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Principles of Physics a Calculus Based Text

Raymond A. Serway, John W. Jewett, Jr.

Chapter 8

Momentum and Collisions - all with Video Answers

Educators

Chapter Questions

01:48

Problem 1

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

Sanjeev Kumar
Sanjeev Kumar
Numerade Educator
03:42

Problem 2

How fast can you set the Earth moving? In particular, when you jump straight up as high as you can, what is the order of magnitude of the maximum recoil speed that you give to the Earth? 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
02:42

Problem 3

In research in cardiology and exercise physiology, it is often important to know the mass of blood pumped by a person's heart in one stroke. This information can be obtained by means of a ballistocardiograph. The instrument works as follows. The subject lies on a horizontal pallet floating on a film of air. Friction on the pallet is negligible. Initially, the momentum of the system is zero. When the heart beats, it expels a mass $m$ of blood into the aorta with speed $v,$ and the body and platform move in the opposite direction with speed $V$. The blood velocity can be determined independently (e.g., by observing the Doppler shift of ultrasound). Assume that it is $50.0 \mathrm{cm} / \mathrm{s}$ in one typical trial. The mass of the subject plus the pallet is 54.0 kg. The pallet moves $6.00 \times 10^{-5} \mathrm{m}$ in $0.160 \mathrm{s}$ after one heartbeat. Calculate the mass of blood that leaves the heart. Assume that the mass of blood is negligible compared with the total mass of the person. (This simplified example illustrates the principle of ballistocardiography, but in practice a more sophisticated model of heart function is used.)

Sanjeev Kumar
Sanjeev Kumar
Numerade Educator
00:55

Problem 4

(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.

Sanjeev Kumar
Sanjeev Kumar
Numerade Educator
00:56

Problem 5

Two blocks with 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. P8.5). 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 potential energy in the spring, taking $M=0.350 \mathrm{kg}$.

Mayukh Banik
Mayukh Banik
Numerade Educator
00:53

Problem 6

A friend claims that as long as he has his seat belt on, he can hold on to a 12.0 -kg child in a $60.0 \mathrm{mi} / \mathrm{h}$ head-on collision with a brick wall in which the car passenger compartment comes to a stop in 0.0500 s. Show that the violent force during the collision will tear the child from his arms. (A child should always be in a toddler seat secured with a seat belt in the back seat of a car.)

Mayukh Banik
Mayukh Banik
Numerade Educator
00:49

Problem 7

An estimated force-time curve for a baseball struck by a bat is shown in Figure $\mathrm{P} 8.7 .$ 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
03:58

Problem 8

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 racquet? (b) What work does the racquet do on the ball?

Learnmore Shenje
Learnmore Shenje
Numerade Educator
03:31

Problem 9

A 3.00 -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. P8.9). 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
13:23

Problem 10

In a slow-pitch softball game, a 0.200 -kg softball crosses the plate at $15.0 \mathrm{m} / \mathrm{s}$ at an angle of $45.0^{\circ}$ below the horizontal. The batter hits the ball toward center field, giving it a velocity of $40.0 \mathrm{m} / \mathrm{s}$ at $30.0^{\circ}$ above the horizontal. (a) Determine the impulse delivered to the ball. (b) If the force on the ball increases linearly for $4.00 \mathrm{ms}$, holds constant for
$20.0 \mathrm{ms},$ and then decreases linearly to zero in another
$4.00 \mathrm{ms},$ what is the maximum force on the ball?

Maria Gabriela Cota Moreira
Maria Gabriela Cota Moreira
Numerade Educator
00:51

Problem 11

A garden hose is held as shown in Figure P8.11. The hose is originally full of motionless water. What additional force is necessary to hold the nozzle stationary after the water flow is turned on if the discharge rate is $0.600 \mathrm{kg} / \mathrm{s}$ with a speed of $25.0 \mathrm{m} / \mathrm{s}$ ?

Sanjeev Kumar
Sanjeev Kumar
Numerade Educator
01:22

Problem 12

A glider of mass $m$ is free to slide along a horizontal air track. It is pushed against a launcher at one end of the track. Model the launcher as a light spring of force constant $k,$ compressed by a distance $x .$ The glider is released from rest. (a) Show that the glider attains a speed $v=x(k / m)^{1 / 2} .$ (b) Does a glider of large or of small mass attain a greater speed? (c) Show that the impulse imparted to the glider is given by the expression $x(k m)^{1 / 2}$. (d) Is a greater impulse imparted to a large or a small mass? (e) Is more work done on a large or a small mass?

Mayukh Banik
Mayukh Banik
Numerade Educator
03:07

Problem 13

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 mechanical energy is lost in the collision?

