Fundamentals of Physics

David Halliday, Robert Resnick, Jearl Walker

Chapter 9

Center of Mass and Linear Momentum - all with Video Answers

Educators

Chapter Questions

Problem 1

A $2.00 \mathrm{~kg}$ particle has the $x y$ coordinates $(-1.20 \mathrm{~m}, 0.500 \mathrm{~m})$, and a $4.00 \mathrm{~kg}$ particle has the $x y$ coordinates $(0.600 \mathrm{~m},-0.750 \mathrm{~m})$. Both lie on a horizontal plane. At what (a) $x$ and (b) $y$ coordinates must you place a $3.00 \mathrm{~kg}$ particle such that the center of mass of the three-particle system has the coordinates $(-0.500 \mathrm{~m},-0.700 \mathrm{~m}) ?$

Salamat Ali

Salamat Ali

Numerade Educator

Problem 2

Figure $9-35$ shows a three-particle system, with masses $m_{1}=3.0$ $\mathrm{kg}, m_{2}=4.0 \mathrm{~kg}$, and $m_{3}=8.0 \mathrm{~kg}$. The scales on the axes are set by $x_{s}=2.0 \mathrm{~m}$ and $y_{s}=2.0 \mathrm{~m}$. What are
(a) the $x$ coordinate and
(b) the $y$ coordinate of the system's center of mass?
(c) If $m_{-}$ is graduallv increased, does the center of mass of the system shift toward or away from that particle, or does it remain stationary?

Meghan Miholics

Meghan Miholics

Numerade Educator

Problem 3

Figure $9-36$ shows a slab with dimensions $d_{1}=11.0 \mathrm{~cm}, d_{2}=$ $2.80 \mathrm{~cm}$, and $d_{3}=13.0 \mathrm{~cm} .$ Half the slab consists of aluminum (density $=2.70 \mathrm{~g} / \mathrm{cm}^{3}$ ) and half consists of iron (density $=7.85 \mathrm{~g} / \mathrm{cm}^{3}$ ). What are (a) the $x$ coordinate, (b) the $y$ coordinate, and (c) the $z$ coordinate of the slab's center of mass?

Salamat Ali

Numerade Educator

Problem 4

In Fig. $9-37$, three uniform thin rods, each of length $L=22 \mathrm{~cm}$, form an inverted U. The vertical rods each have a mass of $14 \mathrm{~g}$; the horizontal rod has a mass of $42 \mathrm{~g}$. What are (a) the $x$ coordinate and (b) the $y$ coordinate of the system's center of mass?

Zachary Warner

Numerade Educator

Problem 5

What are (a) the $x$ coordinate and (b) the $y$ coordinate of the center of mass for the uniform plate shown in Fig. $9-38$ if $L=5.0 \mathrm{~cm} ?$

Salamat Ali

Numerade Educator

Problem 6

Figure $9-39$ shows a cubical box that has been constructed from uniform metal plate of negligible thickness. The box is open at the top and has edge length $L=$ $40 \mathrm{~cm}$. Find (a) the $x$ coordinate, (b) the $\mathrm{y}$ coordinate, and (c) the $z$ coordinate of the center of mass of the box.

Zachary Warner

Numerade Educator

Problem 7

In the ammonia $\left(\mathrm{NH}_{3}\right)$ molecule of Fig. $9-40$, three hydrogen (H) atoms form an equilateral triangle, with the center of the triangle at distance $d=$ $9.40 \times 10^{-11} \mathrm{~m}$ from each hydrogen atom. The nitrogen $(\mathrm{N})$ atom is at the apex of a pyramid, with the three hydrogen atoms forming the base. The nitrogen-to-hydrogen atomic mass ratio is $13.9$, and the nitrogen-to-hydrogen distance is $L=10.14 \times 10^{-11} \mathrm{~m}$. What are the (a) $x$ and (b) $y$ coordinates of the molecule's center of mass?

Salamat Ali

Numerade Educator

Problem 8

A uniform soda can of mass $0.140 \mathrm{~kg}$ is $12.0 \mathrm{~cm}$ tall and filled with $0.354 \mathrm{~kg}$ of soda (Fig. 9-41). Then small holes are drilled in the top and bottom (with negligible loss of metal) to drain the soda. What is the height $h$ of the com of the can and contents (a) initially and (b) after the can loses all the soda?
(c) What happens to $h$ as the soda drains out? (d) If $x$ is the height of the remaining soda at any given instant, find $x$ when the com reaches its lowest point.

Zachary Warner

Numerade Educator

Problem 9

A stone is dropped at $t=0 .$ A second stone, with twice the mass of the first, is dropped from the same point at $t=100 \mathrm{~ms}$. (a) How far below the release point is the center of mass of the two stones at $t=300 \mathrm{~ms}$ ? (Neither stone has yet reached the ground.) (b) How fast is the center of mass of the twostone system moving at that time?

Meghan Miholics

Meghan Miholics

Numerade Educator

Problem 10

A $1000 \mathrm{~kg}$ automobile is at rest at a traffic signal. At the instant the light turns green, the automobile starts to move with a constant acceleration of $4.0 \mathrm{~m} / \mathrm{s}^{2}$. At the same instant a $2000 \mathrm{~kg}$ truck, traveling at a constant speed of $8.0 \mathrm{~m} / \mathrm{s}$, overtakes and passes the automobile. (a) How far is the com of the automobile-truck system from the traffic light at $t=3.0 \mathrm{~s}$ ? (b) What is the speed of the com then?

Zachary Warner

Numerade Educator

Problem 11

A big olive $(m=0.50 \mathrm{~kg})$ lies at the origin of an $x y$ coordinate system, and a big Brazil nut $(M=1.5 \mathrm{~kg})$ lies at the point $(1.0,2.0) \mathrm{m} .$ At $t=0$, a force $\vec{F}_{o}=(2.0 \hat{\mathrm{i}}+3.0 \mathrm{j}) \mathrm{N}$ begins to act on the olive, and a force $\vec{F}_{n}=(-3.0 \hat{\mathrm{i}}-2.0 \hat{\mathrm{j}}) \mathrm{N}$ begins to act on the nut. In unit-vector notation, what is the displacement of the center of mass of the olive-nut system at $t=4.0 \mathrm{~s}$, with respect to its position at $t=0 ?$

Keshav Singh

Numerade Educator

Problem 12

Two skaters, one with mass $65 \mathrm{~kg}$ and the other with mass $40 \mathrm{~kg}$, stand on an ice rink holding a pole of length $10 \mathrm{~m}$ and negligible mass. Starting from the ends of the pole, the skaters pull themselves along the pole until they meet. How far does the 40 kg skater move?

Zachary Warner

Numerade Educator

Problem 13

A shell is shot with an initial velocity $\vec{v}_{0}$ of $20 \mathrm{~m} / \mathrm{s}$, at an angle of $\theta_{0}=60^{\circ}$ with the horizontal. At the top of the trajectory, the shell explodes into two fragments of equal mass (Fig. 9-42). One fragment, whose speed immediately after the explosion is zero, falls vertically. How far from the gun does the other fragment land, assuming that the terrain is level and that air drag is negligible?

Salamat Ali

Numerade Educator

Problem 14

In Figure 9-43, two particles are launched from the origin of the coordinate system at time $t=0$. Particle 1 of mass $m_{1}=5.00 \mathrm{~g}$ is shot directly along the $x$ axis on a frictionless floor, with constant speed $10.0 \mathrm{~m} / \mathrm{s}$. Particle 2 of mass $m_{2}=3.00 \mathrm{~g}$ is shot with a velocity of magnitude $20.0 \mathrm{~m} / \mathrm{s}$, at an upward angle such that it always stays directly above particle $1 .$ (a) What is the maximum height $H_{\operatorname{man}}$ reached by the com of the two-particle system? In unit-vector notation, what are the (b) velocity and (c) acceleration of the com when the com reaches $H_{\max } ?$

Meghan Miholics

Meghan Miholics

Numerade Educator

Problem 15

Figure 9-44 shows an arrangement with an air track, in which a cart is connected by a cord to a hanging block. The cart has mass $m_{1}=0.600 \mathrm{~kg}$, and its center is initially at $x y$ coordinates $(-0.500$ $\mathrm{m}, 0 \mathrm{~m}) ;$ the block has mass $m_{2}=0.400 \mathrm{~kg}$, and its center is initially at $x y$ coordinates $(0,-0.100 \mathrm{~m})$. The mass of the cord and pulley are negligible. The cart is released from rest, and both cart and block move until the cart hits the pulley. The friction between the cart and the air track and between the pulley and its axle is negligible. (a) In unit-vector notation, what is the acceleration of the center of mass of the cart-block system? (b) What is the velocity of the com as a function of time $t ?$ (c) Sketch the path taken by the com. (d) If the path is curved, determine whether it bulges upward to the right or downward to the left, and if it is straight, find the angle between it and the $x$ axis.

