Essential University Physics

Richard Wolfson

Chapter 9 Systems of Particles - all with Video Answers

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Chapter Questions

00:56

Problem 1

What do Newton's apple and the Moon have in common?

Donald Albin

Numerade Educator

01:22

Problem 1

Explain why a high jumper's center of mass need not clear the bar.

Katie Mcalpine

Numerade Educator

00:55

Problem 2

Explain the difference between $G$ and $g$

Donald Albin

Numerade Educator

02:07

The center of mass of a solid sphere is clearly at its center. If the sphere is cut in half and the two halves are stacked as in Fig. 9.23 , is the center of mass at the point where they touch? If not, roughly where is it? Explain.

Mukesh Devi

Numerade Educator

00:00

Problem 3

When you stand on Earth, the distance between you and Earth is zero. So why isn't the gravitational force infinite?

Donald Albin

Numerade Educator

01:18

Problem 3

The momentum of a system of pool balls is the same before and after they are hit by the cue ball. Is it still the same after one of the balls strikes the edge of the table? Explain.

Katie Mcalpine

Numerade Educator

00:52

Problem 4

The force of gravity on an object is proportional to the object's mass, yet all objects fall with the same gravitational acceleration. Why?

Donald Albin

Numerade Educator

01:42

Problem 4

Is it possible to have an inelastic collision in which $a l l$ the kinetic energy of the colliding objects is lost? If so, give an example. If not, why not?

Prabhu Ramji

Numerade Educator

01:18

Problem 5

A friend who knows nothing about physics asks what keeps an orbiting satellite from falling to Earth. Give an answer that will satisfy your friend.

Donald Albin

Numerade Educator

00:46

Problem 5

If you want to stop the neutrons in a reactor, why not use massive nuclei like lead?

Katie Mcalpine

Numerade Educator

00:59

Problem 6

Could you put a satellite in an orbit that keeps it stationary over the south pole? Explain.

Donald Albin

Numerade Educator

01:05

Problem 6

Why don't we need to consider external forces acting on a system as its constituent particles undergo a collision?

Katie Mcalpine

Numerade Educator

02:03

Problem 7

Why are satellites generally launched eastward and from low latitudes? (Hint: Think about Earth's rotation.)

Donald Albin

Numerade Educator

00:59

Problem 7

How is it possible to have a collision between objects that don't ever touch? Give an example of such a collision.

Katie Mcalpine

Numerade Educator

02:06

Problem 8

Given Earth's mass, the Moon's distance and orbital period, and the value of $G,$ could you calculate the Moon's mass? If yes, how? If no, why not?

Donald Albin

Numerade Educator

01:48

Problem 8

A pitched baseball moves no faster than the pitcher's hand. But a batted ball can move much faster than the bat. What's the difference?

Katie Mcalpine

Numerade Educator

02:41

Problem 9

How should a satellite be launched so that its orbit takes it over
every point on the (rotating) Earth?

Donald Albin

Numerade Educator

01:38

Problem 9

Two identical satellites are going in opposite directions in the same circular orbit when they collide head-on. Describe their subsequent motion if the collision is (a) elastic or (b) totally inelastic.

Katie Mcalpine

Numerade Educator

01:49

Problem 10

Does the gravitational force of the Sun do work on a planet in a circular orbit? In an elliptical orbit? Explain.

Donald Albin

Numerade Educator

03:59

Problem 10

A 28 -kg child sits at one end of a 3.5 -m-long seesaw. Where should her $65-\mathrm{kg}$ father sit so the center of mass will be at the center of the seesaw?

Katie Mcalpine

Numerade Educator

02:02

Problem 11

Space explorers land on a planet with the same mass as Earth, but find they weigh twice as much as they would on Earth. What's the planet's radius?

Donald Albin

Numerade Educator

02:46

Problem 11

Two particles of equal mass $m$ are at the vertices of the base of an equilateral triangle. The triangle's center of mass is midway between the base and the third vertex. What's the mass at the third vertex?

Katie Mcalpine

Numerade Educator

04:12

Problem 12

Use data for the Moon's orbit from Appendix E to compute the Moon's acceleration in its circular orbit, and verify that the result is consistent with Newton's law of gravitation.

Donald Albin

Numerade Educator

04:01

Problem 12

Rework Example 9.1 with the origin at the center of the barbell, showing that the physical location of the center of mass doesn't depend on your coordinate system.

Katie Mcalpine

Numerade Educator

02:35

Problem 13

To what fraction of its current radius would Earth have to shrink (with no change in mass) for the gravitational acceleration at its surface to triple?

