Principles 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

Three particles of mass $1.0 \mathrm{~kg}, 2.0 \mathrm{~kg}$, and $3.0 \mathrm{~kg}$ are placed at the vertices $A, B$, and $C$, respectively, of an equilateral triangle $A B C$ of edge $1.0 \mathrm{~m}$ (Fig 9-23). Find the distance of their center of mass from $A$.

Amna Khalid

Amna Khalid

Numerade Educator

Problem 2

Figure 9-24 shows a three-particle system, with masses $m_{1}=2.0 \mathrm{~kg}, m_{2}=4.0$ $\mathrm{kg}$, 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 $m_{3}$ is gradually increased, does the center of mass of the system shift toward or away from that particle, or does it remain stationary?

Amna Khalid

Numerade Educator

Problem 3

Figure 9-25 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 $=$ $\left.7.85 \mathrm{~g} / \mathrm{cm}^{3}\right)$. 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. 9-26, three uniform thin rods, each of length $L=24 \mathrm{~cm}$, 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?

Amna Khalid

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. 9-27 if $L=5.0 \mathrm{~cm}$ ?

Amna Khalid

Numerade Educator

Problem 6

Figure 9-28 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 $L=50 \mathrm{~cm}$. Find (a) the $x$ coordinate, (b) the $y$ coordinate, and (c) the $z$ coordinate of the center of mass of the box.

Amna Khalid

Numerade Educator

Problem 7

In the ammonia $\left(\mathrm{NH}_{3}\right)$ molecule of Fig. 9-29, 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-tohydrogen 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?

Amna Khalid

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. 9-30). 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.

Amna Khalid

Numerade Educator

Problem 9

In the arrangement shown in Fig. $9-31, m_{A}=2.0 \mathrm{~kg}$ and $m_{B}=1.0 \mathrm{~kg}$. The pulley is massless; the string is massless and long. The system is released at $t=0 \mathrm{~s}$. Find (a) the acceleration of the center of mass of the blocks, (b) the displacement of the center of mass at $t=2.0 \mathrm{~s}$, and (c) the speed of the center of mass when $m_{A}$ strikes the floor.

Sahil Kumar

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 $3.0 \mathrm{~m} / \mathrm{s}^{2}$. 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 automobiletruck system from the traffic light at $t=$ $5.0 \mathrm{~s} ?$ (b) What is the speed of the com then?

Amna Khalid

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 $\vec{F}_{o}=(2.0 \hat{\mathrm{i}}+3.0 \hat{\mathrm{j}}) \mathrm{N}$ 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 $t=0$ ?

Salamat Ali

Numerade Educator

Problem 12

Two skaters, one with mass $75 \mathrm{~kg}$ 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 $40 \mathrm{~kg}$ skater move?

Amna Khalid

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. 9-32). 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 9-33, 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 $H_{\max }$ 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 $H_{\max }$ ?

Zachary Warner

Numerade Educator

Problem 15

Figure 9-34 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 $45 \mathrm{~cm}$ horizontally relative to a pier post, what is
(a) Carmelita's mass?

Amna Khalid

Numerade Educator

Problem 17

In Fig. 9-35a, a 4.5 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. 9-35b.)

Amna Khalid

Numerade Educator

Problem 18

A $0.70 \mathrm{~kg}$ ball moving horizontally at $6.0 \mathrm{~m} / \mathrm{s}$ strikes a vertical wall and rebounds with speed $3.5 \mathrm{~m} / \mathrm{s}$. What is the magnitude of the change in its linear momentum?

Sahil Kumar

Numerade Educator

Problem 19

19 A $100 \mathrm{~kg}$ motorbike moves along $A B$ at $10.0 \mathrm{~km} / \mathrm{h}$, and after some time, the motorbike turns to travel along $B C$ at the same speed as shown in Fig. 9-36. Find (a) the change in its kinetic energy and the (b) magnitude and (c) direction (relative to $+x$ ) of the change in its momentum.

Sahil Kumar

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. 9-37 (with $p_{0}=6.0 \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}$ and $p_{1}=$ $4.0 \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s})$. At what initial angle is the ball launched? (Hint: Find a solution that does not require you to read the time corresponding to the low point of the plot.)

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 21

A ball of mass $50 \mathrm{~g}$ moving with a speed of $2.0 \mathrm{~m} / \mathrm{s}$ strikes a wall at an angle of incidence $45^{\circ}$ and is reflected from the wall at the same angle and with the same speed. See the overhead view in Fig. 9-38. Calculate (a) the magnitude of the change $\Delta \vec{p}$ in the momentum of the ball, (b) the change in the magnitude of the momentum $\vec{p}$ of the ball, and (c) the change in the magnitude of the momentum of the wall.