Salamat Ali

Numerade Educator

Problem 16

Ricardo, of mass $80 \mathrm{~kg}$, and Carmelita, who is lighter, are enjoying Lake Merced at dusk in a $30 \mathrm{~kg}$ canoe. When the canoe is at rest in the placid water, they exchange seats, which are $3.0 \mathrm{~m}$ apart and symmetrically located with respect to the canoe's center. If the canoe moves $40 \mathrm{~cm}$ horizontally relative to a pier post, what is Carmelita's mass?

Zachary Warner

Numerade Educator

Problem 17

In Fig. $9-45 a$, a $4.5 \mathrm{~kg}$ dog stands on an $18 \mathrm{~kg}$ flatboat at distance $D=6.1 \mathrm{~m}$ from the shore. It walks $2.4 \mathrm{~m}$ along the boat toward shore and then stops. Assuming no friction between the boat and the water, find how far the dog is then from the shore. (Hint See Fig. $9-45 b$.)

Salamat Ali

Numerade Educator

Problem 18

A $0.70 \mathrm{~kg}$ ball moving horizontally at $5.0 \mathrm{~m} / \mathrm{s}$ strikes a vertical wall and rebounds with speed $2.0 \mathrm{~m} / \mathrm{s}$. What is the magnitude of the change in its linear momentum?

Zachary Warner

Numerade Educator

Problem 19

A $2100 \mathrm{~kg}$ truck traveling north at $41 \mathrm{~km} / \mathrm{h}$ turns east and accelerates to $51 \mathrm{~km} / \mathrm{h} .$ (a) What is the change in the truck's kinetic energy? What are the (b) magnitude and (c) direction of the change in its momentum?

Salamat Ali

Numerade Educator

Problem 20

At time $t=0$, a ball is struck at ground level and sent over level ground. The momentum $p$ versus $t$ during the flight is given by Fig. $9-46$ (with $p_{0}=6.0 \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}$ and
$\left.p_{1}=4.0 \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}\right)$. At what initial angle is the ball launched? (Hint: Find a solution that does not require you to read the time of the low point of the plot.)

Zachary Warner

Numerade Educator

Problem 21

A $0.30 \mathrm{~kg}$ softball has a velocity of $15 \mathrm{~m} / \mathrm{s}$ at an angle of $35^{\circ}$ below the horizontal just before making contact with the bat. What is the magnitude of the change in momentum of the ball while in contact with the bat if the ball leaves with a velocity of (a) $20 \mathrm{~m} / \mathrm{s}$, vertically downward, and (b) $20 \mathrm{~m} / \mathrm{s}$, horizontally back toward the pitcher?

Meghan Miholics

Meghan Miholics

Numerade Educator

Problem 22

Figure 9-47 gives an overhead view of the path taken by a $0.165 \mathrm{~kg}$ cue ball as it bounces from a rail of a pool table. The ball's initial speed is $2.00 \mathrm{~m} / \mathrm{s}$, and the angle $\theta_{1}$ is $30.0^{\circ}$. The bounce reverses the $y$ component of the ball's velocity but does not alter the $x$ component. What are (a) angle $\theta_{2}$ and (b) the change in the ball's linear momentum in unit-vector notation? (The fact that the ball rolls is irrelevant to the problem.)

Meghan Miholics

Meghan Miholics

Numerade Educator

Problem 23

Until his seventies, Henri LaMothe (Fig. $9-48$ ) excited audiences by belly-flopping from a height of $12 \mathrm{~m}$ into $30 \mathrm{~cm}$ of water. Assuming that he stops just as he reaches the bottom of the water and estimating his mass, find the magnitude of the impulse on him from the water.

Meghan Miholics

Meghan Miholics

Numerade Educator

Problem 24

In February 1955 , a paratrooper fell $370 \mathrm{~m}$ from an airplane without being able to open his chute but happened to land in snow, suffering only minor injuries. Assume that his speed at impact was $56 \mathrm{~m} / \mathrm{s}$ (terminal speed), that his mass (including gear) was $85 \mathrm{~kg}$, and that the magnitude of the force on him from the snow was at the survivable limit of $1.2 \times 10^{5} \mathrm{~N}$. What are (a) the minimum depth of snow that would have stopped him safely and (b) the magnitude of the impulse on him from the snow?

Zachary Warner

Numerade Educator

Problem 25

A $1.2 \mathrm{~kg}$ ball drops vertically onto a floor, hitting with a speed of $25 \mathrm{~m} / \mathrm{s} .$ It rebounds with an initial speed of $10 \mathrm{~m} / \mathrm{s}$. (a) What impulse acts on the ball during the contact? (b) If the ball is in contact with the floor for $0.020 \mathrm{~s}$, what is the magnitude of the average force on the floor from the ball?

Salamat Ali

Numerade Educator

Problem 26

In a common but dangerous prank, a chair is pulled away as a person is moving downward to sit on it, causing the victim to land hard on the floor. Suppose the victim falls by $0.50 \mathrm{~m}$, the mass that moves downward is $70 \mathrm{~kg}$, and the collision on the floor lasts $0.082 \mathrm{~s}$. What are the magnitudes of the (a) impulse and (b) average force acting on the victim from the floor during the collision?

Zachary Warner

Numerade Educator

Problem 27

A force in the negative direction of an $x$ axis is applied for $27 \mathrm{~ms}$ to a $0.40 \mathrm{~kg}$ ball initially moving at $14 \mathrm{~m} / \mathrm{s}$ in the positive direction of the axis. The force varies in magnitude, and the impulse has magnitude $32.4 \mathrm{~N} \cdot \mathrm{s}$. What are the ball's (a) speed and (b) direction of travel just after the force is applied? What are (c) the average magnitude of the force and (d) the direction of the impulse on the ball?

Meghan Miholics

Meghan Miholics

Numerade Educator

Problem 28

In tae-kwon-do, a hand is slammed down onto a target at a speed of $13 \mathrm{~m} / \mathrm{s}$ and comes to a stop during the $5.0 \mathrm{~ms}$ collision. Assume that during the impact the hand is independent of the arm and has a mass of $0.70 \mathrm{~kg} .$ What are the magnitudes of the (a) impulse and (b) average force on the hand from the target?

Meghan Miholics

Meghan Miholics

Numerade Educator

Problem 29

Suppose a gangster sprays Superman's chest with $3 \mathrm{~g}$ bullets at the rate of 100 bullets/min, and the speed of each bullet is 500 $\mathrm{m} / \mathrm{s}$. Suppose too that the bullets rebound straight back with no change in speed. What is the magnitude of the average force on Superman's chest?

Salamat Ali

Numerade Educator

Problem 30

Two average forces. A steady stream of $0.250 \mathrm{~kg}$ snowballs is shot perpendicularly into a wall at a speed of $4.00 \mathrm{~m} / \mathrm{s}$. Each ball sticks to the wall. Figure $9-49$ gives the magnitude $F$ of the force on the wall as a function of time $t$ for two of the snowball impacts. Impacts occur with a repetition time interval $\Delta t_{e}=50.0 \mathrm{~ms}$, last a duration time interval $\Delta t_{d}=10 \mathrm{~ms}$, and produce isosceles triangles on the graph, with each impact reaching a force maximum $F_{\max }=200 \mathrm{~N}$ During each impact, what are the magnitudes of (a) the impulse and
(b) the average force on the wall? (c) During a time interval of many impacts, what is the magnitude of the average force on the wall?

Meghan Miholics

Meghan Miholics

Numerade Educator

Problem 31

Jumping up before the elevator hits. After the cable }}$ snaps and the safety system fails, an elevator cab free-falls from a height of $36 \mathrm{~m}$. During the collision at the bottom of the elevator shaft, a $90 \mathrm{~kg}$ passenger is stopped in $5.0 \mathrm{~ms}$. (Assume that neither the passenger nor the cab rebounds) What are the magnitudes of the (a) impulse and (b) average force on the passenger during the collision? If the passenger were to jump upward with a speed of $7.0 \mathrm{~m} / \mathrm{s}$ relative to the cab floor just before the cab hits the bottom of the shaft, what are the magnitudes of the (c) impulse and (d) average force (assuming the same stopping time)?

Keshav Singh

Numerade Educator

Problem 32

A $5.0 \mathrm{~kg}$ toy car can move along an $x$ axis; Fig. $9-50$ gives $F_{x}$ of the force acting on the car, which begins at rest at time $t=0$. The scale on the $F_{x}$ axis is set by $F_{x z}=5.0 \mathrm{~N}$. In unit-vector notation, what is $\vec{p}$ at (a) $t=4.0 \mathrm{~s}$ and $(\mathrm{b}) t=7.0 \mathrm{~s}$, and $(\mathrm{c})$ what is $\vec{v}$ at $t=9.0 \mathrm{~s}$ ?