Donald Albin

Numerade Educator

04:48

Problem 14

Calculate the gravitational acceleration at the surface of (a) Mercury and (b) Saturn's moon Titan.

Donald Albin

Numerade Educator

01:37

Problem 14

How far from Earth's center is the center of mass of the Earth- Moon system? (Hint: Consult Appendix E.)

Prabhu Ramji

Numerade Educator

02:44

Problem 15

Two identical lead spheres with their centers $14 \mathrm{~cm}$ apart attract each other with a $0.25-\mu \mathrm{N}$ force. Find their mass.

Donald Albin

Numerade Educator

03:24

Problem 15

A popcorn kernel at rest in a hot pan bursts into two pieces, with masses $91 \mathrm{mg}$ and $64 \mathrm{mg}$. The more massive piece moves horizontally at $47 \mathrm{~cm} / \mathrm{s} .$ Describe the motion of the second piece.

Katie Mcalpine

Numerade Educator

04:46

Problem 16

A $60-\mathrm{kg}$ skater, at rest on frictionless ice, tosses a $12-\mathrm{kg}$ snowball with velocity $\vec{v}=53.0 \hat{\imath}+14.0 \hat{\jmath} \mathrm{m} / \mathrm{s},$ where the $x$ - and $y$ -axes are in the horizontal plane. Find the skater's subsequent velocity.

Katie Mcalpine

Numerade Educator

10:09

Problem 17

A plutonium- 239 nucleus at rest decays into a uranium- 235 nucleus by emitting an alpha particle $\left({ }^{4} \mathrm{He}\right)$ with kinetic energy 5.15 $\mathrm{MeV}$. Find the speed of the uranium nucleus.

Katie Mcalpine

Numerade Educator

02:43

Problem 18

A toboggan of mass $8.6 \mathrm{~kg}$ is moving horizontally at $23 \mathrm{~km} / \mathrm{h}$. As it passes under a tree, $15 \mathrm{~kg}$ of snow drop onto it. Find its subsequent speed.

Katie Mcalpine

Numerade Educator

02:44

Problem 19

At the peak of its trajectory, a 995 -g fireworks rocket is moving horizontally at $18.6 \mathrm{~m} / \mathrm{s}$. It's a dud, and instead of exploding gloriously, it bursts into two pieces. One of them, with mass $372 \mathrm{~g}$, continues in the original direction at $31.3 \mathrm{~m} / \mathrm{s}$. How much energy did the two pieces gain when the rocket burst?

Nicholas Mogoi

Numerade Educator

11:39

Problem 20

An object with kinetic energy $K$ explodes into two pieces, each of which moves with twice the speed of the original object. Find the ratio of the internal kinetic energy to the center-of-mass energy after the explosion.

Katie Mcalpine

Numerade Educator

01:38

Problem 21

The graph shown with the Application: Crash Tests on page 161 shows the force exerted on a $2000-\mathrm{kg}$ test car as it crashes into a stationary barrier and comes to rest. Take the horizontal axis to extend from 0 to $800 \mathrm{~ms}$ and the vertical axis from 0 to $100 \mathrm{kN}$. Estimate
(a) the impulse imparted to the car and (b) its initial speed.

Prabhu Ramji

Numerade Educator

03:22

Problem 22

High-speed photos of a $220-\mu \mathrm{g}$ flea jumping vertically show that the jump lasts $1.2 \mathrm{~ms}$ and involves an average vertical acceleration of $100 g .$ What (a) average force and (b) impulse does the ground exert on the flea during its jump? (c) What's the change in the flea's momentum during its jump?

Katie Mcalpine

Numerade Educator

01:30

Problem 23

You're working in mission control for an interplanetary space probe. A trajectory correction calls for a rocket firing that imparts an impulse of $5.64 \mathrm{~N} \cdot \mathrm{s}$. If the rocket's average thrust is $135 \mathrm{mN}$, how long should the rocket fire?

Katie Mcalpine

Numerade Educator

06:33

Problem 24

In a railroad switchyard, a 56 -ton freight car is sent at $7.0 \mathrm{mi} / \mathrm{h}$ toward a 31 -ton car moving in the same direction at $2.6 \mathrm{mi} / \mathrm{h}$. (a) What's the speed of the cars after they couple? (b) What fraction of the initial kinetic energy was lost in the collision?

Katie Mcalpine

Numerade Educator

02:24

Problem 25

In a totally inelastic collision between two equal masses, with one initially at rest, show that half the initial kinetic energy is lost.

