69ii. It is found that when a 190-lb father jumps on the bladder, the rocket dart will fly on average a horizontal distance of 500 ft when launched at an angle of 45°. (a) If the rocket dart is instead pointed straight up and the same father jumps on the launcher, what would be the maximum height obtained? (b) What is the initial velocity of the rocket dart as it leaves the launch tube? Would you recommend the use of safety goggles? Ignore air resistance.

70. An archer lines up an arrow on the horizontal exactly dead center on a target as drawn in Figure P4.70. She releases the arrow, and it strikes the target a distance h = 7.6 cm below the center. If the target is 10 m from the archer, (a) how long was the arrow in the air before striking the target? (b) What was the initial velocity of the arrow as it left the bow? Report your answers in meters per second, feet per second, and miles per hour. Neglect air resistance.

71. Consider once again the swimmer in Example 4.7. Assume she can swim at a velocity of 0.30 m/s and the river is 15 m wide. She needs to get across the river as quickly as possible. (a) What direction should her velocity vector make relative to the water to get her across in the shortest time? (b) What is her overall velocity as measured from the shore? (c) What is the direction of her total velocity as seen from the shore? (d) How long does it take her to cross the river? (e) How far downstream does she end up?

72. A bartender slides a mug of root beer with mass m = 2.6 kg down a bar top of length L = 2.0 m to an inattentive patron who lets it fall a height h = 1.1 m to the floor. The bar top (Fig. P4.72) is smooth, but it still has a coefficient of kinetic friction of μ k = 0.080. (a) If the bartender gave the mug an initial velocity of 2.5 m/s, at what distance D from the bottom of the bar will the mug hit the floor? (b) What is the mug’s velocity (magnitude and direction) as it impacts the floor? (c) Draw velocity–time graphs for both the x and y directions for the mug, from the time the bartender lets go of the mug to the time it hits the floor.

73. Two workers must slide a crate designed to be pushed and pulled at the same time as shown in Figure P4.73. Joe can always exert twice as much force as Paul, and Paul can exert 160 N of force. The crate has a mass m = 45 kg, the angle θ = 25°, and the coefficient of kinetic friction between the floor and crate is 0.56. If we want to move the crate as quickly as possible, is it better to have Paul push and Joe pull, or vice versa? Does it matter? Calculate the acceleration for both cases to find out.

74. Inner ear. A student constructs a model of the utricle of the ear by attaching wooden balls (m = 0.080 kg) by strings to the bottom of a fish tank and then submerging them in water so that they float with the strings in a vertical direction. She finds that the wooden balls float with a buoyant force that exerts a tension of 0.20 N on each string as shown in Figure P4.74i. She then has her mother take her on a drive. She notes that as her mother applies the gas pedal, the strings make an angle θ = 16° with respect to the vertical as shown in Figure P4.74ii. (a) What was the acceleration of the car? (b) If the car started from rest, how long would it take to get to freeway speed of 65 mi/h? During the acceleration, the water also makes an angle with the horizontal. Are the wooden balls even needed? Explain. Ignore any forces due to the pressure in the water.

75. A vintage sports car accelerates down a slope of θ = 17° as depicted in Figure P4.75. The driver notices that the strings of the fuzzy dice hanging from his rearview mirror are perpendicular to the roof of the car. What is the car’s acceleration?

76. A spacecraft is traveling through space parallel to the x direction with an initial velocity of magnitude v 0 = 3500 m/s. At t = 0, the pilot turns on the engines, producing an acceleration of magnitude a = 150 m/s 2 along the y direction. Find the speed and the angle the velocity makes with the x direction at (a) t = 15 s and (b) 45 s.

77. A baseball is hit with an initial speed of 45 m/s at an angle of 35° relative to the x (horizontal) axis. (a) Find the speed of the ball at t = 3.2 s. (b) What angle does v make with the x axis at this moment? (c) While the ball is in the air, at what value of t is the speed the smallest?

78. A cannon shell is fired from a battleship with an initial speed of 700 m/s. (a) What is the maximum range of the shell? (b) For the firing angle that gives the maximum range, what is the maximum height reached by the shell? Ignore air drag.