Tanner Manwaring

Tanner Manwaring

Numerade Educator

Problem 33

Figure 9-51 shows a 0.300 $\mathrm{kg}$ baseball just before and just after it collides with a bat. Just before, the ball has velocity $\vec{v}_{1}$ of magnitude $12.0 \mathrm{~m} / \mathrm{s}$ and angle $\theta_{1}=35.0^{\circ}$. Just
after, it is traveling directly upward with velocity $\vec{v}_{2}$ of magnitude $10.0$ $\mathrm{m} / \mathrm{s}$. The duration of the collision is $2.00 \mathrm{~ms}$. What are the (a) magnitude and (b) direction (relative to the positive direction of the $x$ axis) of the impulse on the ball from the bat? What are the (c) magnitude and (d) direction of the average force on the ball from the bat?

Salamat Ali

Numerade Educator

Problem 34

Basilisk lizards can run across the top of a water surface (Fig. 9-52). With each step, a lizard first slaps its foot against the water and then pushes it down into the water rapidly enough to form an air cavity around the top of the foot. To avoid having to pull the foot back up against water drag in order to complete the step, the lizard withdraws the foot before water can flow into the air cavity. If the lizard is not to sink, the average upward impulse on the lizard during this full action of slap, downward push, and withdrawal must match the downward impulse due to the gravitational force. Suppose the mass of a basilisk lizard is $90.0 \mathrm{~g}$, the mass of each foot is $3.00 \mathrm{~g}$, the speed of a foot as it slaps the water is $1.50 \mathrm{~m} / \mathrm{s}$, and the time for a single step is $0.600 \mathrm{~s}$. (a) What is the magnitude of the impulse on the lizard during the slap? (Assume this impulse is directly upward.) (b) During the $0.600 \mathrm{~s}$ duration of a step, what is the downward impulse on the lizard due to the gravitational force? (c) Which action, the slap or the push, provides the primary support for the lizard, or are they approximately equal in their support?

Keshav Singh

Numerade Educator

Problem 35

Figure $9-53$ shows an approximate plot of force magnitude $F$ versus time $t$ during the collision of a $58 \mathrm{~g}$ Superball with a wall. The initial velocity of the ball is $34 \mathrm{~m} / \mathrm{s}$ perpendicular to the wall; the ball rebounds directly back with approximately the same speed, also perpendicular to the wall. What is $F_{\text {mexs }}$ the maximum magnitude of the force on the ball from the wall during the collision?

Meghan Miholics

Meghan Miholics

Numerade Educator

Problem 36

A $0.25 \mathrm{~kg}$ puck is initially stationary on an ice surface with negligible friction. At time $t=0$, a horizontal force begins to move the puck. The force is given by $\vec{F}=\left(12.0-3.00 t^{2}\right) \hat{1}$, with $\vec{F}$ in newtons and $t$ in seconds, and it acts until its magnitude is zero. (a) What is the magnitude of the impulse on the puck from the force between $t=0.500 \mathrm{~s}$ and $t=1.25 \mathrm{~s} ?$ (b) What is the change in momentum of the puck between $t=0$ and the instant at which $F=0 ?$

Meghan Miholics

Meghan Miholics

Numerade Educator

Problem 37

A soccer player kicks a soccer ball of mass $0.45 \mathrm{~kg}$ that is initially at rest. The foot of the player is in contact with the ball for $3.0 \times 10^{-3} \mathrm{~s}$, and the force of the kick is given by $$ F(t)=\left[\left(6.0 \times 10^{6}\right) t-\left(2.0 \times 10^{9}\right) t^{2}\right] \mathrm{N}$$ for $0 \leq t \leq 3.0 \times 10^{-3} \mathrm{~s}$, where $t$ is in seconds. Find the magnitudes of (a) the impulse on the ball due to the kick, (b) the average force on the ball from the player's foot during the period of contact,
(c) the maximum force on the ball from the player's foot during the period of contact, and (d) the ball's velocity immediately after it loses contact with the player's foot.

Salamat Ali

Numerade Educator

Problem 38

In the overhead view of Fig $9-54$, a $300 \mathrm{~g}$ ball with a speed $v$ of $6.0 \mathrm{~m} / \mathrm{s}$ strikes a wall at an angle $\bar{\theta}$ of $30^{\circ}$ and then rebounds with the same speed and angle. It is in contact with the wall for $10 \mathrm{~ms}$. In unitvector notation, what are (a) the impulse on the ball from the wall and (b) the average force on the wall from the ball?

Keshav Singh

Numerade Educator

Problem 39

A $91 \mathrm{~kg}$ man lying on a surface of negligible friction shoves a 68 g stone away from himself, giving it a speed of $4.0 \mathrm{~m} / \mathrm{s}$. What speed does the man acquire as a result?

Salamat Ali

Numerade Educator

Problem 40

A space vehicle is traveling at $4300 \mathrm{~km} / \mathrm{h}$ relative to Earth when the exhausted rocket motor (mass $4 m$ ) is disengaged and sent backward with a speed of $82 \mathrm{~km} / \mathrm{h}$ relative to the command module (mass $m$ ). What is the speed of the command module relative to Earth just after the separation?

Keshav Singh

Numerade Educator

Problem 41

Figure $9-55$ shows a two-ended "rocket" that is initially stationary on a frictionless floor, with its center at the origin of an $x$ axis. The rocket consists of a central block $C$ (of mass $M=6.00 \mathrm{~kg}$ ) and blocks $L$ and $R$ (each of mass $m=2.00 \mathrm{~kg}$ ) on the left and right sides. Small explosions can shoot either of the side blocks away from block $C$ and along the $x$ axis Here is the sequence: $(1)$ At time $t=$ 0 , block $L$ is shot to the left with a speed of $3.00 \mathrm{~m} / \mathrm{s}$ relative to the velocity that the explosion gives the rest of the rocket. (2) Next, at time $t=0.80 \mathrm{~s}$, block $R$ is shot to the right with a speed of $3.00 \mathrm{~m} / \mathrm{s}$ relative to the velocity that block $C$ then has. At $t=2.80 \mathrm{~s}$, what are (a) the velocity of block $C$ and (b) the position of its center?

Salamat Ali

Numerade Educator

Problem 42

An object, with mass $m$ and speed $v$ relative to an observer. explodes into two pieces, one three times as massive as the other; the explosion takes place in deep space. The less massive piece stops relative to the observer. How much kinetic energy is added to the system during the explosion, as measured in the observer's reference frame?

Zachary Warner

Numerade Educator

Problem 43

In the Olympiad of $708 \mathrm{~B}$ the standing long jump used handheld weights called halteres to lengthen their jumps (Fig. $9-56$ ). The weights were swung up in front just before liftoff and then swung down and thrown backward during the flight. Suppose a modern $78 \mathrm{~kg}$ long jumper similarly uses two $5.50 \mathrm{~kg}$ halteres, throwing them horizontally to the rear at his maximum height such that their horizontal velocity is zero relative to the ground. Let his liftoff velocity be $\vec{v}=(9.5 \hat{\mathrm{i}}+4.0 \hat{\mathrm{j}}) \mathrm{m} / \mathrm{s}$
with or without the halteres, and assume that he lands at the liftoff level. What distance would the use of the halteres add to his range?

Keshav Singh

Numerade Educator

Problem 44

In Fig. 9-57, a stationary block explodes into two pieces $\underline{L}$ and $R$ that slide across a frictionless floor and then into regions with friction, where they stop. Piece $L$, with a mass of $2.0 \mathrm{~kg}$, encounters a coefficient of kinetic friction $\mu_{L}=0.40$ and slides to a stop in distance $d_{L}=0.15 \mathrm{~m}$. Piece $R$ encounters a coefficient of kinetic friction $\mu_{R}=$ $0.50$ and slides to a stop in distance $d_{R}=0.25 \mathrm{~m}$. What was the mass of the block?

Zachary Warner

Numerade Educator

Problem 45

A $20.0 \mathrm{~kg}$ body is moving through space in the positive direction of an $x$ axis with a speed of $200 \mathrm{~m} / \mathrm{s}$ when, due to an internal explosion, it breaks into three parts. One part, with a mass of $10.0 \mathrm{~kg}$, moves away from the point of explosion with a speed of $100 \mathrm{~m} / \mathrm{s}$ in the positive $y$ direction. A second part, with a mass of $4.00 \mathrm{~kg}$, moves in the negative $x$ direction with a speed of $500 \mathrm{~m} / \mathrm{s}$. (a) In unit-vector notation, what is the velocity of the third part? (b) How much energy is released in the explosion? Ignore effects due to the gravitational force.

Salamat Ali

Numerade Educator

Problem 46

A $4.0 \mathrm{~kg}$ mess kit sliding on a frictionless surface explodes into two $2.0 \mathrm{~kg}$ parts: $3.0 \mathrm{~m} / \mathrm{s}$, due north, and $5.0 \mathrm{~m} / \mathrm{s}, 30^{\circ}$ north of east. What is the original speed of the mess kit?