Katie Mcalpine

Numerade Educator

04:23

Problem 26

A neutron (mass 1.01 u) strikes a deuteron (mass 2.01 u), and they combine to form a tritium nucleus (mass $3.02 \mathrm{u}$ ). If the neutron's initial velocity was $23.5 \hat{\imath}+14.4 \hat{\jmath} \mathrm{Mm} / \mathrm{s}$ and if the tritium leaves the reaction with velocity $15.1 \hat{\imath}+22.6 \hat{\jmath} \mathrm{Mm} / \mathrm{s},$ what was the deuteron's velocity?

Katie Mcalpine

Numerade Educator

02:17

Problem 27

Two identical trucks have mass $5500 \mathrm{~kg}$ when empty, and the maximum permissible load for each is $8000 \mathrm{~kg}$. The first truck, carrying $3800 \mathrm{~kg},$ is at rest. The second truck plows into it at $65 \mathrm{~km} / \mathrm{h},$ and the pair moves away at $37 \mathrm{~km} / \mathrm{h} .$ As an expert witness, you're asked to determine whether the second truck was overloaded. What do you report?

Prabhu Ramji

Numerade Educator

06:37

Problem 28

An alpha particle $\left({ }^{4} \mathrm{He}\right)$ strikes a stationary gold nucleus $\left({ }^{197} \mathrm{Au}\right)$ head-on. What fraction of the alpha's kinetic energy is transferred to the gold? Assume a totally elastic collision.

Katie Mcalpine

Numerade Educator

05:22

Problem 29

Playing in the street, a child accidentally tosses a ball at $18 \mathrm{~m} / \mathrm{s}$ toward the front of a car moving toward him at $14 \mathrm{~m} / \mathrm{s}$. What's the ball's speed after it rebounds elastically from the car?

Katie Mcalpine

Numerade Educator

02:44

Problem 30

A block of mass $m$ undergoes a one-dimensional elastic collision with a block of mass $M$ initially at rest. If both blocks have the same speed after colliding, how are their masses related?

Katie Mcalpine

Numerade Educator

02:41

Problem 31

A proton moving at $6.9 \mathrm{Mm} / \mathrm{s}$ collides elastically head-on with a second proton moving in the opposite direction at $11 \mathrm{Mm} / \mathrm{s}$. Find their subsequent velocities.

Katie Mcalpine

Numerade Educator

02:26

Problem 32

A head-on, elastic collision between two particles with equal initial speed $v$ leaves the more massive particle $\left(m_{1}\right)$ at rest. Find (a) the ratio of the particle masses and (b) the final speed of the less massive particle.

Katie Mcalpine

Numerade Educator

04:20

Problem 33

A lithium- 5 nucleus $\left({ }^{5} \mathrm{Li}\right)$ is moving at $2.25 \mathrm{Mm} / \mathrm{s}$ when it decays into a proton $\left({ }^{1} \mathrm{H}\right)$ and an alpha particle $\left({ }^{4} \mathrm{He}\right)$. The alpha particle is detected moving at $1.03 \mathrm{Mm} / \mathrm{s}$ at $23.6^{\circ}$ to the original velocity of the ${ }^{5} \mathrm{Li}$ nucleus. Find the magnitude and direction of the proton's velocity.

Prabhu Ramji

Numerade Educator

02:38

Problem 34

A lithium-5 nucleus ( ${ }^{5} \mathrm{Li}$ ) decays into a proton $\left({ }^{1} \mathrm{H}\right)$ and an alpha particle $\left({ }^{4} \mathrm{He}\right) .$ The alpha particle is detected moving at $2.43 \mathrm{Mm} / \mathrm{s}$ at $31.5^{\circ}$ above the $x$ -axis (i.e., with a positive $y$ -component), while the proton is moving at $1.78 \mathrm{Mm} / \mathrm{s}$ at $24.7^{\circ}$ below the $x$ -axis. Find the original velocity of the lithium-5 nucleus, expressed in unit vector notation.

Prabhu Ramji

Numerade Educator

04:12

Problem 35

A spacecraft consists of a 549 -kgorbiteranda 235 -kg lander. It's moving at $81.6 \mathrm{~km} / \mathrm{s}$ relative to a nearby space station. Explosive bolts separate the orbiter and lander, after which the orbiter is moving at $55.2 \mathrm{~km} / \mathrm{s}$ at a $41.4^{\circ}$ angle to the motion of the original composite spacecraft. Find the magnitude and direction of the lander's velocity.

