Problems in Introductory Physics, Study notes of Physics Fundamentals

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Problems in Introductory Physics

B. Crowell and B. Shotwell

Copyright 2016 B. Crowell and B. Shotwell.

This book is licensed under the Creative Com- mons Attribution-ShareAlike license, version 3.0, http://creativecommons.org/licenses/by- sa/3.0/, except for those photographs and drawings of which we are not the author, as listed in the photo credits. If you agree to the license, it grants you certain privileges that you would not otherwise have, such as the right to copy the book, or download the digital version free of charge from www.lightandmatter.com.

• 1 Measurement
  • 1.1 The SI
  • 1.2 Significant figures
  • 1.3 Proportionalities
  • 1.4 Estimation
  • Problems
• 2 Kinematics in one dimension
  • 2.1 Velocity
  • 2.2 Acceleration
  • Problems
• 3 Kinematics in three dimensions
  • 3.1 Vectors
  • 3.2 Motion
  • Problems
• 4 Newton’s laws, part
  • 4.1 Newton’s first law
  • 4.2 Newton’s second law
  • 4.3 Newton’s third law
  • Problems
• 5 Newton’s laws, part
  • 5.1 Classification of forces
  • 5.2 Friction
  • 5.3 Elasticity
  • 5.4 Ropes, pulleys, tension, and simple machines
  • 5.5 Analysis of forces
  • Problems
• 6 Circular motion
  • 6.1 Uniform circular motion
  • 6.2 Rotating frames
  • 6.3 Nonuniform motion
  • 6.4 Rotational kinematics
  • Problems
• 7 Conservation of energy
  • 7.1 Conservation laws
  • 7.2 Work
  • 7.3 Kinetic energy
  • 7.4 Potential energy 4 CONTENTS
  • Problems
• 8 Conservation of momentum
  • 8.1 Momentum: a conserved vector
  • 8.2 Collisions
  • 8.3 The center of mass
  • Problems
• 9 Conservation of angular momentum
  • 9.1 Angular momentum
  • 9.2 Rigid-body dynamics
  • 9.3 Torque
  • 9.4 Statics
  • Problems
• 10 Fluids
  • 10.1 Statics
  • 10.2 Dynamics
  • Problems
• 11 Gravity
  • 11.1 Kepler’s laws
  • 11.2 Newton’s law of gravity
  • 11.3 The shell theorem
  • 11.4 Universality of free fall
  • 11.5 Current status of Newton’s theory
  • 11.6 Energy
  • Problems
• 12 Oscillations
  • 12.1 Periodic motion
  • 12.2 Simple harmonic motion
  • 12.3 Damped oscillations
  • 12.4 Driven oscillations
  • Problems
• 13 Electrical interactions
  • 13.1 Charge and Coulomb’s law
  • 13.2 The electric field
  • 13.3 Conductors and insulators
  • 13.4 The electric dipole
  • 13.5 Gauss’s law
  • Problems

1 Measurement

This is not a textbook. It’s a book of problems meant to be used along with a textbook. Although each chapter of this book starts with a brief sum- mary of the relevant physics, that summary is not meant to be enough to allow the reader to actually learn the subject from scratch. The pur- pose of the summary is to show what material is needed in order to do the problems, and to show what terminology and notation are being used.

1.1 The SI

The Syst`eme International (SI) is a system of measurement in which mechanical quantities are expressed in terms of three basic units: the me- ter (m), the kilogram (kg), and the second (s). Other units can be built out of these. For ex- ample, the SI unit to measure the flow of water through a pipe would be kg/s. To modify the units there is a consistent set of prefixes. The following are common and should be memorized:

prefix meaning nano- n 10 −^9 micro- μ 10 −^6 milli- m 10 −^3 kilo- k 103 mega- M 106

The symbol μ, for micro-, is Greek lowercase mu, which is equivalent to the Latin “m.” There is also centi-, 10−^2 , which is only used in the cen- timeter.

1.2 Significant figures

The international governing body for football (“soccer” in the US) says the ball should have a circumference of 68 to 70 cm. Tak- ing the middle of this range and divid- ing by π gives a diameter of approximately

21. 96338214668155633610595934540698196 cm.