Zachary Warner

Numerade Educator

Problem 47

A vessel at rest at the origin of an $x y$ coordinate system explodes into three pieces. Just after the explosion, one piece, of mass $m$, moves with velocity $(-30 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{i}}$ and a second piece, also of mass $m$, moves with velocity $(-30 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{j}} .$ The third piece has mass $3 \mathrm{~m} .$ Just after the explosion, what are the (a) magnitude and (b) direction of the velocity of the third piece?

Salamat Ali

Numerade Educator

Problem 48

Particle $A$ and particle $B$ are held together with a compressed spring between them. When they are released, the spring pushes them apart, and they then fly off in opposite directions, free of the spring. The mass of $A$ is $2.00$ times the mass of $B$, and the energy stored in the spring was $60 \mathrm{~J}$. Assume that the spring has negligible mass and that all its stored energy is transferred to the particles. Once that transfer is complete, what are the kinetic energies of (a) particle $A$ and (b) particle $\bar{B}$ ?

Zachary Warner

Numerade Educator

Problem 49

A bullet of mass $10 \mathrm{~g}$ strikes a ballistic pendulum of mass 2.0 $\mathrm{kg}$. The center of mass of the pendulum rises a vertical distance of $12 \mathrm{~cm}$. Assuming that the bullet remains embedded in the pendulum, calculate the bullet's initial speed.

Salamat Ali

Numerade Educator

Problem 50

A $5.20 \mathrm{~g}$ bullet moving at $672 \mathrm{~m} / \mathrm{s}$ strikes a $700 \mathrm{~g}$ wooden block at rest on a frictionless surface. The bullet emerges, traveling in the same direction with its speed reduced to $428 \mathrm{~m} / \mathrm{s}$. (a) What is the resulting speed of the block? (b) What is the speed of the bullet-block center of mass?

Keshav Singh

Numerade Educator

Problem 51

In Fig. $9-58 a$, a $3.50 \mathrm{~g}$ bullet is fired horizontally at two blocks at rest on a frictionless table. The bullet passes through block 1 (mass $1.20 \mathrm{~kg}$ ) and embeds itself in block 2 (mass $1.80 \mathrm{~kg}$ ). The blocks end up with speeds $v_{1}=0.630 \mathrm{~m} / \mathrm{s}$ and $v_{2}=1.40 \mathrm{~m} / \mathrm{s}$ (Fig. $9-58 b$ ). Neglecting the material removed from block 1 by the bullet, find the speed of the bullet as it (a) leaves and (b) enters block 1.

Salamat Ali

Numerade Educator

Problem 52

In Fig. $9-59$, a $10 \mathrm{~g}$ bullet moving directly upward at $1000 \mathrm{~m} / \mathrm{s}$ strikes and passes through the center of mass of a $5.0 \mathrm{~kg}$ block initially at rest. The bullet emerges from the block moving directly upward at 400 $\mathrm{m} / \mathrm{s}$. To what maximum height does the block then rise above its initial position?

Salamat Ali

Numerade Educator

Problem 53

In Anchorage, collisions of a vehicle with a moose are so common that they are referred to with the abbreviation MVC. Suppose a $1000 \mathrm{~kg}$ car slides into a stationary $500 \mathrm{~kg}$ moose on a very slippery road, with the moose being thrown through the windshield (a common MVC result). (a) What percent of the original kinetic energy is lost in the collision to other forms of energy? A similar danger occurs in Saudi Arabia because of camel-vehicle collisions (CVC). (b) What percent of the original kinetic energy is lost if the car hits a $300 \mathrm{~kg}$ camel? (c) Generally, does the percent loss increase or decrease if the animal mass decreases?

Salamat Ali

Numerade Educator

Problem 54

A completely inelastic collision occurs between two balls of wet putty that move directly toward each other along a vertical axis. Just before the collision, one ball, of mass $3.0 \mathrm{~kg}$, is moving upward at $20 \mathrm{~m} / \mathrm{s}$ and the other ball, of mass $2.0 \mathrm{~kg}$, is moving downward at $12 \mathrm{~m} / \mathrm{s}$. How high do the combined two balls of putty rise above the collision point? (Neglect air drag.)

Keshav Singh

Numerade Educator

Problem 55

A $5.0 \mathrm{~kg}$ block with a speed of $3.0 \mathrm{~m} / \mathrm{s}$ collides with a 10 kg block that has a speed of $2.0 \mathrm{~m} / \mathrm{s}$ in the same direction. After the collision, the $10 \mathrm{~kg}$ block travels in the original direction with a speed of $2.5 \mathrm{~m} / \mathrm{s}$. (a) What is the velocity of the $5.0 \mathrm{~kg}$ block immediately after the collision? (b) By how much does the total kinetic energy of the system of two blocks change because of the collision? (c) Suppose, instead, that the $10 \mathrm{~kg}$ block ends up with a speed of $4.0 \mathrm{~m} / \mathrm{s}$. What then is the change in the total kinetic energy?
(d) Account for the result you obtained in (c).

Salamat Ali

Numerade Educator

Problem 56

In the "before" part of Fig. 9-60, car $A$ (mass $1100 \mathrm{~kg}$ ) is stopped at a traffic light when it is rear-ended by car $B$ (mass $1400 \mathrm{~kg}$ ). Both cars then slide with locked wheels until the frictional force from the slick road (with a low $\mu_{k}$ of $0.13$ ) stops them, at distances $d_{A}=8.2 \mathrm{~m}$ and $d_{B}=6.1 \mathrm{~m}$. What are the speeds of (a) car $A$ and (b) car $B$ at the start of the sliding, just after the collision? (c) Assuming that linear momentum is conserved during the collision, find the speed of car $B$ just before the collision.
(d) Explain why this assumption may be invalid.

Zachary Warner

Numerade Educator

Problem 57

In Fig. $9-61$, a ball of mass $m=60 \mathrm{~g}$ is shot with speed $v_{i}=22$ $\mathrm{m} / \mathrm{s}$ into the barrel of a spring gun of mass $M=240 \mathrm{~g}$ initially at rest on a frictionless surface. The ball sticks in the barrel at the point of maximum compression of the spring. Assume that the increase in thermal energy due to friction between the ball and the barrel is negligible. (a) What is the speed of the spring gun after the ball stops in the barrel? (b) What fraction of the initial kinetic energy of the ball is stored in the spring?

Keshav Singh

Numerade Educator

Problem 58

In Fig. 9-62, block 2 (mass $1.0$ $\mathrm{kg}$ ) is at rest on a frictionless surface and touching the end of an unstretched spring of spring constant $200 \mathrm{~N} / \mathrm{m}$. The other end of the spring is fixed to a wall. Block 1 (mass $2.0 \mathrm{~kg}$ ), traveling at speed $v_{1}=4.0$ $\mathrm{m} / \mathrm{s}$, collides with block 2 , and the two blocks stick together. When the blocks momentarily stop, by what distance is the spring compressed?

Zachary Warner

Numerade Educator

Problem 59

In Fig. $9-63$, block 1 (mass $2.0 \mathrm{~kg}$ ) is moving rightward at $10 \mathrm{~m} / \mathrm{s}$ and block 2 (mass $5.0 \mathrm{~kg}$ ) is moving rightward at $3.0 \mathrm{~m} / \mathrm{s}$. The surface is frictionless, and a spring with a spring constant of $1120 \mathrm{~N} / \mathrm{m}$ is fixed to block $2 .$ When the blocks collide, the compression of the spring is maximum at the instant the blocks have the same velocity. Find the maximum compression.

Salamat Ali

Numerade Educator

Problem 60

In Fig. 9-64, block $A$ (mass $1.6$ kg) slides into block $B$ (mass $2.4 \mathrm{~kg}$ ), along a frictionless surface. The directions of three velocities before $(i)$ and after $(f)$ the collision are indicated; the corresponding speeds are $v_{A i}=$ $5.5 \mathrm{~m} / \mathrm{s}, v_{B i}=2.5 \mathrm{~m} / \mathrm{s}$, and $v_{B f}=4.9$ $\mathrm{m} / \mathrm{s}$. What are the (a) speed and (b) direction (left or right) of velocity $\vec{v}_{A f} ?$ (c) Is the collision elastic?

Zachary Warner

Numerade Educator

Problem 61

A cart with mass $340 \mathrm{~g}$ moving on a frictionless linear air track at an initial speed of $1.2 \mathrm{~m} / \mathrm{s}$ undergoes an elastic collision with an initially stationary cart of unknown mass. After the collision, the first cart continues in its original direction at $0.66 \mathrm{~m} / \mathrm{s}$. (a) What is the mass of the second cart?
(b) What is its speed after impact? (c) What is the speed of the twocart center of mass?

Keshav Singh

Numerade Educator

Problem 62

Two titanium spheres approach each other head-on with the same speed and collide elastically. After the collision, one of the spheres, whose mass is $300 \mathrm{~g}$, remains at rest. (a) What is the mass of the other sphere? (b) What is the speed of the two-sphere center of mass if the initial speed of each sphere is $2.00 \mathrm{~m} / \mathrm{s}$ ?