Prabhu Ramji

Numerade Educator

02:05

Problem 36

Aspacecraftconsistsofa784-kgorbiteranda392-kg lander. Explosive bolts separate the orbiter and lander, after which the orbiter's velocity is $225 \hat{i}+107 \hat{j} \mathrm{~m} / \mathrm{s}$ and the lander's is $-75.4 \hat{i}-214 \hat{j} \mathrm{~m} / \mathrm{s}$. Find the velocity of the composite spacecraft before the separation.

Prabhu Ramji

Numerade Educator

03:10

Problem 37

Some nuclear reactors, especially in England and Russia, use graphite (pure carbon and nearly all ${ }^{12} \mathrm{C}$ ) for the moderator. When a neutron hits a stationary ${ }^{12} \mathrm{C}$ nucleus in a head-on elastic collision, what percentage of its kinetic energy is transferred to the carbon?

Prabhu Ramji

Numerade Educator

08:18

Problem 38

A neutron undergoes an elastic head-on collision with an initially stationary nucleus, and $48.4 \%$ of the neutron's kinetic energy is transferred to the struck nucleus. How does the mass of the nucleus compare with that of the neutron?

Linda Winkler

Numerade Educator

07:20

Problem 39

$\mathrm{A} 685-\mathrm{g}$ block is sliding on a frictionless surface when it collides elastically and head-on with a stationary block of mass $232 \mathrm{~g}$. What percentage of the more massive block's kinetic energy is transferred to the lighter block?

Linda Winkler

Numerade Educator

04:44

Problem 40

A mass $m_{1}$ collides elastically and head-on with a stationary mass $m_{2},$ and three-fourths of $m_{1}$ 's initial kinetic energy is transferred to $m_{2}$. How are the two masses related?

James Kiss

Numerade Educator

12:16

Problem 41

Find the center of mass of a pentagon with five equal sides of length $a,$ but with one triangle missing (Fig. 9.24). (Hint: See Example $9.3,$ and treat the pentagon as a group of triangles.)

Katie Mcalpine

Numerade Educator

04:36

Problem 42

Wildlife biologists fire 20 -g rubber bullets to stop a rhinoceros charging at $0.81 \mathrm{~m} / \mathrm{s}$. The bullets strike the rhino and drop vertically to the ground. The biologists' gun fires 15 bullets each second, at $73 \mathrm{~m} / \mathrm{s},$ and it takes $34 \mathrm{~s}$ to stop the rhino.
(a) What impulse does each bullet deliver? (b) What's the rhino's mass? Neglect forces between rhino and ground.

Supratim Pal

Numerade Educator

04:51

Problem 43

Three $100-\mathrm{g}$ objects have velocities given by $\vec{v}_{1}=25.0 \hat{i} \mathrm{~m} / \mathrm{s}$ $\vec{v}_{2}=-9.45 \hat{i}+11.6 \hat{j} \mathrm{~m} / \mathrm{s},$ and $\vec{v}_{3}=-3.67 \hat{i}-11.6 \hat{j} \mathrm{~m} / \mathrm{s}$ Find the center of mass and internal kinetic energies of this system.

Linda Winkler

Numerade Educator

07:06

Problem 44

You're with 19 other people on a boat at rest in frictionless water. The group's total mass is $1500 \mathrm{~kg}$, and the boat's mass is $12,000 \mathrm{~kg}$. The entire party walks the 6.5 -m distance from bow to stern. How far does the boat move?

Katie Mcalpine

Numerade Educator

04:01

Problem 45

A hemispherical bowl is at rest on a frictionless counter. A mouse drops onto the bowl's rim from a cabinet directly overhead. The mouse climbs down inside the bowl to eat crumbs at the bottom. If the bowl moves along the counter a distance equal to one-tenth of its diameter, how does the mouse's mass compare with the bowl's mass?

Bret Rosen

Numerade Educator

07:04

Problem 46

Physicians perform needle biopsies to sample tissue from internal organs. A spring-loaded gun shoots a hollow needle into the tissue; extracting the needle brings out the tissue core. A particular device uses 8.3 -mg needles that take $90 \mathrm{~ms}$ to stop in the tissue, which exerts a stopping force of $41 \mathrm{mN}$.
(a) Find the impulse imparted by the tissue.
(b) How far into the tissue does the needle penetrate?

Katie Mcalpine

Numerade Educator

15:04

Problem 47

Find the center of mass of the uniform, solid cone of height $h$, base radius $R,$ and constant density $\rho$ shown in Fig. 9.25. (Hint: Integrate over disk-shaped mass elements of thickness $d y,$ as shown in the figure.)