The digits after the first few are completely meaningless. Since the circumference could have varied by about a centimeter in either direction, the diameter is fuzzy by something like a third of a centimeter. We say that the additional, ran- dom digits are not significant figures. If you write down a number with a lot of gratuitous insignificant figures, it shows a lack of scientific literacy and imples to other people a greater pre- cision than you really have. As a rule of thumb, the result of a calculation has as many significant figures, or “sig figs,” as the least accurate piece of data that went in. In the example with the soccer ball, it didn’t do us any good to know π to dozens of digits, because the bottleneck in the precision of the result was the figure for the circumference, which was two sig figs. The result is 22 cm. The rule of thumb works best for multiplication and division. The numbers 13 and 13. 0 mean different things, because the latter implies higher preci- sion. The number 0.0037 is two significant fig- ures, not four, because the zeroes after the dec- imal place are placeholders. A number like 530 could be either two sig figs or three; if we wanted to remove the ambiguity, we could write it in sci- entific notation as 5. 3 × 102 or 5. 30 × 102.

1.3 Proportionalities

Often it is more convenient to reason about the ratios of quantities rather than their actual val- ues. For example, suppose we want to know what happens to the area of a circle when we triple its radius. We know that A = πr^2 , but the factor of π is not of interest here because it’s present in both cases, the small circle and the large one. Throwing away the constant of proportionality, we can write A ∝ r^2 , where the proportionality symbol ∝, read “is proportional to,” says that the left-hand side doesn’t necessarily equal the right-hand side, but it does equal the right-hand side multiplied by a constant.

PROBLEMS 7

Problems

1-a1 Convert 134 mg to units of kg, writing your answer in scientific notation.

. Solution, p. 153 1-a2 Express each of the following quantities in micrograms: (a) 10 mg, (b) 10^4 g, (c) 10 kg, (d) 100 × 103 g, (e) 1000 ng.

√

1-a3 In the last century, the average age of the onset of puberty for girls has decreased by several years. Urban folklore has it that this is because of hormones fed to beef cattle, but it is more likely to be because modern girls have more body fat on the average and possibly because of estrogen-mimicking chemicals in the environment from the breakdown of pesticides. A hamburger from a hormone-implanted steer has about 0.2 ng of estrogen (about double the amount of natural beef). A serving of peas contains about 300 ng of estrogen. An adult woman produces about 0.5 mg of estrogen per day (note the different unit!). (a) How many hamburgers would a girl have to eat in one day to consume as much estrogen as an adult woman’s daily production? (b) How many servings of peas?

√

1-d1 The usual definition of the mean (aver- age) of two numbers a and b is (a + b)/2. This is called the arithmetic mean. The geometric mean, however, is defined as (ab)^1 /^2 (i.e., the square root of ab). For the sake of definiteness, let’s say both numbers have units of mass. (a) Compute the arithmetic mean of two numbers that have units of grams. Then convert the num- bers to units of kilograms and recompute their mean. Is the answer consistent? (b) Do the same for the geometric mean. (c) If a and b both have units of grams, what should we call the units of ab? Does your answer make sense when you take the square root? (d) Suppose someone pro- poses to you a third kind of mean, called the

superduper mean, defined as (ab)^1 /^3. Is this rea- sonable?

. Solution, p. 153

1-d2 (a) Based on the definitions of the sine, cosine, and tangent, what units must they have? (b) A cute formula from trigonometry lets you find any angle of a triangle if you know the lengths of its sides. Using the notation shown in the figure, and letting s = (a + b + c)/2 be half the perimeter, we have

tan A/2 =

(s − b)(s − c) s(s − a)

Show that the units of this equation make sense. In other words, check that the units of the right- hand side are the same as your answer to part a of the question.

. Solution, p. 153

Problem 1-d2.

1-d3 Jae starts from the formula V = 13 Ah for the volume of a cone, where A is the area of its base, and h is its height. He wants to find an equation that will tell him how tall a conical

8 CHAPTER 1. MEASUREMENT

tent has to be in order to have a certain volume, given its radius. His algebra goes like this:

V =

Ah

A = πr^2

V =

πr^2 h

h =

πr^2 3 V

Use units to check whether the final result makes sense. If it doesn’t, use units to locate the line of algebra where the mistake happened.

. Solution, p. 153

1-d4 The distance to the horizon is given by the expression

2 rh, where r is the radius of the Earth, and h is the observer’s height above the Earth’s surface. (This can be proved using the Pythagorean theorem.) Show that the units of this expression make sense. Don’t try to prove the result, just check its units. (For an example of how to do this, see problem 1-d3 on p. 7, which has a solution given in the back of the book.)

