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Fundamentals of Physics

David Halliday, Robert Resnick, Jearl Walker

Chapter 9

Linear Momentum and Collisions - all with Video Answers

Educators

Chapter Questions

05:00

Problem 1

A $3.00-\mathrm{kg}$ particle has a velocity of $(3.00 \mathbf{i}-4.00 \mathrm{j}) \mathrm{m} / \mathrm{s}$.
(a) Find its $x$ and $y$ components of momentum.
(b) Find the magnitude and direction of its momentum.

Rodrigo Diaz-Meneses
Rodrigo Diaz-Meneses
Numerade Educator
03:53

Problem 2

A $0.100-\mathrm{kg}$ ball is thrown straight up into the air with an initial speed of $15.0 \mathrm{~m} / \mathrm{s}$. Find the momentum of the ball (a) at its maximum height and (b) halfway up to its maximum height.

Nishant Kumar
Nishant Kumar
Numerade Educator
03:44

Problem 3

A $40.0$ kg child standing on a frozen pond throws a $0.500-\mathrm{kg}$ stone to the east with a speed of $5.00 \mathrm{~m} / \mathrm{s}$. Neglecting friction between child and ice, find the recoil velocity of the child.

Rodrigo Diaz-Meneses
Rodrigo Diaz-Meneses
Numerade Educator
02:09

Problem 4

A pitcher claims he can throw a baseball with as much momentum as a $3.00-\mathrm{g}$ bullet moving with a speed of $1500 \mathrm{~m} / \mathrm{s}$. A baseball has a mass of $0.145 \mathrm{~kg}$. What must be its speed if the pitcher's claim is valid?

Rodrigo Diaz-Meneses
Rodrigo Diaz-Meneses
Numerade Educator
03:42

Problem 5

How fast can you set the Earth moving? In particular, when you jump straight up as high as you can, you give the Earth a maximum recoil speed of what order of magnitude? Model the Earth as a perfectly solid object. In your solution, state the physical quantities you take as data and the values you measure or estimate for them.

Sheh Lit Chang
Sheh Lit Chang
University of Washington
06:47

Problem 6

Two blocks of masses $M$ and $3 M$ are placed on a horizontal, frictionless surface. A light spring is attached to one of them, and the blocks are pushed together with the spring between them (Fig. P9.6). A cord initially holding the blocks together is burned; after this, the block of mass $3 M$ moves to the right with a speed of $2.00 \mathrm{~m} / \mathrm{s}$. (a) What is the speed of the block of mass $M$ ?
(b) Find the original elastic energy in the spring if $M=$ $0 . .350 \mathrm{~kg}$

Rodrigo Diaz-Meneses
Rodrigo Diaz-Meneses
Numerade Educator
00:58

Problem 7

(a) A particle of mass $m$ moves with momentum $p$. Show that the kinetic energy of the particle is given by $K=$ $p^{2} / 2 m$. (b) Express the magnitude of the particle's momentum in terms of its kinetic energy and mass.

Sheh Lit Chang
Sheh Lit Chang
University of Washington
03:51

Problem 8

A car is stopped for a traffic signal. When the light turns green, the car accelerates, increasing its speed from zero to $5.20 \mathrm{~m} / \mathrm{s}$ in $0.832 \mathrm{~s}$. What linear impulse and average force does a $70.0-\mathrm{kg}$ passenger in the car experience?

Rodrigo Diaz-Meneses
Rodrigo Diaz-Meneses
Numerade Educator
00:49

Problem 9

An estimated force-time curve for a baseball struck by a bat is shown in Figure $\mathrm{P} 9.9 .$ From this curve, determine (a) the impulse delivered to the ball, (b) the average force exerted on the ball, and (c) the peak force exerted on the ball.

Mayukh Banik
Mayukh Banik
Numerade Educator
03:00

Problem 10

A tennis player receives a shot with the ball $(0.0600 \mathrm{~kg})$ traveling horizontally at $50.0 \mathrm{~m} / \mathrm{s}$ and returns the shot with the ball traveling horizontally at $40.0 \mathrm{~m} / \mathrm{s}$ in the opposite direction. (a) What is the impulse delivered to the ball by the racket? (b) What work does the racket do on the ball?

Averell Hause
Averell Hause
Carnegie Mellon University
03:31

Problem 11

A $3.00-\mathrm{kg}$ steel ball strikes a wall with a speed of $10.0 \mathrm{~m} / \mathrm{s}$ at an angle of $60.0^{\circ}$ with the surface. It bounces off with the same speed and angle (Fig. P9.11). If the ball is in contact with the wall for $0.200 \mathrm{~s}$, what is the average force exerted on the ball by the wall?

Sheh Lit Chang
Sheh Lit Chang
University of Washington
13:26

Problem 12

In a slow-pitch softball game, a $0.200-\mathrm{kg}$ softball crossed the plate at $15.0 \mathrm{~m} / \mathrm{s}$ at an angle of $45.0^{\circ}$ below the horizontal. The ball was hit at $40.0 \mathrm{~m} / \mathrm{s}, 30.0^{\circ}$ above the horizontal. (a) Determine the impulse delivered to the ball. (b) If the force on the ball increased linearly for $4.00 \mathrm{~ms}$, held constant for $20.0 \mathrm{~ms}$, and then decreased to zero linearly in another $4.00 \mathrm{~ms}$, what was the maximum force on the ball?