79. Two blocks of mass m 1 and m 2 are sliding down an inclined plane (Fig. P4.79). (a) If the plane is frictionless, what is the normal force between the two masses? (b) If the coefficient of kinetic friction between m 1 and the plane is μ 1 = 0.25 and between m 2 and the plane is μ 2 = 0.25, what is the normal force between the two masses? (c) If μ 1 = 0.15 and μ 2 = 0.25, what is the normal force?

80. Daredevil stunt. Once upon a time, a stuntman named Evel Knievel boasted that he would jump across the Grand Canyon on a motorcycle. (This stunt was never attempted.) The width of the Grand Canyon varies from place to place, but suppose Mr. Knievel attempted to jump across at a point where the width is about 5 mi and the two rims are at the same altitude. What minimum initial speed would be required to make it safely across the canyon? Express your answer in meters per second and miles per hour. Ignore air drag.

81. Although Evel Knievel never succeeded in jumping over the Grand Canyon (see Problem 80), he was famous for jumping (with the help of his motorcycle) over, among other things, 14 large trucks. This jump covered an approximate distance of 135 ft. What was Mr. Knievel’s minimum initial velocity for this jump? Ignore air drag.

82. A person travels on a ski lift (Fig. P4.82). (a) If the support strut on the ski lift makes an angle θ = 15°, what is the horizontal acceleration of the person? (b) If the person plus the lift chair have a combined mass m = 120 kg, what is the tension force along the support strut? Assume the ski lift is moving horizontally. For simplicity, when considering forces involving the support strut, you can treat the strut in the same way as you would treat a massless cable.

83. A block slides up a frictionless, inclined plane with θ = 25° (Fig. P4.83). (a) If the block reaches a maximum height h = 4.5 m, what was the initial speed of the block? (b) Now consider a second inclined plane with the same tilt angle and for which the coefficient of kinetic friction between the block and the plane is μk = 0.15. What initial speed is now required for the block to reach the same maximum height?

84. A golf ball is hit by a golfer and travels down the fairway. Sketch the ball’s trajectory and show qualitatively the direction and magnitude of the velocity at various points.

85. A car travels along a level road with a speed of v = 25 m/s (about 50 mi/h). The coefficient of kinetic friction between the tires and the pavement is μ k = 0.55. (a) If the driver applies the brakes and the tires “lock up” so that they skid along the road, how far does the car travel before it comes to a stop? (b) If the same car is traveling downhill along a road that makes an angle θ = 12°, how much does the car’s stopping distance increase? (c) What is the stopping distance on an uphill road with θ = 12°?

86. A person riding a bicycle on level ground at a speed of 10 m/s throws a baseball forward at a speed of 15 m/s relative to the bicycle at an angle of 35° relative to the horizontal (x) direction. (a) If the ball is released from a height of 1.5 m, how far does the ball travel horizontally, as measured from the spot where it is released? (b) How far apart are the bicycle and the ball when the ball lands? Ignore air drag.

87. The pilot of an airplane with an open cockpit throws a ball horizontally with a speed of 50 m/s relative to the airplane, and the airplane travels horizontally at a speed of 150 m/s relative to the ground. If the ball’s speed relative to the ground is 175 m/s, what is the angle between the ball’s velocity and the velocity vector of the airplane?

88. A canoe travels across a 40-m-wide river that has a current of speed 5.0 m/s. The canoe has a speed of 0.80 m/s relative to the water (Fig. P4.88). The person paddles the canoe so as to cross the river as fast as possible. (a) What is the canoe’s velocity relative to an observer on shore? Give the magnitude and direction of the velocity vector. (b) How far downstream does the canoe travel as it crosses from one side of the river to the other?

89. Two crates of mass m 1 = 35 kg and m 2 = 15 kg are stacked on the back of a truck (Fig. P4.89). The frictional forces are strong enough that the crates do not slide off the truck. Assume the truck is accelerating with a = 1.7 m/s 2. Draw free-body diagrams for both crates and find the values of all forces in your diagrams.