1-d5 Let the function x be defined by x(t) = Aebt, where t has units of seconds and x has units of meters. (For b < 0, this could be a fairly accurate model of the motion of a bullet shot into a tank of oil.) Show that the Taylor series of this function makes sense if and only if A and b have certain units.

1-g1 In an article on the SARS epidemic, the May 7, 2003 New York Times discusses conflict- ing estimates of the disease’s incubation period (the average time that elapses from infection to the first symptoms). “The study estimated it to be 6.4 days. But other statistical calculations ... showed that the incubation period could be as long as 14.22 days.” What’s wrong here?

1-g2 The photo shows the corner of a bag of pretzels. What’s wrong here?

Problem 1-g2.

1-j1 The one-liter cube in the photo has been marked off into smaller cubes, with linear dimen- sions one tenth those of the big one. What is the volume of each of the small cubes?

. Solution, p. 153

Problem 1-j1.

1-j2 How many cm^2 is 1 mm^2?

. Solution, p. 153 1-j3 Compare the light-gathering powers of a 3-cm-diameter telescope and a 30-cm telescope. . Solution, p. 153 1-j4 The traditional Martini glass is shaped like a cone with the point at the bottom. Sup- pose you make a Martini by pouring vermouth into the glass to a depth of 3 cm, and then adding gin to bring the depth to 6 cm. What are the proportions of gin and vermouth? . Solution, p. 153 1-j5 How many cubic inches are there in a cubic foot? The answer is not 12. (^) √

1-j6 Assume a dog’s brain is twice as great in diameter as a cat’s, but each animal’s brain cells are the same size and their brains are the same shape. In addition to being a far better

10 CHAPTER 1. MEASUREMENT

Problem 1-k5.

1-k6 X-ray images aren’t only used with hu- man subjects but also, for example, on insects and flowers. In 2003, a team of researchers at Argonne National Laboratory used x-ray im- agery to find for the first time that insects, al- though they do not have lungs, do not necessar- ily breathe completely passively, as had been be- lieved previously; many insects rapidly compress and expand their trachea, head, and thorax in order to force air in and out of their bodies. One difference between x-raying a human and an in- sect is that if a medical x-ray machine was used on an insect, virtually 100% of the x-rays would pass through its body, and there would be no contrast in the image produced. Less penetrat- ing x-rays of lower energies have to be used. For comparison, a typical human body mass is about 70 kg, whereas a typical ant is about 10 mg. Es- timate the ratio of the thicknesses of tissue that must be penetrated by x-rays in one case com- pared to the other. (^) √

1-m1 A taxon (plural taxa) is a group of living things. For example, Homo sapiens and Homo neanderthalensis are both taxa — specif- ically, they are two different species within the genus Homo. Surveys by botanists show that the number of plant taxa native to a given contigu- ous land area A is usually approximately pro- portional to A^1 /^3. (a) There are 70 different species of lupine native to Southern California, which has an area of about 200, 000 km^2. The San Gabriel Mountains cover about 1, 600 km^2. Suppose that you wanted to learn to identify all

the species of lupine in the San Gabriels. Ap- proximately how many species would you have to familiarize yourself with?

√

(b) What is the interpretation of the fact that the exponent, 1/3, is less than one? ? 1-m2 The population density of Los Angeles is about 4000 people/km^2. That of San Francisco is about 6000 people/km^2. How many times far- ther away is the average person’s nearest neigh- bor in LA than in San Francisco? The answer is not 1.5. (^) √ ? 1-m3 In Europe, a piece of paper of the stan- dard size, called A4, is a little narrower and taller than its American counterpart. The ratio of the height to the width is the square root of 2, and this has some useful properties. For instance, if you cut an A4 sheet from left to right, you get two smaller sheets that have the same propor- tions. You can even buy sheets of this smaller size, and they’re called A5. There is a whole se- ries of sizes related in this way, all with the same proportions. (a) Compare an A5 sheet to an A in terms of area and linear size. (b) The series of paper sizes starts from an A0 sheet, which has an area of one square meter. Suppose we had a series of boxes defined in a similar way: the B0 box has a volume of one cubic meter, two B boxes fit exactly inside an B0 box, and so on. What would be the dimensions of a B0 box?√ ? 1-p1 Estimate the number of jellybeans in the figure.