Rodrigo Diaz-Meneses
Rodrigo Diaz-Meneses
Numerade Educator
03:55

Problem 13

A garden hose is held in the manner shown in Figure P9.18. The hose is initially full of motionless water. What additional force is necessary to hold the nozzle stationary after the water is turned on if the discharge rate is $0.600 \mathrm{~kg} / \mathrm{s}$ with a speed of $25.0 \mathrm{~m} / \mathrm{s} ?$

Rodrigo Diaz-Meneses
Rodrigo Diaz-Meneses
Numerade Educator
01:18

Problem 14

A professional diver performs a dive from a platform $10 \mathrm{~m}$ above the water surface. Estimate the order of magnitude of the average impact force she experiences in her collision with the water. State the quantities you take as data and their values.

Mayukh Banik
Mayukh Banik
Numerade Educator
00:42

Problem 15

High-speed stroboscopic photographs show that the head of a golf club of mass $200 \mathrm{~g}$ is traveling at $55.0 \mathrm{~m} / \mathrm{s}$ just before it strikes a $46.0-\mathrm{g}$ golf ball at rest on a tee. After the collision, the club head travels (in the same direction) at $40.0 \mathrm{~m} / \mathrm{s}$. Find the speed of the golf ball just after impact.

Mayukh Banik
Mayukh Banik
Numerade Educator
03:49

Problem 16

A $75.0-\mathrm{kg}$ ice skater, moving at $10.0 \mathrm{~m} / \mathrm{s}$, crashes into a stationary skater of equal mass. After the collision, the two skaters move as a unit at $5.00 \mathrm{~m} / \mathrm{s}$. Suppose the average force a skater can experience without breaking a bone is $4500 \mathrm{~N}$. If the impact time is $0.100 \mathrm{~s}$, does a bone break?

Rodrigo Diaz-Meneses
Rodrigo Diaz-Meneses
Numerade Educator
03:02

Problem 17

A $10.0-\mathrm{g}$ bullet is fired into a stationary block of wood $(m=5.00 \mathrm{~kg}) .$ The relative motion of the bullet stops inside the block. The speed of the bullet-plus-wood combination immediately after the collision is measured as $0.600 \mathrm{~m} / \mathrm{s}$. What was the original speed of the bullet?

Rodrigo Diaz-Meneses
Rodrigo Diaz-Meneses
Numerade Educator
08:14

Problem 18

As shown in Figure $\mathrm{P} 9.18$, a bullet of mass $m$ and speed $v$ passes completely through a pendulum bob of mass $M$. The bullet emerges with a speed of $v / 2 .$ The pendulum bob is suspended by a stiff rod of length $\ell$ and negligible mass. What is the minimum value of $z$ such that the pendulum bob will barely swing through a complete vertical circle?

Rodrigo Diaz-Meneses
Rodrigo Diaz-Meneses
Numerade Educator
00:45

Problem 19

A $45.0$ -kg girl is standing on a plank that has a mass of $150 \mathrm{~kg} .$ The plank, originally at rest, is free to slide on a frozen lake, which is a flat, frictionless supporting surface. The girl begins to walk along the plank at a constant speed of $1.50 \mathrm{~m} / \mathrm{s}$ relative to the plank. (a) What is her speed relative to the ice surface? (b) What is the speed of the plank relative to the ice surface?

Mayukh Banik
Mayukh Banik
Numerade Educator
10:47

Problem 20

Gayle runs at a speed of $4.00 \mathrm{~m} / \mathrm{s}$ and dives on a sled, which is initially at rest on the top of a frictionless snowcovered hill. After she has descended a vertical distance
of $5.00 \mathrm{~m}$, her brother, who is initially at rest, hops on her back and together they continue down the hill. What is their speed at the bottom of the hill if the total vertical drop is $15.0 \mathrm{~m}$ ? Gayle's mass is $50.0 \mathrm{~kg}$, the sled has a mass of $5.00 \mathrm{~kg}$ and her brother has a mass of $30.0 \mathrm{~kg}$.

Rodrigo Diaz-Meneses
Rodrigo Diaz-Meneses
Numerade Educator
09:52

Problem 21

A $1200-\mathrm{kg}$ car traveling initially with a speed of $25.0 \mathrm{~m} / \mathrm{s}$ in an easterly direction crashes into the rear end of a $9000-\mathrm{kg}$ truck moving in the same direction at $20.0 \mathrm{~m} / \mathrm{s}$ (Fig. P9.21). The velocity of the car right after the collision is $18.0 \mathrm{~m} / \mathrm{s}$ to the east. (a) What is the velocity of the truck right after the collision? (b) How much mechanical energy is lost in the collision? Account for this loss in energy.

Rodrigo Diaz-Meneses
Rodrigo Diaz-Meneses
Numerade Educator
07:12

Problem 22

A railroad car of mass $2.50 \times 10^{4} \mathrm{~kg}$ is moving with a speed of $4.00 \mathrm{~m} / \mathrm{s} .$ It collides and couples with three other coupled railroad cars, each of the same mass as the single car and moving in the same direction with an initial speed of $2.00 \mathrm{~m} / \mathrm{s}$. (a) What is the speed of the four cars after the collision? (b) How much energy is lost in the collision?

Rodrigo Diaz-Meneses
Rodrigo Diaz-Meneses
Numerade Educator
09:18

Problem 23

Four railroad cars, each of mass $2.50 \times 10^{4} \mathrm{~kg}$, are coupled together and coasting along horizontal tracks at a speed of $v_{i}$ toward the south. A very strong but foolish movie actor, riding on the second car, uncouples the front car and gives it a big push, increasing its speed to $4.00 \mathrm{~m} / \mathrm{s}$ southward. The remaining three cars continue moving toward the south, now at $2.00 \mathrm{~m} / \mathrm{s}$. (a) Find the initial speed of the cars. (b) How much work did the actor do? (c) State the relationship between the process described here and the process in Problem $22 .$

Rodrigo Diaz-Meneses
Rodrigo Diaz-Meneses
Numerade Educator
03:49

Problem 24

A $7.00-\mathrm{kg}$ bowling ball collides head-on with a $2.00-\mathrm{kg}$ bowling pin. The pin flies forward with a speed of $3.00 \mathrm{~m} / \mathrm{s}$. If the ball continues forward with a speed of $1.80 \mathrm{~m} / \mathrm{s}$, what was the initial speed of the ball? Ignore rotation of the ball.