Zachary Warner

Numerade Educator

Problem 63

Block 1 of mass $m_{1}$ slides along a frictionless floor and into a one-dimensional elastic collision with stationary block 2 of mass $m_{2}=3 m_{1}$. Prior to the collision, the center of mass of the twoblock system had a speed of $3.00 \mathrm{~m} / \mathrm{s}$. Afterward, what are the speeds of (a) the center of mass and (b) block 2?

Keshav Singh

Numerade Educator

Problem 64

A steel ball of mass $0.500 \mathrm{~kg}$ is fastened to a cord that is $70.0 \mathrm{~cm}$ long and fixed at the far end. The ball is then released when the cord is horizontal (Fig. $9-65$ ). At the bottom of its path, the ball strikes a $2.50 \mathrm{~kg}$ steel block initially at rest on a frictionless surface. The collision is elastic. Find (a) the speed of the ball and (b) the speed of the block, both just after the collision.

Zachary Warner

Numerade Educator

Problem 65

A body of mass $2.0 \mathrm{~kg}$ makes an elastic collision with another body at rest and continues to move in the original direction but with one-fourth of its original speed. (a) What is the mass of the other body? (b) What is the speed of the two-body center of mass if the initial speed of the $2.0 \mathrm{~kg}$ body was $4.0 \mathrm{~m} / \mathrm{s}$ ?

Salamat Ali

Numerade Educator

Problem 66

Block 1, with mass $m_{1}$ and speed $4.0 \mathrm{~m} / \mathrm{s}$, slides along an $x$ axis on a frictionless floor and then undergoes a one-dimensional elastic collision with stationary block 2, with mass $m_{2}=0.40 m_{1} .$ The two blocks then slide into a region where the coefficient of kinetic friction is $0.50$; there they stop. How far into that region do (a) block 1 and $(b)$ block 2 slide?

Zachary Warner

Numerade Educator

Problem 67

In Fig. $9-66$, particle 1 of mass $m_{1}=0.30 \mathrm{~kg}$ slides rightward along an $x$ axis on a frictionless floor with a speed of $2.0 \mathrm{~m} / \mathrm{s}$. When it reaches $x=$ 0 , it undergoes a one-dimensional elastic collision with stationary particle 2 of mass $m_{2}=0.40 \mathrm{~kg}$. When particle 2 then reaches a wall at $x_{w}=70 \mathrm{~cm}$, it bounces from the wall with no loss of speed. At what position on the $x$ axis does particle 2 then collide with particle $1 ?$

Keshav Singh

Numerade Educator

Problem 68

In Fig. 9-67, block 1 of mass $m_{1}$ slides from rest along a frictionless ramp from height $h=2.50 \mathrm{~m}$ and then collides with stationary block 2, which has mass $m_{2}=2.00 m_{1}$. After the collision. block 2 slides into a region where the coefficient of kinetic friction $\mu_{k}$ is $0.500$ and comes to a stop in distance $d$ within that region. What is the value of distance $d$ if the collision is (a) elastic and (b) completely inelastic?

Zachary Warner

Numerade Educator

Problem 69

A small ball of mass $m$ is aligned above a larger ball of mass $M=0.63 \mathrm{~kg}$ (with a slight separation, as with the baseball and basketball of Fig. $9-68 a$ ), and the two are dropped simultaneously from a height of $h=1.8 \mathrm{~m}$. (Assume the radius of each ball is negligible relative to $h$.) (a) If the larger ball rebounds elastically from the floor and then the small ball rebounds elastically from the larger ball, what value of $m$ results in the larger ball stopping when it collides with the small ball? (b) What height does the small ball then reach (Fig. $9-68 b)$ ?

Salamat Ali

Numerade Educator

Problem 70

In Fig. $9-69$, puck 1 of mass $m_{1}=0.20 \mathrm{~kg}$ is sent sliding across a frictionless lab bench, to undergo a one-dimensional elastic collision with stationary puck $2 .$ Puck 2 then slides off the bench and lands a distance $d$ from the base of the bench. Puck 1 rebounds from the collision and slides off the opposite edge of the bench, landing a distance $2 d$ from the base of the bench. What is the mass of puck 2 ? (Hint: Be careful with signs.)

Zachary Warner

Numerade Educator

Problem 71

In Fig. $9-21$, projectile particle 1 is an alpha particle and target particle 2 is an oxygen nucleus. The alpha particle is scattered at angle $\theta_{1}=64.0^{\circ}$ and the oxygen nucleus recoils with speed $1.20 \times$ $10^{5} \mathrm{~m} / \mathrm{s}$ and at angle $\theta_{2}=51.0^{\circ} .$ In atomic mass units, the mass of the alpha particle is $4.00 \mathrm{u}$ and the mass of the oxygen nucleus is $16.0 \mathrm{u}$. What are the (a) final and (b) initial speeds of the alpha particle?

Salamat Ali

Numerade Educator

Problem 72

Ball $B$, moving in the positive direction of an $x$ axis at speed $v$, collides with stationary ball $A$ at the origin. $A$ and $B$ have different masses. After the collision, $B$ moves in the negative direction of the $y$ axis at speed $v / 2 .$ (a) In what direction does $A$ move?
(b) Show that the speed of $A$ cannot be determined from the given information.

Zachary Warner

Numerade Educator

Problem 73

After a completely inelastic collision, two objects of the same mass and same initial speed move away together at half their initial speed. Find the angle between the initial velocities of the objects.

Salamat Ali

Numerade Educator

Problem 74

Two $2.0 \mathrm{~kg}$ bodies, $A$ and $B$, collide. The velocities before the collision are $\vec{v}_{A}=(15 \hat{\mathrm{i}}+30 \hat{\mathrm{j}}) \mathrm{m} / \mathrm{s}$ and $\vec{v}_{B}=(-10 \hat{\mathrm{i}}+5.0 \hat{\mathrm{j}}) \mathrm{m} / \mathrm{s}$. After the collision, $\vec{v}_{A}^{\prime}=(-5.0 \hat{1}+20 \hat{j}) \mathrm{m} / \mathrm{s} .$ What are (a) the final velocity of $B$ and (b) the change in the total kinetic energy (including sign)?

Keshav Singh

Numerade Educator

Problem 75

A projectile proton with a speed of $500 \mathrm{~m} / \mathrm{s}$ collides elastically with a target proton initially at rest. The two protons then move along perpendicular paths, with the projectile path at $60^{\circ}$ from the original direction. After the collision, what are the speeds of (a) the target proton and (b) the projectile proton?

Salamat Ali

Numerade Educator

Problem 76

A $6090 \mathrm{~kg}$ space probe moving nose-first toward Jupiter at $105 \mathrm{~m} / \mathrm{s}$ relative to the Sun fires its rocket engine, ejecting $80.0 \mathrm{~kg}$ of exhaust at a speed of $253 \mathrm{~m} / \mathrm{s}$ relative to the space probe. What is the final velocity of the probe?

Zachary Warner

Numerade Educator

Problem 77

In Fig. $9-70$, two long barges are moving in the same direction in still water, one with a speed of $10 \mathrm{~km} / \mathrm{h}$ and the other with a speed of $20 \mathrm{~km} / \mathrm{h}$. While they are passing each other, coal is shoveled from the slower to the faster one at a rate of $1000 \mathrm{~kg} / \mathrm{min} .$ How much additional force must be provided by the driving engines of (a) the faster barge and (b) the slower barge if neither is to change speed? Assume that the shoveling is always perfectly sideways and that the frictional forces between the barges and the water do not depend on the mass of the barges.

Salamat Ali

Numerade Educator

Problem 78

Consider a rocket that is in deep space and at rest relative to an inertial reference frame. The rocket's engine is to be fired for a certain interval. What must be the rocket's mass ratio (ratio of ini. tial to final mass) over that interval if the rocket's original speed relative to the inertial frame is to be equal to (a) the exhaust speed (speed of the exhaust products relative to the rocket) and (b) $2.0$ times the exhaust speed?

Keshav Singh

Numerade Educator

Problem 79

A rocket that is in deep space and initially at rest relative to an inertial reference frame has a mass of $2.55 \times 10^{5} \mathrm{~kg}$, of which $1.81 \times 10^{5} \mathrm{~kg}$ is fuel. The rocket engine is then fired for $250 \mathrm{~s}$ while fuel is consumed at the rate of $480 \mathrm{~kg} / \mathrm{s}$. The speed of the exhaust products relative to the rocket is $3.27 \mathrm{~km} / \mathrm{s}$. (a) What is the rocket's thrust? After the $250 \mathrm{~s}$ firing, what are (b) the mass and (c) the speed of the rocket?