Sanjeev Kumar
Sanjeev Kumar
Numerade Educator
06:10

Problem 14

Four railroad cars, each of mass $2.50 \times 10^{4} \mathrm{kg}$, are coupled together and coasting along horizontal tracks at speed $v_{i}$ 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 south, now at
$2.00 \mathrm{m} / \mathrm{s} .$ (a) Find the initial speed of the four cars.
(b) How much work did the actor do? (c) State the relationship between the process described here and the process in Problem 8.13

Rashmi Sinha
Rashmi Sinha
Numerade Educator
00:45

Problem 15

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
07:16

Problem 16

Two blocks are free to slide along the frictionless wooden track $A B C$ shown in Figure $P 8.16 .$ A block of mass $m_{1}=5.00 \mathrm{kg}$ is released from $A .$ Protruding from its front end is the north pole of a strong magnet, repelling the north pole of an identical magnet embedded in the back end of the block of mass $m_{2}=10.0 \mathrm{kg},$ initially at rest. The two blocks never touch. Calculate the maximum height to which $m_{1}$ rises after the elastic collision.

Sheh Lit Chang
Sheh Lit Chang
University of Washington
02:15

Problem 17

Most of us know intuitively that in a head-on collision between a large dump truck and a subcompact car, you are better off being in the truck than in the car. Why? Many people imagine that the collision force exerted on the car is much greater than that experienced by the truck. To substantiate this view, they point out that the car is crushed, whereas the truck is only dented. This idea of unequal forces, of course, is false. Newton's third law tells us that both objects experience forces of the same magnitude. The truck suffers less damage because it is made of stronger metal. What about the two drivers? Do they experience the same forces? To answer this question, suppose each vehicle is initially moving at $8.00 \mathrm{m} / \mathrm{s}$ and they undergo a perfectly inelastic head-on collision. Each driver has mass 80.0 kg. Including the drivers, the total vehicle masses are $800 \mathrm{kg}$ for the car and $4000 \mathrm{kg}$ for the truck. If the collision time is $0.120 \mathrm{s}$, what force does the seat belt exert on each driver?

Mayukh Banik
Mayukh Banik
Numerade Educator
05:09

Problem 18

As shown in Figure $\mathrm{P} 8.18$, 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 $v$ such that the pendulum bob will barely swing through a complete vertical circle?

Learnmore Shenje
Learnmore Shenje
Numerade Educator
03:51

Problem 19

A neutron in a nuclear 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) Assume that 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 nearly 12.0 times the mass of the neutron.)

Sanjeev Kumar
Sanjeev Kumar
Numerade Educator
03:56

Problem 20

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

Sanjeev Kumar
Sanjeev Kumar
Numerade Educator
01:58

Problem 21

A $12.0-\mathrm{g}$ wad of sticky clay is hurled horizontally at a 100 -g wooden block initially at rest on a horizontal surface. The clay sticks to the block. 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 clay immediately before impact?

Anand Jangid
Anand Jangid
Numerade Educator
04:28

Problem 22

(a) Three carts of masses $4.00 \mathrm{kg}, 10.0 \mathrm{kg},$ and $3.00 \mathrm{kg}$ move on a frictionless, horizontal track with speeds of $5.00 \mathrm{m} / \mathrm{s}, 3.00 \mathrm{m} / \mathrm{s},$ and $4.00 \mathrm{m} / \mathrm{s},$ respectively, as shown in
Figure P8.22. Velcro couplers make the carts stick together after colliding. Find the final velocity of the train of three carts. (b) Does your answer require that all the carts collide and stick together at the same time? What if they collide in a different order?

Sheh Lit Chang
Sheh Lit Chang
University of Washington
09:10

Problem 23

A tennis ball of mass $57.0 \mathrm{g}$ is held just above a basketball of mass 590 g. With their centers vertically aligned, both are released from rest at the same time, to fall through a distance of $1.20 \mathrm{m},$ as shown in Figure $\mathrm{P} 8.23 .$ (a) Find the magnitude of the downward velocity with which the basketball reaches the ground. Assume that an elastic collision with the ground instantaneously reverses the velocity of the basketball while the tennis ball is still moving down. Next, the two balls meet in an elastic collision. (b) To what height does the tennis ball rebound?

Sheh Lit Chang
Sheh Lit Chang
University of Washington
10:53

Problem 24

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

Rodrigo Diaz-Meneses
Rodrigo Diaz-Meneses
Numerade Educator
02:06

Problem 25

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. The velocities of the two disks are perpendicular after the collision. Determine the final speed of each disk.

Robert Daine
Robert Daine
Numerade Educator
02:47

Problem 26

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 speed $v_{2 i}$. 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 that he was within the speed limit when the collision occurred. Is he telling the truth?

Averell Hause
Averell Hause
Carnegie Mellon University
01:53

Problem 27

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 $4.33 \mathrm{m} / \mathrm{s}$, 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.

Sanjeev Kumar
Sanjeev Kumar
Numerade Educator
01:37

Problem 28

A proton, moving with a velocity of $v_{i} \hat{\mathbf{i}}$, collides elastically with another proton that is initially at rest. Assuming that 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
01:45

Problem 29

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

Sheh Lit Chang
Sheh Lit Chang
University of Washington
04:23

Problem 30

A 0.300 -kg puck, initially at rest on a horizontal, frictionless surface, is struck by a 0.200 -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 Active Fig. 8.11 ).
(a) Determine the velocity of the 0.300 -kg puck after the collision. (b) Find the fraction of kinetic energy lost in the collision.