. Solution, p. 153 1-p2 Suppose you took enough water out of the oceans to reduce sea level by 1 mm, and you took that water and used it to fill up water bot- tles. Make an order-of-magnitude estimate of how many water bottles could you fill.

1-p3 If you filled up a small classroom with pennies, about much money would be in the room?

PROBLEMS 11

Problem 1-p1.

1-p4 Estimate the mass of one of the hairs in Albert Einstein’s moustache, in units of kg.

1-p5 Estimate the number of blades of grass on a football field.

1-p6 Suppose someone built a gigantic apart- ment building, measuring 10 km × 10 km at the base. Estimate how tall the building would have to be to have space in it for the entire world’s population to live.

1-p7 (a) Using the microscope photo in the figure, estimate the mass of a one cell of the E. coli bacterium, which is one of the most com- mon ones in the human intestine. Note the scale at the lower right corner, which is 1 μm. Each of the tubular objects in the column is one cell. (b) The feces in the human intestine are mostly bacteria (some dead, some alive), of which E. coli is a large and typical component. Estimate the number of bacteria in your intestines, and compare with the number of human cells in your body, which is believed to be roughly on the or- der of 10^13. (c) Interpreting your result from

part b, what does this tell you about the size of a typical human cell compared to the size of a typical bacterial cell?

Problem 1-p7.

1-q1 Estimate the number of man-hours re- quired for building the Great Wall of China.

. Solution, p. 154?

1-q2 Plutonium-239 is one of a small num- ber of important long-lived forms of high-level radioactive nuclear waste. The world’s waste stockpiles have about 10^3 metric tons of pluto- nium. Drinking water is considered safe by U.S. government standards if it contains less than 2 × 10 −^13 g/cm^3 of plutonium. The amount of ra- dioactivity to which you were exposed by drink- ing such water on a daily basis would be very small compared to the natural background radi- ation that you are exposed to every year. Sup- pose that the world’s inventory of plutonium- were ground up into an extremely fine dust and then dispersed over the world’s oceans, thereby becoming mixed uniformly into the world’s wa- ter supplies over time. Estimate the resulting concentration of plutonium, and compare with the government standard. ?

2 Kinematics in one dimension

This is not a textbook. It’s a book of problems meant to be used along with a textbook. Although each chapter of this book starts with a brief sum- mary of the relevant physics, that summary is not meant to be enough to allow the reader to actually learn the subject from scratch. The pur- pose of the summary is to show what material is needed in order to do the problems, and to show what terminology and notation are being used.

2.1 Velocity

The motion of a particle in one dimension can be described using the function x(t) that gives its position at any time. Its velocity is defined by the derivative

v =

dx dt

From the definition, we see that the SI units of velocity are meters per second, m/s. Positive and negative signs indicate the direction of mo- tion, relative to the direction that is arbitrarily called positive when we pick our coordinate sys- tem. In the case of constant velocity, we have

v =

∆x ∆t

where the notation ∆ (Greek uppercase “delta,” like Latin “D”) means “change in,” or “final minus initial.” When the velocity is not con- stant, this equation is false, although the quan- tity ∆x/∆t can be interpreted as a kind of aver- age velocity. Velocity can only be defined if we choose some arbitrary reference point that we consider to be at rest. Therefore velocity is relative, not abso- lute. A person aboard a cruising passenger jet might consider the cabin to be at rest, but some- one on the ground might say that the plane was moving very fast — relative to the dirt. To convert velocities from one frame of ref- erence to another, we add a constant. If, for

example, vAB is the velocity of A relative to B, then

vAC = vAB + vBC. (2.3)

The principle of inertia states that if an object is not acted on by a force, its velocity remains constant. For example, if a rolling soccer ball slows down, the change in its velocity is not be- cause the ball naturally “wants” to slow down but because of a frictional force that the grass exerts on it. A frame of reference in which the principle of inertia holds is called an inertial frame of refer- ence. The earth’s surface defines a very nearly inertial frame of reference, but so does the cabin of a cruising passenger jet. Any frame of refer- ence moving at constant velocity, in a straight line, relative to an inertial frame is also an in- ertial frame. An example of a noninertial frame of reference is a car in an amusement park ride that maneuvers violently.