90. A projectile is fired uphill as sketched in Figure P4.90. If v 0 = 150 m/s and θ = 30°, what is L?

91. When airplanes land or take off, they always travel along a runway in the direction that is “into” the wind because the “lift force” on an airplane wing depends on the speed of the airplane relative to the air (v rel ). We’ll see in Chapter 10 that the lift force is proportional to v 2rel. Consider a case in which a run way is parallel to the x axis and the wind velocity is v wind = 20 mi/h (directed along +x). (a) In what direction should the airplane travel to take off (along +x or –x)? (b) Estimate the speed of a typical airplane (e.g. a Boeing 737) relative to the runway as it takes off. (c) Find the ratio of the lift force for an airplane that travels into the wind to the lift force when it travels in the opposite direction. (d) Is this factor large enough to make a significant difference?

92. Consider again the game of baseball. A pitcher throws a pitch to the catcher with a speed of 100 mi/h, releasing the ball from approximately shoulder height. At what angle with respect to the horizontal (x) should he throw the ball so that it crosses the plate waist high for the batter? The distance from the pitcher to the catcher is 60.5 feet. Ignore air drag. Hint: Assume the angle is small so that the horizontal component of the velocity is approximately equal to the speed.


1. A bicycle wheel of radius 0.30 m is spinning at a rate of 60 revolutions per minute. (a) What is the centripetal acceleration of a point on the edge of the wheel? (b) What is the period of the wheel’s motion?

2. For the bicycle wheel in Problem 1, what is the centripetal acceleration of a point that is 0.10 m from the edge? Explain why this value is different from the answer to part (a) of Problem 1.

3. The Earth has a radius of 6.4 × 10 6 m and completes one revolution about its axis in 24 h. (a) Find the speed of a point at the equator. (b) Find the speed of New York City.

4. In the game of baseball, a pitcher throws a curve ball with as much spin as possible. This spin makes the ball “curve” on its way to the batter. In a typical case, the ball spins at about 30 revolutions per second. What is the maximum centripetal acceleration of a point on the edge of the baseball?

5. A jogger is running around a circular track of circumference 400 m. If the jogger has a speed of 12 km/h, what is the centripetal acceleration of the jogger?

6. Consider the motion of the hand of a mechanical clock. If the minute hand of the clock has a length of 6.0 cm, what is the centripetal acceleration of a point at the end of the hand?

7. If the circular track in Figure 5.1 has a radius of 100 m and the runner has a speed of 5.0 m/s, what is the period of the motion?

8. What is the acceleration of the Moon as it moves in its circular orbit around the Earth? Hint: You will find some useful data in Table 5.1.

9. Consider points on the Earth’s surface as sketched in Figure P5.9. Because of the Earth’s rotation, these points undergo uniform circular motion. Compute the centripetal acceleration of (a) a point at the equator, and (b) at a latitude of 30°.

10. In the days before compact discs (ancient history!), music was recorded in scratches in the surface of vinyl-coated disks called records. In a typical record player, the record rotated with a period of 1.8 s. Find the centripetal acceleration of a point on the edge of the record. Assume a radius of 15 cm.

11. A compact disc spins at 2.5 revolutions per second. An ant is walking on the CD and finds that it just begins to slide off the CD when it reaches a point 3.0 cm from the CD’s center. (a) What is the coefficient of friction between the ant and the CD? (b) Is this the coefficient of static friction or kinetic friction?

12. When a fighter pilot makes a very quick turn, he experiences a centripetal acceleration. When this acceleration is greater than about 8 × g, the pilot will usually lose consciousness (“black out”). Consider a pilot flying at a speed of 900 m/s who wants to make a very sharp turn. What is the minimum radius of curvature he can take without blacking out?

13. At a practice for a recent automobile race, officials found that the drivers were nearly “blacking out,” which led to cancellation of the race. The cars were traveling at about 240 mi/h, and the track was approximately 1.5 mi long. Find the centripetal acceleration during a turn and compare it with the physiological limit of approximately 8 × g discussed in Problem 12. Assume the track was circular.

14. The Daytona 500 stock car race is held on a track that is approximately 2.5 mi long, and the turns are banked at an angle of 31°. It is currently possible for cars to travel through the turns at a speed of about 180 mi/h. Assuming these cars are on the verge of slipping into the outer wall of the racetrack (because they are racing!), find the coefficient of static friction between the tires and the track.