Rodrigo Diaz-Meneses
Rodrigo Diaz-Meneses
Numerade Educator
07:41

Problem 25

A neutron in a reactor makes an elastic head-on collision with the nucleus of a carbon atom initially at rest.
(a) What fraction of the neutron's kinetic energy is transferred to the carbon nucleus? (b) If the initial kinetic energy of the neutron is $1.60 \times 10^{-13} \mathrm{~J}$, find its final kinetic energy and the kinetic energy of the carbon nucleus after the collision. (The mass of the carbon nucleus is about $12.0$ times greater than the mass of the neutron.)

Rodrigo Diaz-Meneses
Rodrigo Diaz-Meneses
Numerade Educator
07:20

Problem 26

Consider a frictionless track $A B C$ as shown in Figure P9.26. A block of mass $m_{1}=5.00 \mathrm{~kg}$ is released from $A$. It makes a head-on elastic collision at $B$ with a block of
mass $m_{2}=10.0 \mathrm{~kg}$ that is initially at rest. Calculate the maximum height to which $m_{1}$ rises after the collision.

Rodrigo Diaz-Meneses
Rodrigo Diaz-Meneses
Numerade Educator
07:28

Problem 27

A $12.0-\mathrm{g}$ bullet is fired into a $100-\mathrm{g}$ wooden block initially at rest on a horizontal surface. After impact, the block slides $7.50 \mathrm{~m}$ before coming to rest. If the coefficient of friction between the block and the surface is $0.650$, what was the speed of the bullet immediately before impact?

Rodrigo Diaz-Meneses
Rodrigo Diaz-Meneses
Numerade Educator
10:19

Problem 28

A $7.00-\mathrm{g}$ bullet, when fired from a gun into a $1.00-\mathrm{kg}$ block of wood held in a vise, would penetrate the block to a depth of $8.00 \mathrm{~cm}$. This block of wood is placed on a frictionless horizontal surface, and a $7.00-\mathrm{g}$ bullet is fired from the gun into the block. To what depth will the bullet penetrate the block in this case?

Rodrigo Diaz-Meneses
Rodrigo Diaz-Meneses
Numerade Educator
10:53

Problem 29

A $90.0-\mathrm{kg}$ fullback running east with a speed of $5.00 \mathrm{~m} / \mathrm{s}$ is tackled by a $95.0-\mathrm{kg}$ opponent running north with a speed of $3.00 \mathrm{~m} / \mathrm{s}$. If the collision is perfectly inelastic,
(a) calculate the speed and direction of the players just after the tackle and (b) determine the energy lost as a result of the collision. Account for the missing energy.

Rodrigo Diaz-Meneses
Rodrigo Diaz-Meneses
Numerade Educator
14:23

Problem 30

The mass of the blue puck in Figure $\mathrm{P} 9.30$ is $20.0 \%$ greater than the mass of the green one. Before colliding, the pucks approach each other with equal and opposite momenta, and the green puck has an initial speed of $10.0 \mathrm{~m} / \mathrm{s}$. Find the speeds of the pucks after the collision if half the kinetic energy is lost during the collision.

Rodrigo Diaz-Meneses
Rodrigo Diaz-Meneses
Numerade Educator
08:13

Problem 31

Two automobiles of equal mass approach an intersection. One vehicle is traveling with velocity $13.0 \mathrm{~m} / \mathrm{s}$ toward the east and the other is traveling north with a speed of $\tau_{2 i} .$ Neither driver sees the other. The vehicles collide in the intersection and stick together, leaving parallel skid marks at an angle of $55.0^{\circ}$ north of east. The speed limit for both roads is $35 \mathrm{mi} / \mathrm{h}$, and the driver of the northward-moving vehicle claims he was within the speed limit when the collision occurred. Is he telling the truth?

Rodrigo Diaz-Meneses
Rodrigo Diaz-Meneses
Numerade Educator
01:37

Problem 32

A proton, moving with a velocity of $v_{i}$, collides elastically with another proton that is initially at rest. If the two protons have equal speeds after the collision, find
(a) the speed of each proton after the collision in terms of $v_{i}$ and (b) the direction of the velocity vectors after the collision.

Sanjeev Kumar
Sanjeev Kumar
Numerade Educator
08:43

Problem 33

A billiard ball moving at $5.00 \mathrm{~m} / \mathrm{s}$ strikes a stationary ball of the same mass. After the collision, the first ball moves at $4.38 \mathrm{~m} / \mathrm{s}$ and at an angle of $30.0^{\circ}$ with respect to the original line of motion. Assuming an elastic collision (and ignoring friction and rotational motion), find the struck ball's velocity.

Rodrigo Diaz-Meneses
Rodrigo Diaz-Meneses
Numerade Educator
13:22

Problem 34

A $0.300-\mathrm{kg}$ puck, initially at rest on a horizontal, frictionless surface, is struck by a $0.200-\mathrm{kg}$ puck moving initially along the $x$ axis with a speed of $2.00 \mathrm{~m} / \mathrm{s}$. After the collision, the $0.200-\mathrm{kg}$ puck has a speed of $1.00 \mathrm{~m} / \mathrm{s}$ at an angle of $\theta=53.0^{\circ}$ to the positive $x$ axis (see Fig. 9.14). (a) Determine the velocity of the $0.300-\mathrm{kg}$ puck after the collision. (b) Find the fraction of kinetic energy lost in the collision.