Guilherme Barros

Numerade Educator

03:08

Problem 48

A firecracker, initially at rest, explodes into two fragments. The first, of mass $14 \mathrm{~g},$ moves in the $+x$ -direction at $48 \mathrm{~m} / \mathrm{s}$. The second moves at $32 \mathrm{~m} / \mathrm{s} .$ Find the second fragment's mass and the direction of its motion.

Katie Mcalpine

Numerade Educator

07:58

Problem 49

An $11,000-\mathrm{kg}$ freight car rests against a spring bumper at the end of a railroad track. The spring has constant $k=0.32 \mathrm{MN} / \mathrm{m} .$ The car is hit by a second car of $9400-\mathrm{kg}$ mass moving at $8.5 \mathrm{~m} / \mathrm{s},$ and the two couple together. Find (a) the maximum compression of the spring and (b) the speed of the two cars when they rebound together from the spring.

Katie Mcalpine

Numerade Educator

04:07

Problem 50

On an icy road, a 1200-kg car moving at $50 \mathrm{~km} / \mathrm{h}$ strikes a $4400-$ kg truck moving in the same direction at $35 \mathrm{~km} / \mathrm{h}$. The pair is soon hit from behind by a 1500 -kg car speeding at $65 \mathrm{~km} / \mathrm{h},$ and all three vehicles stick together. Find the speed of the wreckage.

Katie Mcalpine

Numerade Educator

04:42

Problem 51

Kids are pelting a window with snowballs. On average, two snowballs of roughly $300-\mathrm{g}$ mass hit the window each second, moving horizontally at some $10 \mathrm{~m} / \mathrm{s}$. The snowballs drop vertically to the ground after hitting the window. Estimate the average force exerted on the window.

Linda Winkler

Numerade Educator

06:57

Problem 52

A $1250-\mathrm{kg}$ car is moving with velocity $\vec{v}_{1}=36.2 \hat{\imath}+12.7 \hat{\jmath} \mathrm{m} / \mathrm{s}$ It skids on a frictionless icy patch and collides with a 448 -kg hay wagon with velocity $\vec{v}_{2}=13.8 \hat{\imath}+10.2 \hat{\jmath} \mathrm{m} / \mathrm{s} .$ If the two stay together, what's their velocity?

Katie Mcalpine

Numerade Educator

03:49

Problem 53

Masses $m$ and $3 m$ approach at the same speed $v$ and undergo a head-on elastic collision. Show that mass $3 m$ stops, while mass $m$ rebounds at speed $2 v$

Katie Mcalpine

Numerade Educator

07:27

Problem 54

A ${ }^{238} \mathrm{U}$ nucleus is moving in the $x$ -direction at $5.0 \times 10^{5} \mathrm{~m} / \mathrm{s}$ when it decays into an alpha particle $\left({ }^{4} \mathrm{He}\right)$ and a ${ }^{234} \mathrm{Th}$ nucleus. The alpha moves at $1.4 \times 10^{7} \mathrm{~m} / \mathrm{s}$ at $22^{\circ}$ above the $x$ -axis. Find the recoil velocity of the thorium.

Katie Mcalpine

Numerade Educator

11:18

Problem 55

Find an expression for the center of mass of a solid hemisphere, given as the distance from the center of the flat part of the hemisphere.

Linda Winkler

Numerade Educator

04:38

Problem 56

A 42 -g firecracker is at rest at the origin when it explodes into three pieces. The first, with mass $12 \mathrm{~g}$, moves along the $x$ -axis at $35 \mathrm{~m} / \mathrm{s}$. The second, with mass $21 \mathrm{~g}$, moves along the $y$ -axis at 29 $\mathrm{m} / \mathrm{s} .$ Find the velocity of the third piece.

Katie Mcalpine

Numerade Educator

08:14

Problem 57

A $60-\mathrm{kg}$ astronaut floating in space simultaneously tosses away a 14-kg oxygen tank and a 5.8-kg camera. The tank moves in the $x$ -direction at $1.6 \mathrm{~m} / \mathrm{s},$ and the astronaut recoils at $0.85 \mathrm{~m} / \mathrm{s}$ in a direction $200^{\circ}$ counterclockwise from the $x$ -axis. Find the camera's velocity.

Katie Mcalpine

Numerade Educator

03:37

Problem 58

Assuming equal-mass pieces in Exercise $20,$ find the angles of the two velocities relative to the direction of motion before the explosion.

Guilherme Barros

Numerade Educator

05:33

Problem 59

A 62-kg sprinter stands on the left end of a 190 -kg cart moving leftward at $7.1 \mathrm{~m} / \mathrm{s}$. She runs to the right end and continues horizontally off the cart. What should be her speed relative to the cart so that once she's off the cart, she has no horizontal velocity relative to the ground?