Salamat Ali

Numerade Educator

Problem 80

An object is tracked by a radar station and determined to have a position vector given by $\vec{r}=(3500-160 t) \hat{\mathrm{i}}+2700 \hat{\mathrm{j}}+300 \hat{\mathrm{k}}$, with $\vec{r}$ in meters and $t$ in seconds. The radar station's $x$ axis points east. its $y$ axis north, and its $z$ axis vertically up. If the object is a $250 \mathrm{~kg}$ meteorological missile, what are (a) its linear momentum, (b) its direction of motion, and (c) the net force on it?

Keshav Singh

Numerade Educator

Problem 81

The last stage of a rocket, which is traveling at a speed of $7600 \mathrm{~m} / \mathrm{s}$, consists of two parts that are clamped together: a rocket case with a mass of $290.0 \mathrm{~kg}$ and a payload capsule with a mass of $150.0 \mathrm{~kg}$. When the clamp is released, a compressed spring causes the two parts to separate with a relative speed of $910.0 \mathrm{~m} / \mathrm{s}$. What are the speeds of (a) the rocket case and (b) the payload after they have separated? Assume that all velocities are along the same line. Find the total kinetic energy of the two parts (c) before and (d) after they separate. (e) Account for the difference.

Salamat Ali

Numerade Educator

Problem 82

Pancake collapse of a tall building. In the section of a tall building shown in Fig. $9-71 a$, the infrastructure of any given floor $\bar{K}$ must support the weight $W$ of all higher floors. Normally the infrastructure is constructed with a safety factor $s$ so that it can withstand an even greater downward force of $s W$. If, however, the support columns between $K$ and $L$ suddenly collapse and allow the higher floors to free-fall together onto floor $K$ (Fig. $9-71 b$ ), the force in the collision can exceed $s W$ and. after a brief pause, cause $K$ to collapse onto floor $J$, which collapses on floor $I$, and so on until the ground is reached. Assume that the floors are separated by $d=4.0 \mathrm{~m}$ and have the same mass. Also assume that when the floors above $K$ free-fall onto $K$, the collision lasts $1.5 \mathrm{~ms}$. Under these simplified conditions, what value must the safety factor $s$ exceed to prevent pancake collapse of the building?

Keshav Singh

Numerade Educator

Problem 83

"Relative" is an important word. In Fig. $9-72$, block $L$ of mass $m_{L}=1.00 \mathrm{~kg}$ and block $R$ of mass $m_{R}=0.500 \mathrm{~kg}$ are held in place with a compressed spring between them.When the blocks are released, the spring sends them sliding across a frictionless floor. (The spring has negligible mass and falls to the floor after the blocks leave it.) (a) If the spring gives block $L$ a release speed of $1.20 \mathrm{~m} / \mathrm{s}$ relative to the floor, how far does block $R$ travel in the next $0.800 \mathrm{~s} ?$ (b) If, instead, the spring gives block $L$ a release speed of $1.20 \mathrm{~m} / \mathrm{s}$ relative to the velocity that the spring gives block $R$, how far does block $R$ travel in the next $0.800 \mathrm{~s}$ ?

Salamat Ali

Numerade Educator

Problem 84

Figure $9-73$ shows an overhead view of two particles sliding at constant velocity over a frictionless surface. The particles have the same mass and the same initial speed $v=4.00 \mathrm{~m} / \mathrm{s}$, and they collide where their paths intersect. An $x$ axis is arranged to bisect the angle between their incoming paths, such that $\theta=40.0^{\circ} .$ The region to the right of the collision is divided into four lettered sections by the $x$ axis and four numbered dashed lines. In what region or along what line do the particles travel if the collision is (a) completely inelastic, (b) elastic, and (c) inelastic? What are their final speeds if the collision is (d) completely inelastic and (e) elastic?

Keshav Singh

Numerade Educator

Problem 85

Speed deamplifier. In Fig. $9-74$, block 1 of mass $m_{1}$ slides along an $x$ axis on a frictionless floor at speed $4.00 \mathrm{~m} / \mathrm{s}$. Then it undergoes a one-dimensional elastic collision with stationary block 2 of mass $m_{2}=$ $2.00 m_{1} .$ Next, block 2 undergoes a one-dimensional elastic collision with stationary block 3 of mass $m_{3}=2.00 m_{2}$. (a) What then is the speed of block 3 ? Are (b) the speed, (c) the kinetic energy, and (d) the momentum of block 3 greater than, less than, or the same as the initial values for block 1 ?

Salamat Ali

Numerade Educator

Problem 86

Speed amplifier. In Fig. $9-75$, block 1 of mass $m_{1}$ slides along an $x$ axis on a frictionless floor with a speed of $v_{1 i}=4.00 \mathrm{~m} / \mathrm{s}$. Then it undergoes a one-dimensional elastic collision with stationary block 2 of mass $m_{2}=0.500 m_{1} .$ Next, block 2 undergoes a one-dimensional elastic collision with stationary block 3 of mass $m_{3}=0.500 m_{2}$. (a) What then is the speed of block 3 ? Are (b) the speed, (c) the kinetic energy, and (d) the momentum of block 3 greater than, less than, or the same as the initial values for block $1 ?$

Keshav Singh

Numerade Educator

Problem 87

A ball having a mass of $150 \mathrm{~g}$ strikes a wall with a speed of $5.2 \mathrm{~m} / \mathrm{s}$ and rebounds with only $50 \%$ of its initial kinetic energy. (a) What is the speed of the ball immediately after rebounding? (b) What is the magnitude of the impulse on the wall from the ball? (c) If the ball is in contact with the wall for $7.6 \mathrm{~ms}$, what is the magnitude of the average force on the ball from the wall during this time interval?

Salamat Ali

Numerade Educator

Problem 88

A spacecraft is separated into two parts by detonating the explosive bolts that hold them together. The masses of the parts are $1200 \mathrm{~kg}$ and $1800 \mathrm{~kg} ;$ the magnitude of the impulse on each part from the bolts is $300 \mathrm{~N} \cdot \mathrm{s}$. With what relative speed do the two parts separate because of the detonation?

Keshav Singh

Numerade Educator

Problem 89

A $1400 \mathrm{~kg}$ car moving at $5.3 \mathrm{~m} / \mathrm{s}$ is initially traveling north along the positive direction of a $y$ axis. After completing a $90^{\circ}$ right-hand turn in $4.6 \mathrm{~s}$, the inattentive operator drives into a tree, which stops the car in $350 \mathrm{~ms}$. In unit-vector notation, what is the impulse on the car (a) due to the turn and (b) due to the collision? What is the magnitude of the average force that acts on the car (c) during the turn and (d) during the collision? (e) What is the direction of the average force during the turn?

Salamat Ali

Numerade Educator

Problem 90

A certain radioactive (parent) nucleus transforms to a different (daughter) nucleus by emitting an electron and a neutrino. The parent nucleus was at rest at the origin of an $x y$ coordinate system. The electron moves away from the origin with linear momentum $\left(-1.2 \times 10^{-22} \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}\right) \hat{i}$; the neutrino moves away from the origin with linear momentum $\left(-6.4 \times 10^{-23} \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}\right) \hat{\mathrm{j}} .$ What are the (a) magnitude and (b) direction of the linear momentum of the daughter nucleus? (c) If the daughter nucleus has a mass of $5.8 \times$ $10^{-26} \mathrm{~kg}$, what is its kinetic energy?

Zachary Warner

Numerade Educator

Problem 91

A $75 \mathrm{~kg}$ man rides on a $39 \mathrm{~kg}$ cart moving at a velocity of $2.3 \mathrm{~m} / \mathrm{s}$ He jumps off with zero horizontal velocity relative to the ground. What is the resulting change in the cart's velocity, including sign?

Salamat Ali

Numerade Educator

Problem 92

Two blocks of masses $1.0 \mathrm{~kg}$ and $3.0 \mathrm{~kg}$ are connected by a spring and rest on a frictionless surface. They are given velocities toward each other such that the $1.0 \mathrm{~kg}$ block travels initially at $1.7 \mathrm{~m} / \mathrm{s}$ toward the center of mass, which remains at rest. What is the initial speed of the other block?

Zachary Warner

Numerade Educator

Problem 93

A railroad freight car of mass $3.18 \times 10^{4} \mathrm{~kg}$ collides with a stationary caboose car. They couple together, and $27.0 \%$ of the initial kinetic energy is transferred to thermal energy, sound, vibrations, and so on. Find the mass of the caboose.

Salamat Ali

Numerade Educator

Problem 94

An old Chrysler with mass $2400 \mathrm{~kg}$ is moving along a straight stretch of road at $80 \mathrm{~km} / \mathrm{h}$. It is followed by a Ford with mass 1600 kg moving at $60 \mathrm{~km} / \mathrm{h}$. How fast is the center of mass of the two cars moving?