15. Consider again the problem of a car traveling along a banked turn. Sometimes roads have a “reversed” banking angle. That is, the road is tilted “away” from the center of curvature of the road. If the coefficient of static friction between the tires and the road is μ s = 0.50, the radius of curvature is 15 m, and the banking angle is 10°, what is the maximum speed at which a car can safely navigate such a turn?

16. Consider the motion of a rock tied to a string of length 0.50 m. The string is spun so that the rock travels in a vertical circle as shown in Figure P5.16. The mass of the rock is 1.5 kg, and it is twirling at constant speed with a period of 0.33 s. (a) Draw free-body diagrams for the rock when it is at the top and when it is at the bottom of the circle. Your diagrams should include the tension in the string, but the value of T is not yet known. (b) What is the total force on the rock directed toward the center of the circle? (c) Find the tension in the string when the rock is at the top and when it is at the bottom of the circle.

17. Consider the motion of the rock in Figure P5.16. What is the minimum speed the rock can have without the string becoming “slack”?

18. A stone of mass 0.30 kg is tied to a string of length 0.75 m and is swung in a horizontal circle with speed v. The string has a breaking-point force of 50 N. What is the largest value v can have without the string breaking? Ignore any effects due to gravity.

19. The track near the top of your favorite roller coaster has a circular shape with a diameter of 20 m. When you are at the top, you feel as if your weight is only one-third your true weight. What is the speed of the roller coaster?

20. A roller coaster track is designed so that the car travels upside down on a certain portion of the track as shown in Figure P5.20. What is the minimum speed the roller coaster can have without falling from the track? Assume the track has a radius of curvature of 30 m.

21. A car of mass 1000 kg is traveling over the top of a hill as shown in Figure P5.21. (a) If the hill has a radius of curvature of 40 m and the car is traveling at 15 m/s, what is the normal force between the hill and the car at the top of the hill? (b) If the driver increases her speed sufficiently, the car will leave the ground at the top of the hill. What is the speed required to make that happen?

22. On a popular amusement park ride, the rider sits in a chair suspended by a cable as shown in Figure P5.22. The top end of the cable is tied to a rotating frame that spins, hence moving the chair in a horizontal circle with r = 10 m. The ride makes one complete revolution every 10 s. (a) Draw pictures showing the path followed by the chair. Give both a side view and a top view. (b) In your diagrams in part (a), indicate all the forces on the chair. Also draw a free-body diagram for the chair. (c) Find the components of the forces in the vertical direction and in the direction toward the center of the chair’s circular path. Express your answers in terms of the tension in the cable, the angle u, and the mass of the chair m. (d) Apply Newton’s second law in both the vertical and radial directions. What is the acceleration of the chair along y? (e) Find the angle u the cable makes with the vertical.

23. Consider a roller coaster as it travels near the bottom of its track as sketched in Figure P5.23. At this point, the normal force on the roller coaster is three times its weight. If the speed of the roller coaster is 20 m/s, what is the radius of curvature of the track?

24. A coin is sitting on a record as sketched in Figure P5.24. It is found that the coin slips off the record when the rotation rate is 0.30 rev/s. What is the coefficient of static friction between the coin and the record?

25. A rock is tied to a string and spun in a circle of radius 1.5 m as shown in Figure P5.25. The speed of the rock is 10 m/s. (a) Draw a picture giving both a top view and a side view of the motion of the rock. (b) What are all the forces acting on the rock? Add them to your pictures in part (a). Then draw a free-body diagram for the rock. (c) What is the total force on the rock directed toward the center of its circular path? Express your answer in terms of the (unknown) tension in the string T. (d) Apply Newton’s second law along both the vertical and the horizontal direction and find the angle u the string makes with the horizontal.