2.2 Acceleration

The acceleration of a particle is defined as the time derivative of the velocity, or the second derivative of the position with respect to time:

a =

dv dt

d^2 x dt^2

It measures the rate at which the velocity is changing. Its units are m/s/s, more commonly written as m/s^2. Unlike velocity, acceleration is not just a mat- ter of opinion. Observers in different inertial frames of reference agree on accelerations. An acceleration is caused by the force that one ob- ject exerts on another. In the case of constant acceleration, simple al-

14 CHAPTER 2. KINEMATICS IN ONE DIMENSION

gebra and calculus give the following relations:

a =

∆v ∆t

x = x 0 + v 0 t +

at^2 (2.6)

v^2 f = v 02 + 2a∆x, (2.7)

where the subscript 0 (read “nought”) means ini- tial, or t = 0, and f means final.

Free fall

Galileo showed by experiment that when the only force acting on an object is gravity, the ob- ject’s acceleration has a value that is indepen- dent of the object’s mass. This is because the greater force of gravity on a heavier object is exactly compensated for by the object’s greater inertia, meaning its tendency to resist a change in its motion. For example, if you stand up now and drop a coin side by side with your shoe, you should see them hit the ground at almost the same time, despite the huge disparity in mass. The magnitude of the acceleration of falling ob- jects is notated g, and near the earth’s surface g is approximately 9.8 m/s^2. This number is a measure of the strength of the earth’s gravita- tional field.

The inclined plane

If an object slides frictionlessly on a ramp that forms an angle θ with the horizontal, then its acceleration equals g sin θ. This can be shown based on a looser, generalized statement of the principle of inertia, which leads to the conclusion that the gain in speed on a slope depends only on the vertical drop.^1 For θ = 90◦, we recover the case of free fall.

(^1) For details of this argument, see Crowell, Mechanics, lightandmatter.com, section 3.6.

Graphs of position, velocity, and acceler-

ation

The motion of an object can be represented vi- sually by a stack of graphs of x versus t, v versus t, and a versus t. Figure ?? shows two examples. The slope of the tangent line at a given point on one graph equals the value of the function at the same time in the graph below.

16 CHAPTER 2. KINEMATICS IN ONE DIMENSION

Problems

2-a1 You’re standing in a freight train, and have no way to see out. If you have to lean to stay on your feet, what, if anything, does that tell you about the train’s velocity? Explain.

2-a2 Interpret the general rule vAB = −vBA in words.

2-a3 Wa-Chuen slips away from her father at the mall and walks up the down escalator, so that she stays in one place. Write this in terms of symbols, using the notation with two subscripts introduced in section 2.1.

2-a4 Driving along in your car, you take your foot off the gas, and your speedometer shows a reduction in speed. Describe an inertial frame of reference in which your car was speeding up during that same period of time. ? 2-b1 (a) Translate the following information into symbols, using the notation with two sub- scripts introduced in section 2.1. Eowyn is riding on her horse at a velocity of 11 m/s. She twists around in her saddle and fires an arrow back- ward. Her bow fires arrows at 25 m/s. (b) Find the velocity of the arrow relative to the ground.

2-b2 An airport has a moving walkway to help people move across and/or between termi- nals quickly. Suppose that you’re walking north on such a walkway, where the walkway has speed 3 .0 m/s relative to the airport, and you walk at a speed of 2.0 m/s. You pass by your friend, who is off the walkway, traveling south at 1.5 m/s. (a) What is the magnitude of your velocity with respect to the moving walkway?

√

(b) What is the magnitude of your velocity with respect to the airport?

√

(c) What is the magnitude of your velocity with respect to your friend?

√

(d) If it takes you 45 seconds to get across the air- port terminal, how long does it take your friend?√

2-b3 On a 20 km bike ride, you ride the first 10 km at an average speed of 8 km/hour. What average speed must you have over the next 10 km if your average speed for the whole ride is to be 12 km/hour? (^) √

2-b4 (a) In a race, you run the first half of the distance at speed u, and the second half at speed v. Find the over-all speed, i.e., the total distance divided by the total time.

√

(b) Check the units of your equation. (c) Check that your answer makes sense in the case where u = v. (d) Show that the dependence of your result on u and v makes sense. That is, first check whether making u bigger makes the result big- ger, or smaller. Then compare this with what you expect physically.

2-b5 An object starts moving at t = 0, and its position is given by x = At^5 − Bt^2 , where t is in seconds and x is in meters. A is a non-zero constant. (a) Infer the units of A and B. (b) Find the velocity as a function of t.