Amna Khalid

Numerade Educator

Problem 22

Figure 9-39 gives an overhead view of the path taken by a $0.150 \mathrm{~kg}$ 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.)

Amna Khalid

Numerade Educator

Problem 23

Until his seventies, Henri LaMothe (Fig. 9-40) 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 $5.00 \mathrm{~g}$ bullet moving at $100 \mathrm{~m} / \mathrm{s}$ strikes a log. Assume that the bullet undergoes a uniform deceleration and stops after penetrating $6.00 \mathrm{~cm}$. Find (a) the time taken by the bullet to stop, (b) the impulse on the log, and (c) the magnitude of the average force experienced by the log.

Sahil Kumar

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 $75 \mathrm{~kg}$, and the collision on the floor lasts $0.088 \mathrm{~s}$. What are the magnitudes of the (a) impulse and (b) average force acting on the victim from the floor during the collision?

Amna Khalid

Numerade Educator

Problem 27

A $3.00 \mathrm{~kg}$ block slides on a frictionless horizontal surface, first moving to the left at $50.0 \mathrm{~m} / \mathrm{s}$. It collides with a spring whose other end is fixed to a wall, compresses the spring, and is brought to rest momentarily. Then it continues to be accelerated toward the right by the force of the compressed spring. The block acquires a final speed of $40.0 \mathrm{~m} / \mathrm{s}$. It is in contact with the spring for $0.020 \mathrm{~s}$. Find (a) the magnitude and (b) the direction of the impulse of the spring force on the block. (c) What is the magnitude of the spring's average force on the block?

Khoobchandra Agrawal

Khoobchandra Agrawal

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 $5.5 \mathrm{~ms}$ 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?

Amna Khalid

Numerade Educator

Problem 29

A ball of mass $1.00 \mathrm{~kg}$ is attached to a loose string fixed to a ceiling. The ball is released from rest and falls $2.00 \mathrm{~m}$, where the string suddenly stops it. Find the impulse on it from the string.

Sahil Kumar

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 9-41 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 $\Delta t_{r}=50.0 \mathrm{~ms}$, 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 $F_{\max }=$ $160 \mathrm{~N}$. 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?

Amna Khalid

Numerade Educator

Problem 31

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)?

Amna Khalid

Numerade Educator

Problem 32

A $2.5 \mathrm{~kg}$ toy car can move along an $x$ axis; Fig. 9-42 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 $F_{x s}=5.0 \mathrm{~N}$. In unit-vector notation, what is $\vec{p}$ at (a) $t=4.0 \mathrm{~s}$ and (b) $t=7.0 \mathrm{~s}$, and (c) what is $\vec{v}$ at $t=9.0 \mathrm{~s}$ ?

Sahil Kumar

Numerade Educator

Problem 33

Figure 9-43 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. 9-44). 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 9-45 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 $F_{\max }$, the maximum magnitude of the force on the ball from the wall during the collision?

Amna Khalid

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 $\vec{F}=\left(12.0-3.00 t^{2}\right) \hat{i}$, 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 $t=0.750 \mathrm{~s}$ and $t=1.25 \mathrm{~s}$ ? (b) What is the change in momentum of the puck between $t=0$ and the instant at which $F=0$ ?

Amna Khalid

Numerade Educator

Problem 37

A particle of unknown mass is acted upon by a force $\vec{F}=\left(100 e^{-2 \hat{\mathrm{i}}}\right) \mathrm{N}$. If at $t=0.00 \mathrm{~s}$ the particle is at rest, for the time interval $t=0.00 \mathrm{~s}$ to $t=2.00 \mathrm{~s}$ find (a) the impulse on the particle and (b) the average force on the particle.

Sahil Kumar

Numerade Educator

Problem 38

In the overhead view of Fig. $9-46$, a $300 \mathrm{~g}$ ball with a speed $v$ of $6.0 \mathrm{~m} / \mathrm{s}$ strikes a wall at an angle $\theta$ 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 man of mass $m_{1}=80 \mathrm{~kg}$ is standing on a platform of mass $m_{2}=20 \mathrm{~kg}$ that lies on a frictionless horizontal surface. The man starts moving on the platform with a velocity $v_{r}=10 \mathrm{~m} / \mathrm{s}$ relative to the platform. Find the recoil speed of the platform.

Sahil Kumar

Numerade Educator

Problem 40

A space vehicle is traveling at $4800 \mathrm{~km} / \mathrm{h}$ relative to Earth when the exhausted rocket motor (mass $4 \mathrm{~m}$ ) 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?