Katie Mcalpine

Numerade Educator

03:57

Problem 60

You're a production engineer in a cookie factory, where mounds of dough drop vertically onto a conveyor belt at the rate of one 12-g mound every 2 s. You're asked to design a mechanism that will keep the conveyor belt moving at a constant $50 \mathrm{~cm} / \mathrm{s}$. What average force must the mechanism exert on the belt?

Linda Winkler

Numerade Educator

09:01

Problem 61

Mass $m,$ moving at speed $2 v,$ approaches mass $4 m,$ moving at speed $v$. The two collide elastically head-on. Find expressions for their subsequent speeds.

Katie Mcalpine

Numerade Educator

03:34

Problem 62

Verify explicitly that kinetic energy is conserved in the collision of the preceding problem.

Katie Mcalpine

Numerade Educator

22:22

Problem 63

While standing on frictionless ice, you (mass $65.0 \mathrm{~kg}$ ) toss a $4.50-\mathrm{kg}$ rock with initial speed $12.0 \mathrm{~m} / \mathrm{s} .$ If the rock is $15.2 \mathrm{~m}$ from you when it lands, (a) at what angle did you toss it? (b) How fast are you moving?

Katie Mcalpine

Numerade Educator

04:39

Problem 64

You're an accident investigator at a scene where a drunk driver in a $1600-\mathrm{kg}$ car has plowed into a $1300-\mathrm{kg}$ parked car with its brake set. You measure skid marks showing that the combined wreckage moved $25 \mathrm{~m}$ before stopping, and you determine a frictional coefficient of 0.77 . What do you report for the drunk driver's speed just before the collision?

Katie Mcalpine

Numerade Educator

10:18

Problem 65

A fireworks rocket is launched vertically upward at $40 \mathrm{~m} / \mathrm{s}$. At the peak of its trajectory, it explodes into two equal-mass fragments. One reaches the ground $2.87 \mathrm{~s}$ after the explosion. When does the second reach the ground?

Katie Mcalpine

Numerade Educator

06:29

Problem 66

Two objects moving in opposite directions with the same speed $v$ undergo a totally inelastic collision, and half the initial kinetic energy is lost. Find the ratio of their masses.

Katie Mcalpine

Numerade Educator

03:43

Problem 67

Explosive bolts separate a $950-\mathrm{kg}$ communications satellite from its 640 -kg booster rocket, imparting a $350-\mathrm{N} \cdot \mathrm{s}$ impulse. At what relative speed do satellite and booster separate?

Linda Winkler

Numerade Educator

03:55

Problem 68

You're working in quality control for a model rocket manufacturer, testing a class-D rocket whose specifications call for an impulse between 10 and $20 \mathrm{~N} \cdot \mathrm{s}$. The rocket's burn time is $\Delta t=2.8 \mathrm{~s},$ and its thrust during that time is $F(t)=a t(t-\Delta t)$ where $a=-4.6 \mathrm{~N} / \mathrm{s}^{2} .$ Does the rocket meet its specs?

Linda Winkler

Numerade Educator

09:55

Problem 69

You're investigating a crash in which a 1640 -kg Nissan Leaf electric car and a 3220 -kg Toyota Land Cruiser SUV collided at right angles in an intersection. The combined wreckage skidded $17.6 \mathrm{~m}$ before stopping. You measure the coefficient of friction between tires and road and find it to be $0.697 .$ Show that at least one car must have exceeded the $70-\mathrm{km} / \mathrm{h}$ speed limit at the intersection. You'll need to consider each car separately, assuming that it was at the speed limit and finding the other car's speed, which you should report in your answer.

Linda Winkler

Numerade Educator

03:13

Problem 70

A 400 -mg popcorn kernel is skittering across a nonstick frying pan at $8.2 \mathrm{~cm} / \mathrm{s}$ when it pops and breaks into two equal-mass pieces. If one piece ends up at rest, how much energy was released in the popping?

Supratim Pal

Numerade Educator

04:03

Problem 71

Two identical objects with the same initial speed collide and stick together. If the composite object moves with half the initial speed of either object, what was the angle between the initial velocities?

Guilherme Barros

Numerade Educator

13:31

Problem 72

A proton (mass 1 u) moving at $6.90 \mathrm{Mm} / \mathrm{s}$ collides elastically head-on with a second particle moving in the opposite direction at $2.80 \mathrm{Mm} / \mathrm{s}$. After the collision, the proton is moving opposite its initial direction at $8.62 \mathrm{Mm} / \mathrm{s}$. Find the mass and final velocity of the second particle.