Rodrigo Diaz-Meneses
Rodrigo Diaz-Meneses
Numerade Educator
05:00

Problem 35

A $3.00-\mathrm{kg}$ mass with an initial velocity of $5.00 \mathrm{i} \mathrm{m} / \mathrm{s}$ collides with and sticks to a $2.00-\mathrm{kg}$ mass with an initial velocity of $-3.00 \mathbf{j} \mathrm{m} / \mathrm{s}$. Find the final velocity of the composite mass.

Rodrigo Diaz-Meneses
Rodrigo Diaz-Meneses
Numerade Educator
08:52

Problem 36

Two shuffleboard disks of equal mass, one orange and the other yellow, are involved in an elastic, glancing collision. The yellow disk is initially at rest and is struck by the orange disk moving with a speed of $5.00 \mathrm{~m} / \mathrm{s}$. After the collision, the orange disk moves along a direction that makes an angle of $37.0^{\circ}$ with its initial direction of motion, and the velocity of the yellow disk is perpendicular to that of the orange disk (after the collision). Determine the final speed of each disk.

Rodrigo Diaz-Meneses
Rodrigo Diaz-Meneses
Numerade Educator
08:55

Problem 37

Two shuffleboard disks of equal mass, one orange and the other yellow, are involved in an elastic, glancing collision. The yellow disk is initially at rest and is struck by the orange disk moving with a speed $v_{i}$. After the collision, the orange disk moves along a direction that makes an angle $\theta$ with its initial direction of motion, and the velocity of the yellow disk is perpendicular to that of the orange disk (after the collision). Determine the final speed of each disk.

Rodrigo Diaz-Meneses
Rodrigo Diaz-Meneses
Numerade Educator
09:08

Problem 38

During the battle of Gettysburg, the gunfire was so intense that several bullets collided in midair and fused together. Assume a $5.00-g$ Union musket ball was moving to the right at a speed of $250 \mathrm{~m} / \mathrm{s}, 20.0^{\circ}$ above the horizontal, and that a $3.00-\mathrm{g}$ Confederate ball was moving to the left at a speed of $280 \mathrm{~m} / \mathrm{s}, 15.0^{\circ}$ above the horizontal. Immediately after they fuse together, what is their velocity?

Rodrigo Diaz-Meneses
Rodrigo Diaz-Meneses
Numerade Educator
13:35

Problem 39

An unstable nucleus of mass $17.0 \times 10^{-27} \mathrm{~kg}$ initially at rest disintegrates into three particles. One of the particles, of mass $5.00 \times 10^{-27} \mathrm{~kg}$, moves along the $y$ axis with a velocity of $6.00 \times 10^{6} \mathrm{~m} / \mathrm{s}$. Another particle, of mass $8.40 \times 10^{-27} \mathrm{~kg}$, moves along the $x$ axis with a speed of $4.00 \times 10^{6} \mathrm{~m} / \mathrm{s}$. Find (a) the velocity of the third particle and (b) the total kinetic energy increase in the process.

Rodrigo Diaz-Meneses
Rodrigo Diaz-Meneses
Numerade Educator
05:38

Problem 40

Four objects are situated along the $y$ axis as follows: $\mathrm{A}$ $2.00-\mathrm{kg}$ object is at $+3.00 \mathrm{~m}$, a $3.00-\mathrm{kg}$ object is at $+2.50 \mathrm{~m}$, a $2.50-\mathrm{kg}$ object is at the origin, and a $4.00-\mathrm{kg}$ object is at $-0.500 \mathrm{~m} .$ Where is the center of mass of these objects?

Rodrigo Diaz-Meneses
Rodrigo Diaz-Meneses
Numerade Educator
02:58

Problem 41

A uniform piece of sheet steel is shaped as shown in Figure P9.41. Compute the $x$ and $y$ coordinates of the center of mass of the piece.

Sheh Lit Chang
Sheh Lit Chang
University of Washington
04:16

Problem 42

The mass of the Earth is $5.98 \times 10^{24} \mathrm{~kg}$, and the mass of the Moon is $7.36 \times 10^{22} \mathrm{~kg} .$ The distance of separation, measured between their centers, is $8.84 \times 10^{8} \mathrm{~m}$. Locate the center of mass of the Earth-Moon system as measured from the center of the Earth.

Rodrigo Diaz-Meneses
Rodrigo Diaz-Meneses
Numerade Educator
01:34

Problem 43

A water molecule consists of an oxygen atom with two hydrogen atoms bound to it (Fig. P9.43). The angle between the two bonds is $106^{\circ}$. If the bonds are $0.100 \mathrm{~nm}$ long, where is the center of mass of the molecule?

Mayukh Banik
Mayukh Banik
Numerade Educator
08:24

Problem 44

A $0.400-\mathrm{kg}$ mass $m_{1}$ has position $\mathbf{r}_{1}=12.0 \mathbf{j} \mathrm{cm} . \mathrm{A} 0.800-$
kg mass $m_{2}$ has position $\mathbf{r}_{2}=-12.0 \mathbf{i} \mathrm{cm}$. Another $0.800-\mathrm{kg}$ mass $m_{3}$ has position $\mathbf{r}_{3}=(12.0 \mathbf{i}-12.0 \mathbf{j}) \mathrm{cm}$.
Make a drawing of the masses. Start from the origin and, to the scale $1 \mathrm{~cm}=1 \mathrm{~kg} \cdot \mathrm{cm}$, construct the vector $m_{1} \mathbf{r}_{1}$, then the vector $m_{1} \mathbf{r}_{1}+m_{2} \mathbf{r}_{2}$, then the vector $m_{1} \mathbf{r}_{1}$
$+m_{2} \mathbf{r}_{2}+m_{3} \mathbf{r}_{3}$, and at last $\mathbf{r}_{\mathrm{CM}}=\left(m_{1} \mathbf{r}_{1}+m_{2} \mathbf{r}_{2}+\right.$
$\left.m_{3} \mathbf{r}_{3}\right) /\left(m_{1}+m_{2}+m_{3}\right) .$ Observe that the head of the vector $\mathbf{r}_{\mathrm{CM}}$ indicates the position of the center of mass.