√

(c) What is the average velocity from 0 to t as a function of time?

√

(d) At what time t (t > 0) is the velocity at t equal to the average velocity from 0 to t? (^) √

2-b6 (a) Let R be the radius of the Earth and T the time (one day) that it takes for one rotation. Find the speed at which a point on the equator moves due to the rotation of the earth.

√

(b) Check the units of your equation. (c) Check that your answer to part a makes sense in the case where the Earth stops rotating completely, so that T is infinitely long. (d) Nairobi, Kenya, is very close to the equator. Plugging in numbers to your answer from part a, find Nairobi’s speed in meters per second. See the table in the back of the book for the relevant data. For comparison, the speed of sound is about 340 m/s.

√

PROBLEMS 17

2-c1 In running races at distances of 800 meters and longer, runners do not have their own lanes, so in order to pass, they have to go around their opponents. Suppose we adopt the simplified geometrical model suggested by the figure, in which the two runners take equal times to trace out the sides of an isoceles triangle, deviating from parallelism by the angle θ. The runner going straight runs at speed v, while the one who is passing must run at a greater speed. Let the difference in speeds be ∆v. (a) Find ∆v in terms of v and θ.

√

(b) Check the units of your equation. (c) Check that your answer makes sense in the special case where θ = 0, i.e., in the case where the runners are on an extremely long straightaway. (d) Suppose that θ = 1.0 degrees, which is about the smallest value that will allow a runner to pass in the distance available on the straight- away of a track, and let v = 7.06 m/s, which is the women’s world record pace at 800 meters. Plug numbers into your equation from part a to determine ∆v, and comment on the result.

√

2-c2 In 1849, Fizeau carried out the first terrestrial measurement of the speed of light; previous measurements by Roemer and Bradley had involved astronomical observation. The fig- ure shows a simplified conceptual representation of Fizeau’s experiment. A ray of light from a bright source was directed through the teeth at the edge of a spinning cogwheel. After traveling a distance L, it was reflected from a mirror and returned along the same path. The figure shows the case in which the ray passes between two teeth, but when it returns, the wheel has rotated by half the spacing of the teeth, so that the ray is blocked. When this condition is achieved, the observer looking through the teeth toward the far-off mirror sees it go completely dark. Fizeau adjusted the speed of the wheel to achieve this condition and recorded the rate of rotation to be f rotations per second. Let the number of teeth on the wheel be n.

Problem 2-c1.

(a) Find the speed of light c in terms of L, n, and f.

√

(b) Check the units of your equation. (Here f ’s units of rotations per second should be taken as inverse seconds, s−^1 , since the number of rotations in a second is a unitless count.) (c) Imagine that you are Fizeau trying to design this experiment. The speed of light is a huge number in ordinary units. Use your equation from part a to determine whether increasing c requires an increase in L, or a decrease. Do the same for n and f. Based on this, decide for each of these variables whether you want a value that is as big as possible, or as small as possible. (d) Fizeau used L = 8633 m, f = 12.6 s−^1 , and n = 720. Plug in to your equation from part

PROBLEMS 19

lower limit; if there really was a hole that deep, the fall would actually take a longer time than the one you calculate, both because there is air friction and because gravity gets weaker as you get deeper (at the center of the earth, g is zero, because the earth is pulling you equally in every direction at once). (^) √

2-f5 You shove a box with initial velocity 2 .0 m/s, and it stops after sliding 1.3 m. What is the magnitude of the deceleration, assuming it is constant? (^) √

2-f6 You’re an astronaut, and you’ve arrived on planet X, which is airless. You drop a hammer from a height of 1.00 m and find that it takes 350 ms to fall to the ground. What is the acceleration due to gravity on planet X? (^) √

2-i1 Mr. Whiskers the cat can jump 2.0 me- ters vertically (undergoing free-fall while in the air). (a) What initial velocity must he have in order to jump this high?

√

(b) How long does it take him to reach his max- imum height from the moment he leaves the ground?

√

(c) From the start of the jump to the time when he lands on the ground again, how long is he in the air? (^) √

2-i2 A baseball pitcher throws a fastball. Her hand accelerates the ball from rest to 45.0 m/s over a distance 1.5 meters. For the purposes of this problem, we will make the simplifying as- sumption that this acceleration is constant. (a) What is the ball’s acceleration?