Zachary Warner

Numerade Educator

Problem 95

In the arrangement of Fig. 9-21, billiard ball 1 moving at a speed of $2.2 \mathrm{~m} / \mathrm{s}$ undergoes a glancing collision with identical billiard ball 2 that is at rest. After the collision, ball 2 moves at speed $1.1 \mathrm{~m} / \mathrm{s}$, at an angle of $\theta_{2}=60^{\circ}$. What are (a) the magnitude and (b) the direction of the velocity of ball 1 after the collision? (c) Do the given data suggest the collision is elastic or inelastic?

Salamat Ali

Numerade Educator

Problem 96

A rocket is moving away from the solar system at a speed of $6.0 \times 10^{3} \mathrm{~m} / \mathrm{s}$. It fires its engine, which ejects exhaust with a speed of $3.0 \times 10^{3} \mathrm{~m} / \mathrm{s}$ relative to the rocket. The mass of the rocket at this time is $4.0 \times 10^{4} \mathrm{~kg}$, and its acceleration is $2.0 \mathrm{~m} / \mathrm{s}^{2}$. (a) What is the thrust of the engine? (b) At what rate, in kilograms per second, is exhaust ejected during the firing?

Zachary Warner

Numerade Educator

Problem 97

The three balls in the overhead view of Fig. $9-76$ are identical. Balls 2 and 3 touch each other and are aligned perpendicular to the path of ball 1 The velocity of ball 1 has magnitude $v_{0}=10 \mathrm{~m} / \mathrm{s}$ and is directed at the contact point of balls 1 and 2 . After the collision, what are the
(a) speed and (b) direction of the velocity of ball 2, the (c) speed and (d) direction of the velocity of ball 3 , and the (e) speed and (f) direction of the velocity of ball 1? (Hint: With friction absent, each impulse is directed along the line connecting the centers of the colliding balls, normal to the colliding surfaces.)

Salamat Ali

Numerade Educator

Problem 98

A $0.15 \mathrm{~kg}$ ball hits a wall with a velocity of $(5.00 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{i}}+(6.50$
$\mathrm{m} / \mathrm{s}) \hat{\mathrm{j}}+(4.00 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{k}}$. It rebounds from the wall with a velocity of $(2.00 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{i}}+(3.50 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{j}}+(-3.20 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{k}} .$ What are
(a) the change in the ball's momentum, (b) the impulse on the ball, and (c) the impulse on the wall?

Zachary Warner

Numerade Educator

Problem 99

In Fig. 9-77, two identical containers of sugar are connected by a cord that passes over a frictionless pulley. The cord and pulley have negligible mass, each container and its sugar together have a mass of $500 \mathrm{~g}$, the centers of the containers are separated by $50 \mathrm{~mm}$, and the containers are held fixed at the same height. What is the horizontal distance between the center of container 1 and the center of mass of the two-container system (a) initially and (b) after $20 \mathrm{~g}$ of sugar is transferred from container 1 to container 2? After the transfer and after the containers are released, (c) in what direction and (d) at what acceleration magnitude does the center of mass move?

Salamat Ali

Numerade Educator

Problem 100

In a game of pool, the cue ball strikes another ball of the same mass and initially at rest. After the collision, the cue ball moves at $3.50 \mathrm{~m} / \mathrm{s}$ along a line making an angle of $22.0^{\circ}$ with the cue ball's original direction of motion, and the second ball has a speed of $2.00 \mathrm{~m} / \mathrm{s}$. Find (a) the angle between the direction of motion of the second ball and the original direction of motion of the cue ball and (b) the original speed of the cue ball. (c) Is kinetic energy (of the centers of mass, don't consider the rotation) conserved?

Zachary Warner

Numerade Educator

Problem 101

In Fig. $9-78$, a $3.2 \mathrm{~kg}$ box of running shoes slides on a horizontal frictionless table and collides with a $2.0 \mathrm{~kg}$ box of ballet slippers initially at rest on the edge of the table, at height $h=0.40 \mathrm{~m}$. The speed of the $3.2 \mathrm{~kg}$ box is $3.0 \mathrm{~m} / \mathrm{s}$ just before the collision. If the two boxes stick together because of packing tape on their sides, what is their kinetic energy just before they strike the floor?

Salamat Ali

Numerade Educator

Problem 102

In Fig. $9-79$, an $80 \mathrm{~kg}$ man is on a ladder hanging from a balloon that has a total mass of $320 \mathrm{~kg}$ (including the basket passenger). The balloon is initially stationary relative to the ground. If the man on the ladder begins to climb at $2.5 \mathrm{~m} / \mathrm{s}$ relative to the ladder, (a) in what direction and (b) at what speed does the balloon move? (c) If the man then stops climbing. what is the speed of the balloon?

Zachary Warner

Numerade Educator

Problem 103

In Fig. $9-80$, block 1 of mass $m_{1}=6.6 \mathrm{~kg}$ is at rest on a long frictionless table that is up against a wall. Block 2 of mass $m_{2}$ is placed between block 1 and the wall and sent sliding to the left, toward block 1 , with constant speed $v_{2 j}$. Find the value of $m_{2}$ for which both blocks move with the same velocity after block 2 has collided once with block 1 and once with the wall. Assume all collisions are elastic (the collision with the wall does not change the speed of block 2).

Keshav Singh

Numerade Educator

Problem 104

The script for an action movie calls for a small race car (of mass $1500 \mathrm{~kg}$ and length $3.0 \mathrm{~m}$ ) to accelerate along a flattop boat (of mass $4000 \mathrm{~kg}$ and length $14 \mathrm{~m}$ ), from one end of the boat to the other, where the car will then jump the gap between the boat and a somewhat lower dock. You are the technical advisor for the movie. The boat will initially touch the dock, as in Fig. 9-81; the boat can slide through the water without significant resistance; both the car and the boat can be approximated as uniform in their mass distribution. Determine what the width of the gap will be just as the car is about to make the jump.

Keshav Singh

Numerade Educator

Problem 105

A $3.0 \mathrm{~kg}$ object moving at $8.0 \mathrm{~m} / \mathrm{s}$ in the positive direction of an $x$ axis has a one-dimensional elastic collision with an object of mass $M$, initially at rest. After the collision the object of mass $M$ has a velocity of $6.0 \mathrm{~m} / \mathrm{s}$ in the positive direction of the axis. What is mass $M ?$

Vishal Gupta

Numerade Educator

Problem 106

A $2140 \mathrm{~kg}$ railroad flatcar, which can move with negligible friction, is motionless next to a platform. A $242 \mathrm{~kg}$ sumo wrestler runs at $5.3 \mathrm{~m} / \mathrm{s}$ along the platform (parallel to the track) and then jumps onto the flatcar. What is the speed of the flatcar if he then
(a) stands on it, (b) runs at $5.3 \mathrm{~m} / \mathrm{s}$ relative to it in his original direction, and (c) turns and runs at $5.3 \mathrm{~m} / \mathrm{s}$ relative to the flatcar opposite his original direction?

Keshav Singh

Numerade Educator

Problem 107

A $6100 \mathrm{~kg}$ rocket is set for vertical firing from the ground. If the exhaust speed is $1200 \mathrm{~m} / \mathrm{s}$, how much gas must be ejected each second if the thrust (a) is to equal the magnitude of the gravitational force on the rocket and (b) is to give the rocket an initial upward acceleration of $21 \mathrm{~m} / \mathrm{s}^{2}$ ?

Salamat Ali

Numerade Educator

Problem 108

A $500.0 \mathrm{~kg}$ module is attached to a $400.0 \mathrm{~kg}$ shuttle craft, which moves at $1000 \mathrm{~m} / \mathrm{s}$ relative to the stationary main spaceship. Then a small explosion sends the module backward with speed $100.0 \mathrm{~m} / \mathrm{s}$ relative to the new speed of the shuttle craft. As measured by someone on the main spaceship, by what fraction did the kinetic energy of the module and shuttle craft increase because of the explosion?

Zachary Warner

Numerade Educator

Problem 109

(a) How far is the center of mass of the Earth-Moon system from the center of Earth? (Appendix C gives the masses of Earth and the Moon and the distance between the two.) (b) What percentage of Earth's radius is that distance?

Salamat Ali

Numerade Educator

Problem 110

A $140 \mathrm{~g}$ ball with speed $7.8 \mathrm{~m} / \mathrm{s}$ strikes a wall perpendicularly and rebounds in the opposite direction with the same speed. The collision lasts $3.80 \mathrm{~ms}$. What are the magnitudes of the (a) impulse and (b) average force on the wall from the ball during the elastic collision?

Zachary Warner

Numerade Educator

Problem 111

A rocket sled with a mass of $2900 \mathrm{~kg}$ moves at $250 \mathrm{~m} / \mathrm{s}$ on a set of rails. At a certain point, a scoop on the sled dips into a trough of water located between the tracks and scoops water into an empty tank on the sled. By applying the principle of conservation of linear momentum, determine the speed of the sled after $920 \mathrm{~kg}$ of water has been scooped up. Ignore any retarding force on the scoop.