Rodrigo Diaz-Meneses
Rodrigo Diaz-Meneses
Numerade Educator
04:29

Problem 45

A rod of length $30.0 \mathrm{~cm}$ has linear density (mass-perlength) given by
$$
\lambda=50.0 \mathrm{~g} / \mathrm{m}+20.0 x \mathrm{~g} / \mathrm{m}^{2}
$$
where $x$ is the distance from one end, measured in meters. (a) What is the mass of the rod? (b) How far from the $x=0$ end is its center of mass?

Suhas Katkar
Suhas Katkar
Numerade Educator
13:26

Problem 46

Consider a system of two particles in the $x y$ plane:
$m_{1}=2.00 \mathrm{~kg}$ is at $\mathbf{r}_{1}=(1.00 \mathbf{i}+2.00 \mathbf{j}) \mathrm{m}$ and has ve-
locity $(3.00 \mathbf{i}+0.500 \mathbf{j}) \mathrm{m} / \mathrm{s} ; m_{2}=3.00 \mathrm{~kg}$ is at $\mathbf{r}_{2}=$
$(-4.00 \mathbf{i}-3.00 \mathbf{j}) \mathrm{m}$ and has velocity $(3.00 \mathbf{i}-2.00 \mathbf{j}) \mathrm{m} / \mathrm{s}$
(a) Plot these particles on a grid or graph paper. Draw their position vectors and show their velocities. (b) Find the position of the center of mass of the system and mark it on the grid. (c) Determine the velocity of the center of mass and also show it on the diagram. (d) What is the total linear momentum of the system?

Ryan Williams
Ryan Williams
Numerade Educator
01:28

Problem 47

Romeo $(77.0 \mathrm{~kg})$ entertains Juliet (55.0 kg) by playing his guitar from the rear of their boat at rest in still water, $2.70 \mathrm{~m}$ away from Juliet who is in the front of the boat. After the serenade, Juliet carefully moves to the rear of the boat (away from shore) to plant a kiss on Romeo's cheek. How far does the $80.0$ -kg boat move toward the shore it is facing?

Sanjeev Kumar
Sanjeev Kumar
Numerade Educator
17:09

Problem 48

Two masses, $0.600 \mathrm{~kg}$ and $0.300 \mathrm{~kg}$, begin uniform motion at the same speed, $0.800 \mathrm{~m} / \mathrm{s}$, from the origin at $t=0$ and travel in the directions shown in Figure $\mathrm{P} 9.48 .$
(a) Find the velocity of the center of mass in unitvector notation. (b) Find the magnitude and direction of the velocity of the center of mass. (c) Write the position vector of the center of mass as a function of time.

Rodrigo Diaz-Meneses
Rodrigo Diaz-Meneses
Numerade Educator
06:06

Problem 49

A $2.00-\mathrm{kg}$ particle has a velocity of $\left(2.00^{1}-3.00 \mathrm{j}\right) \mathrm{m} / \mathrm{s}$, and a $3.00-\mathrm{kg}$ particle has a velocity of $(1.00 \mathbf{i}+6.00 \mathbf{j})$ $\mathrm{m} / \mathrm{s}$. Find (a) the velocity of the center of mass and
(b) the total momentum of the system.

Rodrigo Diaz-Meneses
Rodrigo Diaz-Meneses
Numerade Educator
08:38

Problem 50

A ball of mass $0.200 \mathrm{~kg}$ has a velocity of $1.50 \mathrm{i} \mathrm{m} / \mathrm{s}$; a ball of mass $0.300 \mathrm{~kg}$ has a velocity of $-0.400 \mathbf{i} \mathrm{m} / \mathrm{s}$. They meet in a head-on elastic collision. (a) Find their velocities after the collision. (b) Find the velocity of their center of mass before and after the collision.

Rodrigo Diaz-Meneses
Rodrigo Diaz-Meneses
Numerade Educator
04:25

Problem 51

The first stage of a Saturn $\mathrm{V}$ space vehicle consumes fuel and oxidizer at the rate of $1.50 \times 10^{4} \mathrm{~kg} / \mathrm{s}$, with an exhaust speed of $2.60 \times 10^{3} \mathrm{~m} / \mathrm{s}$. (a) Calculate the thrust produced by these engines. (b) Find the initial acceleration of the vehicle on the launch pad if its initial mass is $3.00 \times 10^{6} \mathrm{~kg} .[$ Hint : You must include the force of gravity to solve part (b).]

Rodrigo Diaz-Meneses
Rodrigo Diaz-Meneses
Numerade Educator
05:04

Problem 52

A large rocket with an exhaust speed of $v_{e}=3000 \mathrm{~m} / \mathrm{s}$ develops a thrust of $24.0$ million newtons. (a) How much mass is being blasted out of the rocket exhaust per second? (b) What is the maximum speed the rocket can attain if it starts from rest in a force-free environment with $v_{e}=8.00 \mathrm{~km} / \mathrm{s}$ and if $90.0 \%$ of its initial mass is fuel and oxidizer?