26. Spin out! An interesting amusement park activity involves a cylindrical room that spins about a vertical axis (Fig. P5.26). Participants in the “ride” are in contact with the wall of the room, and the circular motion of the room results in a normal force from the wall on the riders. When the room spins sufficiently fast, the floor is retracted and the frictional force from the wall keeps the people “stuck” to the wall. Assume the room has a radius of 2.0 m and the coefficient of static friction between the people and the wall is μ s = 0.50. (a) Draw pictures showing the motion of a “rider.” Give both a side view and a top view. (b) What are all the forces acting on a rider? Add them to your pictures in part (a). Then draw a free-body diagram for a rider. (c) What are the components of all the forces directed toward the center of the circle (the radial direction)? (d) Apply Newton’s second law along both the vertical and the radial directions. Find the minimum rotation rate for which the riders do not slip down the wall.

27. Consider the circular space station in Figure 5.13. Suppose the station has a radius of 15 m and is designed to have an acceleration due to “artificial gravity” of g/2. Find the speed of the rim of the space station.

28. A rock of mass m = 2.5 kg is tied to the end of a string of length L = 1.2 m. The other end of the string is fastened to a ceiling, and the rock is set into motion so that it travels in a horizontal circle of radius r = 0.70 m as sketched in Figure P5.28. (a) Draw a picture showing the motion of the rock. Give both a top view and a side view. (b) What are all the forces on the rock? Add them to your pictures in part (a). Then draw a free-body diagram for the rock. (c) Find the vertical and horizontal components of all the forces on the rock. (d) Apply Newton’s second law in both the vertical (y) and the horizontal directions. What is the acceleration along y? Find the tension in the string.

29. A car of mass 1700 kg is traveling without slipping on a flat, curved road with a radius of curvature of 35 m. If the car’s speed is 12 m/s, what is the frictional force between the road and the tires?

30. Consider a Ferris wheel in which the chairs hang down from the main wheel via a cable. The cable is 2.0 m long, and the radius of the wheel is 12 m (see Fig. P5.30). When a chair is in the orientation shown in Figure P5.30 (the “3 o’clock” position), the cable attached to the chair makes an angle of θ = 20° with the vertical. Find the speed of the chair.

31. A rock of mass m = 1.5 kg is tied to a string of length L = 2.0 m and is twirled in a vertical circle as shown in Figure 5.10. The speed v of the rock is constant; that is, it is the same at the top and the bottom of the circle. If the tension in the string is zero when the rock is at its highest point (so that the string just barely goes slack), what is the tension when the rock is at the bottom?

32. We saw in Example 5.6 how a centrifuge can be used to separate cells from a liquid. To increase the rate at which objects can be separated from solution, it is useful to make the centrifuge’s speed as large as possible. If you want to design a centrifuge of diameter 50 cm to have a force of 10 6 times the force of Earth’s gravity, what is the speed of the outer edge of the centrifuge? Such a device is called an ultracentrifuge.

33. A centrifuge can be used to separate DNA molecules from solution. Estimate how long it will take the centrifuge in Example 5.6 to separate a DNA molecule from water. Assume the centrifuge tube is 2.0 cm long. For this case, the drag coefficient in Stokes’s law (Eq. 5.14) is C = 0.020 N· s/m 2 .

34. NASA has built centrifuges to enable astronauts to train in conditions in which the acceleration is very large. The device in Figure P5.34 shows one of these “human centrifuges.” If the device has a radius of 8.0 m and attains accelerations as large as 5.0 × g, what is the rotation rate?

35. Two small objects of mass 20 kg and 30 kg are a distance 1.5 m apart. What is the gravitational force of one of these objects on the other?

36. If the masses of the objects in Problem 35 are both increased by a factor of √5, by what factor does the gravitational force change? Do not use a calculator to solve this problem!

37. Three lead balls of mass m 1 = 15 kg, m 2 = 25 kg, and m 3 = 9.0 kg are arranged as shown in Figure P5.37. Find the total gravitational force exerted by balls 1 and 2 on ball 3. Be sure to give the magnitude and the direction of this force.

38. Travel and lose pounds! Your apparent weight is the force you feel on the bottoms of your feet when you are standing. Due to the Earth’s rotation, your apparent weight is slightly more when you are at the South Pole than when you are at the equator. What is the ratio of your apparent weights at these two locations? Carry three significant figures in your calculation.