√

(b) How much time does it take for the pitcher to accelerate the ball?

√

(c) If home plate is 18.0 meters away from where the pitcher releases the baseball, how much total time does the baseball take to get there, assum- ing it moves with constant velocity as soon as it leaves the pitcher’s hand? Include both the time required for acceleration and the time the ball spends on the fly. (^) √

2-i3 The photo shows Apollo 16 astronaut John Young jumping on the moon and saluting at the top of his jump. The video footage of the jump shows him staying aloft for 1.45 seconds. Gravity on the moon is 1/6 as strong as on the earth. Compute the height of the jump. (^) √

Problem 2-i3.

2-i4 Find the error in the following calcula- tion. A student wants to find the distance trav- eled by a car that accelerates from rest for 5.0 s with an acceleration of 2.0 m/s^2. First he solves a = ∆v/∆t for ∆v = 10 m/s. Then he multiplies to find (10 m/s)(5.0 s) = 50 m. Do not just re- calculate the result by a different method; if that was all you did, you’d have no way of knowing which calculation was correct, yours or his.

2-i5 A naughty child drops a golf ball from the roof of your apartment building, and you see it drop past your window. It takes the ball time T to traverse the window’s height H. Find the initial speed of the ball when it first came into view. (^) √

2-i6 Objects A and B, with v(t) graphs shown in the figure, both leave the origin at time t = 0 s. When do they cross paths again? (^) √

20 CHAPTER 2. KINEMATICS IN ONE DIMENSION

Problem 2-i6.

2-i7 An elevator is moving upward at con- stant speed of 2.50 m/s. A bolt in the elevator’s ceiling, 3.00 m above the floor, works loose and falls. (a) How long does it take for the bolt to fall to the floor?

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(b) What is the speed of the bolt just as it hits the floor, according to an observer in the elevator?

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(c) What is the speed of the bolt just as it hits the elevator’s floor, according to an observer standing on one of the floor landings of the building?

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2-i8 You’re in your Honda, cruising on the freeway at velocity u, when, up ahead at dis- tance L, a Ford pickup truck cuts in front of you while moving at constant velocity u/2. Like half the speed that any reasonable person would go! Let positive x be the direction of motion, and let your position be x = 0 at t = 0. To avoid a col- lision, you immediately slam on the brakes and start decelerating with acceleration −a, where a is a positive constant. (a) Write an equation for xF(t), the position of the Ford as a function of time, as they trundle onward obliviously.

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(b) Write an equation for xH(t), the position of your Honda as a function of time.

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(c) By subtracting one from the other, find an expression for the distance between the two ve- hicles as a function of time, d(t).

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(d) Find the minimum value of a that avoids a

collision.

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(e) Show that your answer to part e has units that make sense. (f) Show that the dependence of your answer on the variables makes sense physically.

2-i9 You’re in your Honda on the freeway traveling behind a Ford pickup truck. The truck is moving at a steady speed of 30.0 m/s, you’re speeding at 40.0 m/s, and you’re cruising 45 me- ters behind the Ford. At t = 0, the Ford slams on his/her brakes, and decelerates at a rate of 5 .0 m/s^2. You don’t notice this until t = 1.0 s, where you begin decelerating at 10.0 m/s^2. Let positive x be the direction of motion, and let your position be x = 0 at t = 0. The goal is to find the motion of each vehicle and determine whether there is a collision. (a) Doing this entire calculation purely numer- ically would be very cumbersome, and it would be difficult to tell whether you had made mis- takes. Translate the given information into alge- bra symbols, and find an equation for xF(t), the position of the Ford as a function of time.

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(b) Write a similar symbolic equation for xH(t) (for t > 1 s), the position of the Honda as a function of time. Why isn’t this formula valid for t < 1 s?

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(c) By subtracting one from the other, find an expression for the distance between the two ve- hicles as a function of time, d(t) (valid for t > 1 s until the truck stops). Does the equation d(t) = 0 have any solutions? What does this tell you?

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(d) Because this is a fairly complicated calcula- tion, we will find the result in two different ways and check them against each other. Plug num- bers back in to the results of parts a and b, re- placing the symbols in the constant coefficients, and graph the two functions using a graphing cal- culator or an online utility such as desmos.com. (e) As you should have discovered in parts c and d, the two vehicles do not collide. At what time does the minimum distance occur, and what is that distance? (^) √