Salamat Ali

Numerade Educator

Problem 112

A pellet gun fires ten $2.0 \mathrm{~g}$ pellets per second with a speed of $500 \mathrm{~m} / \mathrm{s}$. The pellets are stopped by a rigid wall. What are
(a) the magnitude of the momentum of each pellet, (b) the kinetic energy of each pellet, and (c) the magnitude of the average force on the wall from the stream of pellets? (d) If each pellet is in contact with the wall for $0.60 \mathrm{~ms}$, what is the magnitude of the average force on the wall from each pellet during contact? (e) Why is this average force so different from the average force calculated in (c)?

Zachary Warner

Numerade Educator

Problem 113

A railroad car moves under a grain elevator at a constant speed of $3.20 \mathrm{~m} / \mathrm{s}$. Grain drops into the car at the rate of $540 \mathrm{~kg} / \mathrm{min} .$ What is the magnitude of the force needed to keep the car moving at constant speed if friction is negligible?

Salamat Ali

Numerade Educator

Problem 114

Figure $9-82$ shows a uniform square plate of edge length $6 d=6.0 \mathrm{~m}$ from which a square piece of edge length $2 d$ has been removed. What are (a) the $x$ coordinate and (b) the $y$ coordinate of the center of mass of the remaining piece?

Zachary Warner

Numerade Educator

Problem 115

At time $t=0$, force $\vec{F}_{1}=(-4.00 \hat{\mathrm{i}}+5.00 \hat{\mathrm{j}}) \mathrm{N}$ acts on an
initially stationary particle of mass $2.00 \times 10^{-3} \mathrm{~kg}$ and force $\vec{F}_{2}=(2.00 \hat{i}-4.00 \mathrm{j}) \mathrm{N}$ acts on an initially stationary particle of mass $4.00 \times 10^{-3} \mathrm{~kg}$. From time $t=0$ to $t=2.00 \mathrm{~ms}$, what are the
(a) magnitude and (b) angle (relative to the positive direction of the $x$ axis) of the displacement of the center of mass of the twoparticle system? (c) What is the kinetic energy of the center of mass at $t=2.00 \mathrm{~ms}$ ?

Tanner Manwaring

Tanner Manwaring

Numerade Educator

Problem 116

Two particles $P$ and $Q$ are released from rest $1.0 \mathrm{~m}$ apart. $P$ has a mass of $0.10 \mathrm{~kg}$, and $Q$ a mass of $0.30 \mathrm{~kg} . P$ and $Q$ attract each other with a constant force of $1.0 \times 10^{-2} \mathrm{~N}$. No external forces act on the system. (a) What is the speed of the center of mass of $P$ and $Q$ when the separation is $0.50 \mathrm{~m}$ ? (b) At what distance from $P$ 's original position do the particles collide?

Zachary Warner

Numerade Educator

Problem 117

A collision occurs between a $2.00 \mathrm{~kg}$ particle traveling with velocity $\vec{v}_{1}=(-4.00 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{i}}+(-5.00 \mathrm{~m} / \mathrm{s}) \mathrm{j}$ and a $4.00 \mathrm{~kg}$ particle
traveling with velocity $\vec{v}_{2}=(6.00 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{i}}+(-2.00 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{j}} .$ The collision connects the two particles. What then is their velocity in (a) unit-vector notation and as a (b) magnitude and (c) angle?

Keshav Singh

Numerade Educator

Problem 118

In the two-sphere arrangement of Fig. $9-20$, assume that sphere 1 has a mass of $50 \mathrm{~g}$ and an initial height of $h_{1}=9.0 \mathrm{~cm}$, and that sphere 2 has a mass of $85 \mathrm{~g}$. After sphere 1 is released and collides elastically with sphere 2, what height is reached by (a) sphere 1 and (b) sphere 2 ? After the next (elastic) collision, what height is reached by (c) sphere 1 and (d) sphere 2 ? (Hint: Do not use rounded-off values.)

Zachary Warner

Numerade Educator

Problem 119

In Fig. $9-83$, block 1 slides along an $x$ axis on a frictionless floor with a speed of $0.75 \mathrm{~m} / \mathrm{s}$. When it reaches stationary block 2, the two blocks undergo an elastic collision. The following table gives the mass and length of the (uniform) blocks and a so the locations of their centers at time $t=0 .$ Where is the center of mass of the two-block system located (a) at $t=0,(\mathrm{~b})$ when the two blocks first touch, and $(\mathrm{c})$ at $t=4.0 \mathrm{~s}$ ?

Tanner Manwaring

Tanner Manwaring

Numerade Educator

Problem 120

A body is traveling at $2.0 \mathrm{~m} / \mathrm{s}$ along the positive direction of an $x$ axis; no net force acts on the body. An internal explosion separates the body into two parts, each of $4.0 \mathrm{~kg}$, and increases the total kinetic energy by $16 \mathrm{~J}$. The forward part continues to move in the original direction of motion. What are the speeds of (a) the rear part and (b) the forward part?

Zachary Warner

Numerade Educator

Problem 121

An electron undergoes a one-dimensional elastic collision with an initially stationary hydrogen atom. What percentage of the electron's initial kinetic energy is transferred to kinetic energy of the hydrogen atom? (The mass of the hydrogen atom is 1840 times the mass of the electron.)

Salamat Ali

Numerade Educator

Problem 122

A man (weighing $915 \mathrm{~N}$ ) stands on a long railroad flatcar (weighing $2415 \mathrm{~N}$ ) as it rolls at $18.2 \mathrm{~m} / \mathrm{s}$ in the positive direction of an $x$ axis, with negligible friction. Then the man runs along the flatcar in the negative $x$ direction at $4.00 \mathrm{~m} / \mathrm{s}$ relative to the flatcar. What is the resulting increase in the speed of the flatcar?

Zachary Warner

Numerade Educator

Problem 123

An unmanned space probe (of mass $m$ and speed $v$ relative to the Sun) approaches the planet Jupiter (of mass $M$ and speed $V_{J}$ relative to the Sun) as shown in Fig. $9-84$. The spacecraft rounds the planet and departs in the opposite direction. What is its speed (in kilometers per second), relative to the Sun, after this slingshot encounter, which can be analyzed as a collision? Assume $v=10.5 \mathrm{~km} / \mathrm{s}$ and $V_{J}=13.0 \mathrm{~km} / \mathrm{s}$ (the orbital speed of Jupiter). The mass of Jupiter is very much greater than the mass of the spacecraft $(M \gg m)$.

Salamat Ali

Numerade Educator

Problem 124

A $0.550 \mathrm{~kg}$ ball falls directly down onto concrete, hitting it with a speed of $12.0 \mathrm{~m} / \mathrm{s}$ and rebounding directly upward with a speed of $3.00 \mathrm{~m} / \mathrm{s}$. Extend a $y$ axis upward. In unit-vector notation, what are (a) the change in the ball's momentum, (b) the impulse on the ball, and (c) the impulse on the concrete?

Zachary Warner

Numerade Educator

Problem 125

An atomic nucleus at rest at the origin of an $x y$ coordinate system transforms into three particles. Particle 1, mass $16.7 \times 10^{-27}$ $\mathrm{kg}$, moves away from the origin at velocity $\left(6.00 \times 10^{6} \mathrm{~m} / \mathrm{s}\right) \hat{\mathrm{i}}$; particle 2, mass $8.35 \times 10^{-27} \mathrm{~kg}$, moves away at velocity $\left(-8.00 \times 10^{6} \mathrm{~m} / \mathrm{s}\right)$ )
(a) In unit-vector notation, what is the linear momentum of the third particle, mass $11.7 \times 10^{-27} \mathrm{~kg} ?$ (b) How much kinetic energy appears in this transformation?

Salamat Ali

Numerade Educator

Problem 126

Particle 1 of mass $200 \mathrm{~g}$ and speed $3.00 \mathrm{~m} / \mathrm{s}$ undergoes a onedimensional collision with stationary particle 2 of mass $400 \mathrm{~g}$. What is the magnitude of the impulse on particle 1 if the collision is (a) elastic and (b) completely inelastic?

Keshav Singh

Numerade Educator

Problem 127

During a lunar mission, it is necessary to increase the speed of a spacecraft by $2.2 \mathrm{~m} / \mathrm{s}$ when it is moving at $400 \mathrm{~m} / \mathrm{s}$ relative to the Moon. The speed of the exhaust products from the rocket engine is $1000 \mathrm{~m} / \mathrm{s}$ relative to the spacecraft. What fraction of the initial mass of the spacecraft must be burned and ejected to accomplish the speed increase?

Salamat Ali

Numerade Educator

Problem 128

A cue stick strikes a stationary pool ball, with an average force of $32 \mathrm{~N}$ over a time of $14 \mathrm{~ms}$. If the ball has mass $0.20 \mathrm{~kg}$, what speed does it have just after impact?

Zachary Warner

Numerade Educator