Rodrigo Diaz-Meneses
Rodrigo Diaz-Meneses
Numerade Educator
11:03

Problem 53

A rocket for use in deep space is to have the capability of boosting a total load (payload plus rocket frame and engine) of $3.00$ metric tons to a speed of $10000 \mathrm{~m} / \mathrm{s}$.
(a) It has an engine and fuel designed to produce an exhaust speed of $2000 \mathrm{~m} / \mathrm{s}$. How much fuel plus oxidizer is required? (b) If a different fuel and engine design could give an exhaust speed of $5000 \mathrm{~m} / \mathrm{s}$, what amount of fuel and oxidizer would be required for the same task?

Ryan Williams
Ryan Williams
Numerade Educator
09:01

Problem 54

A rocket car has a mass of $2000 \mathrm{~kg}$ unfueled and a mass of $5000 \mathrm{~kg}$ when completely fueled. The exhaust velocity is $2500 \mathrm{~m} / \mathrm{s}$. (a) Calculate the amount of fuel used to accelerate the completely fueled car from rest to $225 \mathrm{~m} / \mathrm{s}$ (about $500 \mathrm{mi} / \mathrm{h}) .$ (b) If the burn rate is constant at $30.0 \mathrm{~kg} / \mathrm{s}$, calculate the time it takes the car to reach this speed. Neglect friction and air resistance.

Ryan Williams
Ryan Williams
Numerade Educator
03:25

Problem 55

A $60.0$ -kg person running at an initial speed of $4.00 \mathrm{~m} / \mathrm{s}$ jumps onto a $120-\mathrm{kg}$ cart initially at rest (Fig. $\mathrm{P} 9.55$ ). The person slides on the cart's top surface and finally comes to rest relative to the cart. The coefficient of kinetic friction between the person and the cart is $0.400 .$ Friction between the cart and ground can be neglected. (a) Find the final velocity of the person and cart relative to the ground. (b) Find the frictional force acting on the person while he is sliding across the top surface of the cart. (c) How long does the frictional force act on the person? (d) Find the change in momentum of the person and the change in momentum of the cart. (e) Determine the displacement of the person relative to the ground while he is sliding on the cart. (f) Determine the displacement of the cart relative to the ground while the person is sliding. (g) Find the change in kinetic energy of the person. (h) Find the change in kinetic energy of the cart.
(i) Explain why the answers to parts $(g)$ and $(\mathrm{h})$ differ. (What kind of collision is this, and what accounts for the loss of mechanical energy?)

Ummatul Choudary
Ummatul Choudary
Numerade Educator
07:01

Problem 56

A golf ball $(m=46.0 \mathrm{~g})$ is struck a blow that makes an angle of $45.0^{\circ}$ with the horizontal. The ball lands $200 \mathrm{~m}$ away on a flat fairway. If the golf club and ball are in contact for $7.00 \mathrm{~ms}$, what is the average force of impact? (Neglect air resistance.)

Ryan Williams
Ryan Williams
Numerade Educator
07:58

Problem 57

An $8.00-\mathrm{g}$ bullet is fired into a $2.50-\mathrm{kg}$ block that is initially at rest at the edge of a frictionless table of height $1.00 \mathrm{~m}$ (Fig. P9.57). The bullet remains in the block, and after impact the block lands $2.00 \mathrm{~m}$ from the bottom of the table. Determine the initial speed of the bullet.

Ryan Williams
Ryan Williams
Numerade Educator
06:00

Problem 58

A bullet of mass $m$ is fired into a block of mass $M$ that is initially at rest at the edge of a frictionless table of height $h$ (see Fig. P9.57). The bullet remains in the block, and after impact the block lands a distance $d$ from the bottom of the table. Determine the initial
speed of the bullet.

Ryan Williams
Ryan Williams
Numerade Educator
05:13

Problem 59

An $80.0-\mathrm{kg}$ astronaut is working on the engines of his ship, which is drifting through space with a constant velocity. The astronaut, wishing to get a better view of the Universe, pushes against the ship and much later finds himself $30.0 \mathrm{~m}$ behind the ship and at rest with respect to it. Without a thruster, the only way to return to the ship is to throw his $0.500-\mathrm{kg}$ wrench directly away from the ship. If he throws the wrench with a speed of $20.0 \mathrm{~m} / \mathrm{s}$ relative to the ship, how long does it take the astronaut to reach the ship?

Ryan Williams
Ryan Williams
Numerade Educator
07:34

Problem 60

A small block of mass $m_{1}=0.500 \mathrm{~kg}$ is released from rest at the top of a curve-shaped frictionless wedge of mass $m_{2}=3.00 \mathrm{~kg}$, which sits on a frictionless horizontal surface, as shown in Figure P9.60a. When the block leaves the wedge, its velocity is measured to be $4.00 \mathrm{~m} / \mathrm{s}$ to the right, as in Figure P9.60b. (a) What is the velocity of the wedge after the block reaches the horizontal surface? (b) What is the height $h$ of the wedge?

Ryan Williams
Ryan Williams
Numerade Educator
10:01

Problem 61

Tarzan, whose mass is $80.0 \mathrm{~kg}$, swings from a $3.00-\mathrm{m}$ vine that is horizontal when he starts. At the bottom of his arc, he picks up $60.0-\mathrm{kg}$ Jane in a perfectly inelastic collision. What is the height of the highest tree limb they can reach on their upward swing?

Ryan Williams
Ryan Williams
Numerade Educator
01:21

Problem 62

A jet aircraft is traveling at $500 \mathrm{mi} / \mathrm{h}(228 \mathrm{~m} / \mathrm{s})$ in horizontal flight. The engine takes in air at a rate of $80.0 \mathrm{~kg} / \mathrm{s}$ and burns fuel at a rate of $3.00 \mathrm{~kg} / \mathrm{s}$. If the exhaust gases are ejected at $600 \mathrm{~m} / \mathrm{s}$ relative to the aircraft, find the thrust of the jet engine and the delivered horsepower.