39. Find the gravitational force of the Sun on the Earth.

40. Calculate the acceleration due to gravity on the surface of Mars.

41. Estimate the gravitational force between two bowling balls that are nearly touching.

42. Suppose the bowling balls in Problem 41 are increased in size (radius) by a factor of two, but their density does not change. By what factor does the gravitational force change? Hint: When the radius is changed, both the mass and the separation will change.

43. When a spacecraft travels from Earth to the Moon, the gravitational force from Earth initially opposes this journey. Eventually, the spacecraft reaches a point where the Moon’s gravitational attraction overcomes the Earth’s gravity. How far from Earth must the spacecraft be for the gravitational forces from the Moon and Earth to just cancel?

44. Find the ratio of your weight on Earth to your weight on the surface of the Sun.

45. Some communications and television towers are much taller than any buildings. These towers have been used to study how the Earth’s gravitational force varies with distance from the center of the Earth. Calculate the ratio of the acceleration due to Earth’s gravity at the top of a tower that is 600 m tall to the value of g at the Earth’s surface. Hint: Keep four significant figures in your calculation.

46. Suppose the density of the Earth was somehow reduced from its actual value to 1000 kg/m 3 (the density of water). Find the value of g, the acceleration due to gravity, on this new planet. Assume the radius does not change.

47. You are an astronaut (m = 95 kg) and travel to a planet that is the same radius and mass size as Earth, but it has a rotational period of only 2 h. What is your apparent weight at the equator of this planet?

48. In Section 5.4, we showed that the radius of a geosynchronous orbit about the Earth is 4.2 × 10 7 m, compared with the radius of the Earth, which is 6.4 × 10 6 m. By what factor is the force of gravity smaller when you are in geosynchronous orbit than when you are on the Earth’s surface?

49. Saturn makes one complete orbit of the Sun every 29.4 years. Calculate the radius of the orbit of Saturn. Hint: It is a very good approximation to assume this orbit is circular.

50. The region of the solar system between Mars and Jupiter contains many asteroids that orbit the Sun. Consider an asteroid in a circular orbit of radius 5.0 × 10 11 m. Find the period of the orbit.

51. A newly discovered planet is found to have a circular orbit, with a radius equal to 27 times the radius of Earth’s orbit. How long does this planet take to complete one orbit around the Sun?

52. In recent years, a number of nearby stars have been found to possess planets. Suppose the orbital radius of such a planet is found to be 4.0 × 10 11 m, with a period of 1100 days. Find the mass of the star.

53. Mars has two moons, Phobos and Deimos. It is known that the larger moon, Phobos, has an orbital radius of 9.4 × 10 6 m and a mass of 1.1 × 10 16 kg. Find its orbital period.

54. In our derivation of Kepler’s laws, we assumed the only force on a planet is due to the Sun. In a real solar system, however, the gravitational forces from the other planets can sometimes be important. Calculate the gravitational force of Jupiter on the Earth and compare it to the magnitude of the force from the Sun. Do the calculations for the cases when Jupiter is both closest to and farthest from Earth (Fig. P5.54).

55. What is the speed of a satellite in a geosynchronous orbit about Earth? Compare it with the speed of the Earth as it orbits the Sun.

56. Syzygy. We have seen that the normal tides are due to the gravitational force exerted by the Moon on Earth’s oceans. When the Moon, Sun, and Earth are aligned as shown in Figure P5.56, the magnitude of the tide increases due to the gravitational force exerted by the Sun on the oceans (at the times of the “new” Moon and the “full” Moon during the course of a month). Calculate the ratio of the gravitational force of the Sun to that of the Moon on the oceans. This “extra” force from the Sun does make a difference!

57. In Figure 5.29, we saw that the tides on Earth are due to the variation of the Moon’s gravitational force with distance. Find the approximate ratio of the Moon’s gravitational force on two portions of the ocean, one nearest the Moon and one on the opposite side of the Earth.

58. Your weight is due to the gravitational attraction of the Earth. The Moon, though, also exerts a gravitational force on you, and when it is overhead, your weight decreases by a small amount. Calculate the effect of the Moon on your weight. Express your result as a percentage change for the cases of the Moon overhead and the Moon on the opposite side of the Earth.