Amna Khalid

Numerade Educator

Problem 41

Figure 9-47 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

A $15.0 \mathrm{~kg}$ package is moving at a speed of $10.0 \mathrm{~m} / \mathrm{s}$ vertically upward along a $y$ axis when it explodes into three fragments: a $2.00$ $\mathrm{kg}$ fragment is shot upward with an initial speed of $20.0 \mathrm{~m} / \mathrm{s}$ and a $3.00 \mathrm{~kg}$ fragment is shot in the positive direction of a horizontal $x$ axis with an initial speed of $5.00 \mathrm{~m} / \mathrm{s}$. Find (a) the speed of the third fragment right after the explosion and (b) the total kinetic energy provided by the explosion.

Amna Khalid

Numerade Educator

Problem 43

In the Olympiad of 708 B.C., some athletes competing in the standing long jump used handheld weights called halteres to lengthen their jumps (Fig. 9-48). 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 $\vec{v}=(9.5 \hat{i}+4.0 \mathrm{j}) \mathrm{m} / \mathrm{s}$ 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?

Amna Khalid

Numerade Educator

Problem 44

In Fig. 9-49, a stationary block explodes into two pieces $L$ 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 $\mu_{L}=0.35$ 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 $d_{R}=0.30 \mathrm{~m}$. What was the mass of the block?

Amna Khalid

Numerade Educator

Problem 45

A vase of mass $m$ falls onto a floor and breaks into three pieces that then slide across the frictionless floor. One piece of mass $0.25 m$ moves at speed $v$ along an $x$ axis. The second piece of the same mass and speed moves along the $y$ axis. Find the speed of the third piece.

Khoobchandra Agrawal

Khoobchandra Agrawal

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 $6.0 \mathrm{~m} / \mathrm{s}, 30^{\circ}$ north of east. What is the original speed of the mess kit?

Amna Khalid

Numerade Educator

Problem 47

A particle of mass $2.0 \mathrm{~m}$ is projected at an angle of $45^{\circ}$ with the horizontal with a speed of $20 \sqrt{2} \mathrm{~m} / \mathrm{s}$. After $1.0 \mathrm{~s}$, an explosion occurs and the particle is broken into two equal pieces. One piece is momentarily at rest before it falls. Find the maximum height attained by the other piece.

Amna Khalid

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 $80 \mathrm{~J}$. 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 $B$ ?

Amna Khalid

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 $700 \mathrm{~m} / \mathrm{s}$ 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 $450 \mathrm{~m} / \mathrm{s}$. (a) What is the resulting speed of the block? (b) What is the speed of the bullet-block center of mass?

Amna Khalid

Numerade Educator

Problem 51

In Fig. $9-50 a$, 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. $9-50 b) .$ 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. 9-51, 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 300 $\mathrm{m} / \mathrm{s}$. To what maximum height does the block then rise above its initial position?

Amna Khalid

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 percentage 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 percentage of the original kinetic energy is lost if the car hits a $300 \mathrm{~kg}$ camel? (c) Generally, does the percentage loss increase or decrease if the animal mass decreases?

Amna Khalid

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 $10 \mathrm{~m} / \mathrm{s}$. How high do the combined two balls of putty rise above the collision point? (Neglect air drag.)

Amna Khalid

Numerade Educator

Problem 55

Block 1 of mass $3.0 \mathrm{~kg}$ is sliding across a floor with speed $v_{1}=2.0 \mathrm{~m} / \mathrm{s}$ when it makes a head-on, one-dimensional, elastic collision with initially stationary block 2 of mass $2.0 \mathrm{~kg}$. The coefficient of kinetic friction between the blocks and the floor is $\mu_{k}=$ $0.30$. Find the speeds of (a) block 1 and (b) block 2 just after the collision. Also find (c) their final separation after friction has stopped them and $(\mathrm{d})$ the energy lost to thermal energy because of the friction.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 56

In the "before" part of Fig. 9-52, 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 $0.10$ ) 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.

Amna Khalid

Numerade Educator

Problem 57

In Fig. 9-53, 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. 9-54, block 2 (mass 1.0 kg) is at rest on a frictionless surface and touching the end of an unstretched spring of spring constant $230 \mathrm{~N} / \mathrm{m}$. 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?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 59

In Fig. 9-55, 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. 9-56, block $A$ (mass $1.6$ $\mathrm{kg}$ ) 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 $v_{A i}=5.5 \mathrm{~m} / \mathrm{s}, \quad v_{B i}=2.5 \mathrm{~m} / \mathrm{s}$, 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

Two bodies of masses $m=0.30 \mathrm{~kg}$ and $2 m$ are connected by a long string of negligible mass. The string is looped over a pulley and, with the string taut, the bodies are released at time $t=0$ so that the heavier one descends and the lighter one ascends. At time $t=4.0 \mathrm{~s}$, the lighter one undergoes a fully inelastic collision with a third body of mass $m$. Because the first two bodies move in rigid fashion, the collision is effectively between the third body and the system of the first two bodies. (a) Just after the collision, what is the speed of the three bodies? (b) By how much was the kinetic energy of the descending body decreased because of the collision?