Ummatul Choudary
Ummatul Choudary
Numerade Educator
03:29

Problem 63

A $75.0-\mathrm{kg}$ firefighter slides down a pole while a constant frictional force of $900 \mathrm{~N}$ retards her motion. A horizon-
tal $20.0-\mathrm{kg}$ platform is supported by a spring at the bottom of the pole to cushion the fall. The firefighter starts from rest $4.00 \mathrm{~m}$ above the platform, and the spring constant is $4000 \mathrm{~N} / \mathrm{m}$. Find (a) the firefighter's speed just before she collides with the platform and (b) the maximum distance the spring is compressed. (Assume the frictional force acts during the entire motion.)

Ummatul Choudary
Ummatul Choudary
Numerade Educator
00:49

Problem 64

A cannon is rigidly attached to a carriage, which can move along horizontal rails but is connected to a post by a large spring, initially unstretched and with force constant $k=2.00 \times 10^{4} \mathrm{~N} / \mathrm{m}$, as shown in Figure P9.64. The cannon fires a $200-\mathrm{kg}$ projectile at a velocity of $125 \mathrm{~m} / \mathrm{s}$ directed $45.0^{\circ}$ above the horizontal. (a) If the mass of the cannon and its carriage is $5000 \mathrm{~kg}$, find the recoil speed of the cannon. (b) Determine the maximum extension of the spring. (c) Find the maximum force the spring exerts on the carriage. (d) Consider the system consisting of the cannon, carriage, and shell. Is the momentum of this system conserved during the firing? Why or why not?

Mayukh Banik
Mayukh Banik
Numerade Educator
03:33

Problem 65

A chain of length $L$ and total mass $M$ is released from rest with its lower end just touching the top of a table, as shown in Figure $\mathrm{P} 9.65 \mathrm{a}$. Find the force exerted by the table on the chain after the chain has fallen through a distance $x$, as shown in Figure P9.65b. (Assume each link comes to rest the instant it reaches the table.)

Ummatul Choudary
Ummatul Choudary
Numerade Educator
02:33

Problem 66

Two gliders are set in motion on an air track. A spring of force constant $k$ is attached to the near side of one glider. The first glider of mass $m_{1}$ has a velocity of $\mathbf{v}_{1}$, and the second glider of mass $m_{2}$ has a velocity of $\mathbf{v}_{2}$, as shown in Figure $\mathrm{P} 9.66\left(v_{1}>v_{2}\right) .$ When $m_{1}$ collides with the spring attached to $m_{2}$ and compresses the spring to its maximum compression $x_{m}$, the velocity of the gliders is $\mathbf{v}$. In terms of $\mathbf{v}_{1}, \mathbf{v}_{2}, m_{1}, m_{2}$, and $k$, find $($ a) the velocity $\mathbf{v}$ at maximum compression, (b) the maximum compression $x_{m}$, and $(c)$ the velocities of each glider after $m_{1}$ has lost contact with the spring.

Mayukh Banik
Mayukh Banik
Numerade Educator
01:26

Problem 67

Sand from a stationary hopper falls onto a moving conveyor belt at the rate of $5.00 \mathrm{~kg} / \mathrm{s}$, as shown in Figure P9. 67. The conveyor belt is supported by frictionless rollers and moves at a constant speed of $0.750 \mathrm{~m} / \mathrm{s}$ under the action of a constant horizontal external force $\mathbf{F}_{\text {ext }}$ supplied by the motor that drives the belt. Find
(a) the sand's rate of change of momentum in the horizontal direction, (b) the force of friction exerted by the belt on the sand, (c) the external force $\mathbf{F}_{\text {ext }}$, (d) the work done by $\mathbf{F}_{\mathrm{ext}}$ in $1 \mathrm{~s}$, and (e) the kinetic energy acquired by the falling sand each second due to the change in its horizontal motion. (f) Why are the answers to parts (d) and (e) different?

Ummatul Choudary
Ummatul Choudary
Numerade Educator
08:09

Problem 68

A rocket has total mass $M_{i}=360 \mathrm{~kg}$, including $390 \mathrm{~kg}$ of fuel and oxidizer. In interstellar space it starts from rest, turns on its engine at time $t=0$, and puts out exhaust with a relative speed of $v_{e}=1500 \mathrm{~m} / \mathrm{s}$ at the constant rate $k=2.50 \mathrm{~kg} / \mathrm{s}$. Although the fuel will last for an actual burn time of $380 \mathrm{~kg} /(2.5 \mathrm{~kg} / \mathrm{s})=132 \mathrm{~s}$, de-
fine a "projected depletion time" as $T_{p}=M_{i} / k=$ $360 \mathrm{~kg} /(2.5 \mathrm{~kg} / \mathrm{s})=144 \mathrm{~s}$. (This would be the burn
time if the rocket could use its payload, fuel tanks, and even the walls of the combustion chamber as fuel.)
(a) Show that during the burn the velocity of the rocket is given as a function of time by
$$
v(t)=-v_{e} \ln \left(1-t / T_{p}\right)
$$
(b) Make a graph of the velocity of the rocket as a function of time for times running from 0 to $132 \mathrm{~s}$. (c) Show that the acceleration of the rocket is
$$
a(t)=v_{e} /\left(T_{p}-t\right)
$$
(d) Graph the acceleration as a function of time.
(e) Show that the displacement of the rocket from its initial position at $t=0$ is
$$
x(t)=v_{e}\left(T_{p}-t\right) \ln \left(1-t / T_{p}\right)+v_{e} t
$$
(f) Graph the displacement during the burn.