59. During an eclipse, the Sun, Earth, and Moon are arranged in a line as shown in Figure P5.59. There are two types of eclipse: (a) a lunar eclipse, when Earth is between the Sun and the Moon, and (b) a solar eclipse, when the Moon is between the Sun and Earth. Calculate the percentage change in your weight when going from one type of eclipse to the other.

60. In the film Mission to Mars (released in 2000), the spacecraft (see Fig. P5.60) features a rotating section to provide artificial gravity for the long voyage. A physicist viewing a scene from the interior of the spacecraft notices that the diameter of the rotating portion of ship is about five times the height of an astronaut walking in that section (or about 10 m). Later, in a scene showing the spacecraft from the exterior, she notices that the living quarters of the ship rotate with a period of about 30 s. Did the movie get the physics right? Compare the centripetal acceleration of a 1.7-m-tall astronaut at his feet to that at his head. Compare these accelerations to g.

61. Will your apparent weight at the top of Mount Everest (altitude = 8850 m = 29,035 ft) be more or less than at sea level at the same latitude (27.98° N)? What is the ratio of your apparent weight at these two locations? For simplicity, consider only the effect of altitude and ignore the spinning of the Earth.

62. A man stands 6.0 ft tall at sea level on the North Pole as shown in Figure P5.62. (a) What is the difference in the value of g (the gravitational acceleration) between his head and his feet? (b) The man is now put in a space suit and transported to a location one Earth radius away from a black hole of mass equal to 20 times the mass of our Sun. If the man’s feet are pointing in the direction of the black hole, what is the difference in the gravitational acceleration between his head and his feet? Would this difference in acceleration be harmful, just noticeable, or somewhere in between?

63. Proponents of astrology claim that the positions of the planets at the time of a baby’s birth will affect the life of that person in important ways. Some assert that this effect is due to gravity. Examine this claim with Newton’s law of gravity. Calculate the maximum force exerted on the baby by the planet Mars and compare that with the force of gravity exerted by the mass of the doctor’s head as she delivers the baby. Assume Mars is at its closest approach to Earth at the time. You will need to estimate the mass of a newborn as well as the mass and distance between the doctor’s head and the baby.

64. An ancient and deadly weapon, a sling consists of two braided cords, each about half an arm’s length long, attached to a leather pocket. The pocket is loaded with a projectile made of lead, carved rock, or clay and made to swing in a vertical circle as shown in Figure P5.64. The projectile is released by letting go of one end of the cord. (a) If a Roman soldier can swing the sling at a rate of 7.5 rotations per second, what is the maximum range of his 100-g projectile? (Ignore air drag.) (b) What is the maximum tension in each cord during the rotation?

65. A popular circus act features daredevil motorcycle riders encased in the “Globe of Death” (Fig. P5.65), a spherical metal cage of diameter 16 ft. (a) A rider of mass 65 kg on a 125-cc (95-kg) motorcycle keeps his bike horizontal as he rides around the “equator” of the globe. What coefficient of friction is needed between his tire and the cage to keep him in place? (b) How many loops will the rider make per second? (c) The same rider performs vertical loops in the globe. What force does the cage need to withstand at the top and the bottom of the rider’s loop? Assume a speed of 20 mi/h for both tricks.

66. Asteroid satellite. While on its way to Jupiter in 1993, the Galileo spacecraft made a flyby of asteroid Ida. Images captured (Fig. P5.66) of Ida showed that the asteroid has a tiny moon of its own, since given the name Dactyl. Measurements found Ida to be about 56 × 24 × 21 km (35 × 15 × 13 mi) in size, and Dactyl’s orbital period and radius are approximately 27 h and 95 km, respectively. From these data, determine Ida’s approximate mass and density.

67. The death spiral. An Olympic pair figure- skating routine features an element called the death spiral shown in Figure P5.67. In this routine, the male skater swings his female partner in a circle. If the rotation rate is three-fourths a rotation per second, estimate the tension in the arms that the skaters’ grip must withstand when performing this element. Hint: Approximate the female skater as a point mass located at her waist. It may be useful to take measurements directly from Figure P5.67 and scale appropriately.

68. Consider a hypothetical extrasolar world, planet Tungsten, that has twice the radius of the Earth and twice its density. (a) What is the acceleration due to gravity on the surface of planet Tungsten? (b) An interstellar astronaut lands on the equator of this planet and finds that his apparent weight matches his weight on Earth. What is the period of rotation of planet Tungsten?