Katie Mcalpine

Numerade Educator

10:40

Problem 73

Two objects, one initially at rest, undergo a one-dimensional elastic collision. If half the kinetic energy of the initially moving object is transferred to the other object, what is the ratio of their masses?

Katie Mcalpine

Numerade Educator

09:53

Problem 74

Blocks $B$ and $C$ have masses $2 m$ and $m,$ respectively, and are at rest on a frictionless surface. Block $A,$ also of mass $m,$ is heading at speed $v$ toward block $B$ as shown in Fig. 9.26 Determine the final velocity of each block after all subsequent collisions are over. Assume all collisions are elastic.

Guilherme Barros

Numerade Educator

04:50

Problem 75

Derive Equation $9.15 \mathrm{~b}$.

Guilherme Barros

Numerade Educator

04:34

Problem 76

An object collides elastically with an equal-mass object initially at rest. If the collision isn't head-on, show that the final velocity vectors are perpendicular.

Guilherme Barros

Numerade Educator

02:09

Problem 77

A proton (mass 1 u) collides elastically with a stationary deuteron (mass $2 \mathrm{u}$ ). If the proton is deflected $37^{\circ}$ from its original direction, what fraction of its kinetic energy does it transfer to the deuteron?

Penny Riley

Numerade Educator

08:21

Problem 78

Two identical billiard balls are initially at rest when they're struck symmetrically by a third identical ball moving with velocity $\vec{v}_{0}=v_{0} \hat{\imath}$ (Fig. 9.27 ). Find the velocities of all three balls after this elastic collision.

Sanat Mukherjee

Numerade Educator

04:50

Problem 79

A 114-g Frisbee is lodged on a tree branch $7.65 \mathrm{~m}$ above the ground. To free it, you lob a $240-\mathrm{g}$ dirt clod vertically upward. The dirt leaves your hand at a point $1.23 \mathrm{~m}$ above the ground, moving at $17.7 \mathrm{~m} / \mathrm{s}$. It sticks to the Frisbee. Find (a) the maximum height reached by the Frisbee-dirt combination and (b) the speed with which the combination hits the ground.

Ajay Singhal

Numerade Educator

08:11

Problem 80

You set a small ball of mass $m$ atop a large ball of mass $M \gg m$ and drop the pair from height $h .$ Assuming the balls are perfectly elastic, show that the smaller ball rebounds to height $9 h$.

Katie Mcalpine

Numerade Educator

08:37

Problem 81

A car moving at speed $v$ undergoes a one-dimensional collision with an identical car initially at rest. The collision is neither elastic nor fully inelastic; $5 / 18$ of the initial kinetic energy is lost. Find the velocities of the two cars after the collision.

Guilherme Barros

Numerade Educator

16:47

Problem 82

A $200-\mathrm{g}$ block is released from rest at a height of $25 \mathrm{~cm}$ on a frictionless $30^{\circ}$ incline. It slides down the incline and then along a frictionless surface until it collides elastically with an $800-\mathrm{g}$ block at rest $1.4 \mathrm{~m}$ from the bottom of the incline (Fig. 9.28 ). How much later do the two blocks collide again?

Guilherme Barros

Numerade Educator

07:35

Problem 83

A rocket of mass $M$ moving at speed $v$ ejects an infinitesimal mass $d m$ out its exhaust nozzle at speed $v_{e x}$. (a) Show that conservation of momentum implies that $M d v=v_{\text {ex }} d m,$ where $d v$ is the change in the rocket's speed. (b) Integrate this equation from some initial speed $v_{\mathrm{i}}$ and mass $M_{\mathrm{i}}$ to a final speed $v_{\mathrm{f}}$ and mass $M_{\mathrm{f}}$ to show that the rocket's final velocity is given by the expression $v_{\mathrm{f}}=v_{\mathrm{i}}+v_{\mathrm{ex}} \ln \left(M_{\mathrm{i}} / M_{\mathrm{f}}\right)$

Linda Winkler

Numerade Educator

08:55

Problem 84

A block of mass $m_{1}$ undergoes a one-dimensional elastic collision with an initially stationary block of mass $m_{2}$. Find an expression for the fraction of the initial kinetic energy transferred to the second block, and plot your result for mass ratios $m_{1} / m_{2}$ from 0 to 20 .

Guilherme Barros

Numerade Educator

08:27

Problem 85

Two objects of unequal mass, one initially at rest, undergo a one-dimensional elastic collision. For a given mass ratio, show that the fraction of the initial energy transferred to the initially stationary object doesn't depend on which object it is.