Sheh Lit Chang
Sheh Lit Chang
University of Washington
03:54

Problem 69

A $40.0-\mathrm{kg}$ child stands at one end of a $70.0 \mathrm{~kg}$ boat that is $4.00 \mathrm{~m}$ in length (Fig. P9.69). The boat is initially $3.00 \mathrm{~m}$ from the pier. The child notices a turtle on a rock near the far end of the boat and proceeds to walk to that end to catch the turtle. Neglecting friction between the boat and the water, (a) describe the subsequent motion of the system (child plus boat). (b) Where is the child relative to the pier when he reaches the far end of the boat? (c) Will he catch the turtle? (Assume he can reach out $1.00 \mathrm{~m}$ from the end of the boat.)

Ummatul Choudary
Ummatul Choudary
Numerade Educator
08:29

Problem 70

A student performs a ballistic pendulum experiment, using an apparatus similar to that shown in Figure 9.11b. She obtains the following average data: $h=$ $8.68 \mathrm{~cm}, m_{1}=68.8 \mathrm{~g}$, and $m_{2}=263 \mathrm{~g} .$ The symbols re-
fer to the quantities in Figure 9.11a. (a) Determine the initial speed $v_{1 i}$ of the projectile. (b) In the second part of her experiment she is to obtain $v_{1 i}$ by firing the same projectile horizontally (with the pendulum removed from the path) and measuring its horizontal displacement $x$ and vertical displacement $y$ (Fig. P9.70). Show that the initial speed of the projectile is related to $x$ and y through the relationship
$$
v_{1 i}=\frac{x}{\sqrt{2 y / g}}
$$
What numerical value does she obtain for $v_{1 i}$ on the basis of her measured values of $x=257 \mathrm{~cm}$ and $y=$ $85.3 \mathrm{~cm} ?$ What factors might account for the difference in this value compared with that obtained in part (a)?

Ryan Williams
Ryan Williams
Numerade Educator
09:06

Problem 71

A $5.00-\mathrm{g}$ bullet moving with an initial speed of $400 \mathrm{~m} / \mathrm{s}$ is fired into and passes through a $1.00-\mathrm{kg}$ block, as shown in Figure $\mathrm{P} 9.71$. The block, initially at rest on a frictionless, horizontal surface, is connected to a spring of force constant $900 \mathrm{~N} / \mathrm{m}$. If the block moves $5.00 \mathrm{~cm}$ to the right after impact, find (a) the speed at which the bullet emerges from the block and (b) the energy lost in the collision.

Rodrigo Diaz-Meneses
Rodrigo Diaz-Meneses
Numerade Educator
08:40

Problem 72

Two masses $m$ and $3 m$ are moving toward each other along the $x$ axis with the same initial speeds $v_{i}$. Mass $m$ traveling to the left, while mass $3 \mathrm{~m}$ is traveling to the right. They undergo a head-on elastic collision and each rebounds along the same line as it approached. Find the final speeds of the masses.

Rodrigo Diaz-Meneses
Rodrigo Diaz-Meneses
Numerade Educator
14:31

Problem 73

Two masses $m$ and $3 m$ are moving toward each other along the $x$ axis with the same initial speeds $v_{i} .$ Mass $m$ traveling to the left, while mass $3 m$ is traveling to the right. They undergo an elastic glancing collision such that mass $m$ is moving downward after the collision at right angles from its initial direction. (a) Find the final speeds of the two masses. (b) What is the angle $\theta$ at which the mass $3 m$ is scattered?

Ryan Williams
Ryan Williams
Numerade Educator
00:48

Problem 74

Review Problem. There are (one can say) three coequal theories of motion: Newton's second law, stating that the total force on an object causes its acceleration; the work-kinetic energy theorem, stating that the total work on an object causes its change in kinetic energy; and the impulse-momentum theorem, stating that the total impulse on an object causes its change in momentum. In this problem, you compare predictions of the three theories in one particular case. A $3.00-\mathrm{kg}$ object has a velocity of $7.00 \mathrm{j} \mathrm{m} / \mathrm{s}$. Then, a total force $12.0 \mathrm{i} \mathrm{N}$ acts on the object for $5.00 \mathrm{~s}$. (a) Calculate the object's final velocity, using the impulse-momentum theorem.
(b) Calculate its acceleration from $\mathbf{a}=\left(\mathbf{v}_{f}-\mathbf{v}_{i}\right) / t$
(c) Calculate its acceleration from $\mathbf{a}=\Sigma \mathbf{F} / m \cdot(\mathrm{d})$ Find the object's vector displacement from $\mathbf{r}=\mathbf{v}_{i} t+\frac{1}{2} \mathbf{a} t^{2}$
(e) Find the work done on the object from $W=\mathbf{F} \cdot \mathbf{r}$.
(f) Find the final kinetic energy from $\frac{1}{2} m v_{f}^{2}=\frac{1}{2} m \mathbf{v}_{f} \cdot \mathbf{v}_{f}$
(g) Find the final kinetic energy from $\frac{1}{2} m v_{i}^{2}+W$.

Mayukh Banik
Mayukh Banik
Numerade Educator
01:30

Problem 75

A rocket has a total mass of $M_{i}=360 \mathrm{~kg}$, including $390 \mathrm{~kg}$ of fuel and oxidizer. In interstellar space it starts from rest. Its engine is turned on at time $t=0$, and it puts out exhaust with a relative speed of $v_{e}=1500 \mathrm{~m} / \mathrm{s}$ at the constant rate $2.50 \mathrm{~kg} / \mathrm{s}$. The burn lasts until the fuel runs out at time $330 \mathrm{~kg} /(2.5 \mathrm{~kg} / \mathrm{s})=132 \mathrm{~s}$. Set up
and carry out a computer analysis of the motion according to Euler's method. Find (a) the final velocity of the rocket and (b) the distance it travels during the burn.

Mayukh Banik
Mayukh Banik
Numerade Educator