Khoobchandra Agrawal

Khoobchandra Agrawal

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 $250 \mathrm{~g}$, remains at rest. (a) What is the mass of the other sphere? (b) What is the speed of the twosphere center of mass if the initial speed of each sphere is $2.00 \mathrm{~m} / \mathrm{s}$ ?

Amna Khalid

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 $0.600 \mathrm{~kg}$ 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. 9-57). At the bottom of its path, the ball strikes a $2.80 \mathrm{~kg}$ 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.

Amna Khalid

Numerade Educator

Problem 65

Particle 1 with mass $m$ and velocity $v$ and particle 2 with mass $2 m$ and velocity $-2 v$ are moving toward each other along an $x$ axis when they undergo a one-dimensional elastic collision. After the collision, what are the velocities of (a) particle 1 and (b) particle $2 ?$ What is the velocity of the center of mass of the two-particle system (c) before and (d) after the collision?

Amna Khalid

Numerade Educator

Problem 66

Block 1 , with mass $m_{1}$ and speed $3.0 \mathrm{~m} / \mathrm{s}$, 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?

Amna Khalid

Numerade Educator

Problem 67

In Fig. 9-58, 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. 9-59, block 1 of mass $m_{1}$ slides from rest along a frictionless ramp from height $h=3.00 \mathrm{~m}$ 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 $0.450$ 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?

Amna Khalid

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. $9-60 a$ ), 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. 9-60b)?

Amna Khalid

Numerade Educator

Problem 70

In Fig. 9-61, puck 1 of mass $m_{1}=0.25 \mathrm{~kg}$ 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.)

Amna Khalid

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

In the two-dimensional collision in Fig. 9-21, the projectile particle has mass $m_{1}=m$, initial speed $v_{1 i}=3 v_{0}$, and final speed $v_{1 f}$ $=\sqrt{5} v_{0}$. The initially stationary target particle has mass $m_{1}=2 m$ and final speed $v_{2 f}=v_{2}$. The projectile is scattered at an angle given by $\tan \theta_{1}=2.0$. (a) Find angle $\theta_{2}$. (b) Find $v_{2}$ in terms of $v_{0}$. (c) Is the collision elastic?

Khoobchandra Agrawal

Khoobchandra Agrawal

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

A force $\vec{F}$ acts on two particles of masses $m$ and $4.0 m$ moving at the same speed but at right angles to each other, as shown in Fig. 9-62. The force acts on both the particles for a time $T$. Consequently, the particle of mass $m$ moves with a velocity $4 v$ in its original direction. (a) Find the new velocity $v^{\prime}$ of the other particle. (b) Also find the change in the kinetic energy of the system.

Amna Khalid

Numerade Educator

Problem 75

In Fig. 9-63, a bob of mass $10 \mathrm{~m}$ is suspended from an inextensible string with negligible mass. When the bob is in equilibrium (at rest), two particles each of mass $m$ strike it simultaneously with the speeds indicated. The particles stick to the bob. Find (a) the magnitude of the net impulse on the string due to the collision, (b) the velocity of the system just after the collision, and (c) the mechanical energy lost in the collision.

Amna Khalid

Numerade Educator

Problem 76

A $6090 \mathrm{~kg}$ space probe moving nose-first toward Jupiter at $120 \mathrm{~m} / \mathrm{s}$ relative to the Sun fires its rocket engine, ejecting $70.0 \mathrm{~kg}$ of exhaust at a speed of $253 \mathrm{~m} / \mathrm{s}$ relative to the space probe. What is the final velocity of the probe?

Amna Khalid

Numerade Educator

Problem 77

In Fig. 9-64, 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 initial 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?

Zachary Warner

Numerade Educator

Problem 79

A rocket that is set for a vertical launch has a mass of $50.0 \mathrm{~kg}$ and contains $450 \mathrm{~kg}$ of fuel. The rocket can have a maximum exhaust velocity of $2.00 \mathrm{~km} / \mathrm{s}$. What should be the minimum rate of fuel consumption (a) to just lift it off the launching pad and (b) to give it an acceleration of $20.0 \mathrm{~m} / \mathrm{s}^{2} ?$ (c) If the consumption rate is set at $10.0 \mathrm{~kg} / \mathrm{s}$, what is the rocket speed at the moment when the fuel is fully consumed?

Amna Khalid

Numerade Educator