69. The movie 2001: A Space Odyssey (released in 1968) features a massive rotating space station of radius 100 m, similar to the one in Figure 5.13B. (a) What period of rotation is needed to provide an artificial gravity of g at the rim? (b) At what speed is the rim moving? (c) What is your apparent weight if you run along the rim at 4.2 m/s opposite the rotation direction? (d) What is your apparent weight if you instead run in the direction of rotation? (e) In which direction would you run (either with or against the rotation) to get the best workout? Would it matter?

70. An astronaut stands on the surface of Vesta, which, with an average radius of 270 m, makes it the third largest object in the asteroid belt. The astronaut picks up a rock and drops it from a height of 1.5 m. He times the fall and finds that the rock strikes the ground after 3.2 s. (a) Determine the acceleration due to gravity at the surface of Vesta. (b) Find the mass of Vesta. (c) If the astronaut can jump to a height of 82 cm while wearing his space suit on Earth, how high could he jump on Vesta? Assume any rotation of Vesta can be ignored.

71. The International Space Station orbits at an average height of 350 km above sea level. (a) Determine the acceleration due to gravity at that height and find the orbital velocity and the period of the space station. (b) The Hubble Space Telescope orbits at 600 km. What is the telescope’s orbital velocity and period?

72. Oil exploration. When searching for gold, measurements of g can be used to find regions within the Earth where the density is larger than that of normal soil. Such measurements can also be used to find regions in which the density of the Earth is smaller than normal soil; these regions might contain a valuable fluid (oil). Consider a deposit of oil that is 300 m in diameter and just below the surface of the Earth. For simplicity, assume the deposit is spherical. Estimate the change in the acceleration due to gravity on the surface above this deposit. Assume the density of the oil is 1500 kg/m 3 and the density of normal soil and rock is 2000 kg/m 3. Note: Companies that search for valuable minerals actually use this method.

73. A rock of mass m is tied to a string of length L and swung in a horizontal circle of radius r. The string can withstand a maximum tension Tmax before it breaks. (a) What is the maximum speed v max the rock can have without the string breaking? (b) The speed of the rock is now increased to 3v max. The original single string is then replaced by N pieces that are all identical to the original string. What is the minimum value of N required so that the strings do not break? Ignore the force of gravity on the rock.

74. On which of the planets in our solar system would you weigh the most?

75. Experiments have shown that riders in a car begin to feel uncomfortable while traveling around a turn if their acceleration is greater than about 0.40 × g. Use this fact to calculate the minimum radius of curvature for turns at (a) 10 m/s (appropriate for driving in town) and (b) 30 m/s (highway driving).

76. In Insight 5.2, we discussed how, because of the force of gravity from the Moon, the Earth moves in an orbit around a point that lies between it and the Moon. (a) Find the radius of the Earth’s orbit. (b) Where does this point lie relative to the Earth’s surface? That is, does it lie inside or outside the Earth itself?

77. Gravitational tractor. In some science fiction stories, a “tractor beam” is used to pull an object from one point in space to another. That may not be just science fiction. It has been proposed that a “gravitational tractor” could be used to “tow” an asteroid. In theory, this tractor could be used to deflect asteroids that would otherwise collide with the Earth. A typical asteroid that could significantly damage life on Earth might have a radius 100 m and density 2000 kg/m 3. The “tractor” would be just a very massive spacecraft; in current designs, it would have a mass of about 2 × 10 4 kg. (a) What is the maximum gravitational force the tractor could exert on the asteroid? (b) What would be the acceleration of the asteroid? (c) If the tractor stayed near the asteroid for 1 year, what would be the deflection of the asteroid? (d) Consider an asteroid initially headed straight for Earth with a speed of 3 × 10 4 m/s. If the tractor first comes close to the asteroid when it is 6 × 10 11 m from Earth (near the orbit of Jupiter), will the asteroid hit or miss the Earth? For simplicity, ignore the gravitational force of the Earth on the asteroid. Note: NASA is seriously considering the development of such a gravitational tractor.

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