Linda Winkler

Numerade Educator

07:26

Problem 86

In Figure $9.6,$ the uniform semicircular wire has radius $R .$ How far above the center of the semicircle is its center of mass?

Katie Mcalpine

Numerade Educator

11:18

Problem 87

Find the center of mass of a uniform slice of pizza with radius $R$ and angular width $\theta$.

Katie Mcalpine

Numerade Educator

08:52

Problem 88

In a ballistic pendulum demonstration gone bad, a $0.52-\mathrm{g}$ pellet, fired horizontally with kinetic energy $3.25 \mathrm{~J}$, passes straight through a 400 -g Styrofoam pendulum block. If the pendulum rises a maximum height of $0.50 \mathrm{~mm}$, how much kinetic energy did the pellet have after emerging from the Styrofoam?

Katie Mcalpine

Numerade Educator

02:38

Problem 89

An $80-\mathrm{kg}$ astronaut has become detached from the safety line connecting her to the International Space Station. She's $200 \mathrm{~m}$ from the station, at rest relative to it, and has 4 min of air remaining. To get herself back, she tosses a $10-\mathrm{kg}$ tool kit away from the station at $8.0 \mathrm{~m} / \mathrm{s}$. Will she make it back in time?

Katie Mcalpine

Numerade Educator

03:34

Problem 90

Astronomers detect extrasolar planets by measuring the slight movement of stars around the center of mass of the star-planet system. Considering just the Sun and Jupiter, determine the radius of the circular orbit the Sun makes about the Sun-Jupiter center of mass.

Guilherme Barros

Numerade Educator

07:02

Problem 91

A thin rod extends from $x=0$ to $x=L$. It carries a nonuniform mass per unit length $\mu=M x^{a} / L^{1+a},$ where $M$ is a constant with units of mass, and $a$ is a non-negative dimensionless constant. Find expressions for (a) the rod's mass and (b) the location of its center of mass.(c) Are your results what you expect when $a=0 ?$

Katie Mcalpine

Numerade Educator

07:57

Problem 92

Model rocket motors are specified by giving the impulse they provide, in $\mathrm{N} \cdot \mathrm{s},$ over the entire time the rocket is firing. The table below shows the results of rocket-motor tests with differ-
ent motors used to launch rockets of different masses. Determine
two data-based quantities that, when plotted against each other, should give a straight line and whose slope should allow you to determine $g .$ Plot the data, establish a best-fit line, and determine
g. Assume that the maximum height is much greater than the distance over which the rocket motor is firing, so you can neglect the latter. You're also neglecting air resistance-but explain how that affects your experimentally determined value for $g$.
$$\begin{array}{|l|r|r|r|r|r|}\hline \text { Impulse, } J(\mathrm{~N} \cdot \mathrm{s}) & 4.5 & 7.8 & 4.5 & 7.8 & 11 \\\hline \text { Rocket mass }(\mathrm{g}) \text { (including motor) } & 180 & 485 & 234 & 234 & 485 \\\hline \text { Maximum height achieved }(\mathrm{m}) & 22 & 13 & 19 & 51 & 23 \\\hline\end{array}$$

Linda Winkler

Numerade Educator

16:12

Problem 93

A block of mass $M$ is moving at speed $v_{0}$ on a frictionless surface that ends in a rigid wall, heading toward a stationary block of mass $n M,$ where $n \geq 1$ (Fig. 9.29). Collisions between the two blocks or the left-hand block and the wall are elas- tic and one-dimensional. (a) Show that the blocks will undergo only one collision with each other if $n \leq 3$. (b) Show that the blocks will undergo two collisions with each other if $n=4 .$ (c) How many collisions will the blocks undergo if $n=10,$ and what will be their final speeds?

Luis Garcia

Numerade Educator

01:58

Problem 94

The collisions between ball and floor are
a. totally elastic.
b. totally inelastic.
c. neither totally elastic nor totally inelastic.

Linda Winkler

Numerade Educator

01:40

Problem 95

The fraction of the ball's mechanical energy that's lost in the second collision is
a. about $10 \%$.
b. a little less than half.

Linda Winkler

Numerade Educator

03:08

Problem 96

The component of the ball's velocity whose magnitude is most affected by the collisions is
a. horizontal.
b. vertical.
c. Both are affected equally.

Linda Winkler

Numerade Educator

02:31

Problem 97

Compared with the time between bounces, the duration of each collision is
a. a tiny fraction of the time between bounces.
b. a significant fraction of the time between bounces.
c. much longer than the time between bounces.

Linda Winkler

Numerade Educator