Sanjeev Kumar
Sanjeev Kumar
Numerade Educator
03:50

Problem 31

An unstable atomic 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.

Narayan Hari
Narayan Hari
Numerade Educator
02:31

Problem 32

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

Robert Daine
Robert Daine
Numerade Educator
02:58

Problem 33

A uniform piece of sheet steel is shaped as shown in Figure P8.33. Compute the $x$ and $y$ coordinates of the center of mass of the piece.

Sheh Lit Chang
Sheh Lit Chang
University of Washington
03:13

Problem 34

A water molecule consists of an oxygen atom with two hydrogen atoms bound to it (Fig. P8.34). 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?

Sanjeev Kumar
Sanjeev Kumar
Numerade Educator
02:10

Problem 35

(a) Consider an extended object whose different portions have different elevations. Assume that the free fall acceleration is uniform over the object. Prove that the gravitational potential energy of the object-Earth system is given by $U_{g}=M g y_{\mathrm{CM}},$ where $M$ is the total mass of the object and $y_{\mathrm{CM}}$ is the elevation of its center of mass above the chosen reference level. (b) Calculate the gravitational potential energy associated with a ramp constructed on level ground with stone with density $3800 \mathrm{kg} / \mathrm{m}^{3}$ and everywhere $3.60 \mathrm{m}$ wide (Fig. $\mathrm{P} 8.35$ ). In a side view, the ramp appears as a right triangle with height $15.7 \mathrm{m}$ at the top end and base $64.8 \mathrm{m}$

Mayukh Banik
Mayukh Banik
Numerade Educator
04:29

Problem 36

A rod of length $30.0 \mathrm{cm}$ has linear density (mass-perlength) given by
$$\lambda=50.0 \mathrm{g} / \mathrm{m}+20.0 \times \mathrm{g} / \mathrm{m}^{2}$$
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
01:26

Problem 37

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

Sanjeev Kumar
Sanjeev Kumar
Numerade Educator
02:10

Problem 38

Consider a system of two particles in the $x y$ plane:
$m_{1}=2.00 \mathrm{kg}$ is at the location $\overrightarrow{\mathbf{r}}_{1}=(1.00 \hat{\mathbf{i}}+2.00 \hat{\mathbf{j}}) \mathrm{m}$
and has a velocity of $(3.00 \hat{\mathbf{i}}+0.500 \hat{\mathbf{j}}) \mathrm{m} / \mathrm{s} ; m_{2}=3.00 \mathrm{kg}$
is at $\overrightarrow{\mathbf{r}}_{2}=(-4.00 \hat{\mathbf{i}}-3.00 \hat{\mathbf{j}}) \mathrm{m}$ and has velocity $(3.00 \hat{\mathbf{i}}-2.00 \hat{\mathbf{j}}) \mathrm{m} / \mathrm{s} .$ (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?

Mayukh Banik
Mayukh Banik
Numerade Educator
01:28

Problem 39

Romeo $(77.0 \mathrm{kg})$ entertains Juliet $(55.0 \mathrm{kg})$ 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
05:57

Problem 40

A ball of mass $0.200 \mathrm{~kg}$ has a velocity of $1.50 \hat{\mathbf{i}} \mathrm{m} / \mathrm{s}$; a ball of mass $0.300 \mathrm{~kg}$ has a velocity of $-0.400 \hat{\mathbf{i}} \mathrm{m} / \mathrm{s}$. 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.

Sheh Lit Chang
Sheh Lit Chang
University of Washington
01:06

Problem 41

The first stage of a Saturn $V$ space vehicle consumed 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 acceleration of the vehicle just as it lifted off the launch pad on the Earth, taking the vehicle's initial mass as $3.00 \times 10^{6} \mathrm{kg} .$ You must include the gravitational force to solve part (b).

Mayukh Banik
Mayukh Banik
Numerade Educator
03:07

Problem 42

Model rocket engines are sized by thrust, thrust duration, and total impulse, among other characteristics. A size $\mathrm{C} 5$ model rocket engine has an average thrust of $5.26 \mathrm{N}$, a fuel mass of $12.7 \mathrm{g},$ and an initial mass of $25.5 \mathrm{g}$. The duration of its burn is $1.90 \mathrm{s}$. (a) What is the average exhaust speed of the engine? (b) If this engine is placed in a rocket body of mass $53.5 \mathrm{g}$, what is the final velocity of the rocket if it is fired in outer space? Assume that the fuel burns at a constant rate.

Suhas Katkar
Suhas Katkar
Numerade Educator
02:07

Problem 43

A rocket for use in deep space is to be capable 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?

Mayukh Banik
Mayukh Banik
Numerade Educator
08:09

Problem 44

Rocket science. A rocket has total mass $M_{i}=360 \mathrm{kg}$, including
$330 \mathrm{kg}$ of fuel and oxidizer. In interstellar space, it starts from rest at the position $x=0,$ turns on its engine at time $t=0,$ and puts out exhaust with relative speed $v_{e}=$ $1500 \mathrm{m} / \mathrm{s}$ at the constant rate $k=2.50 \mathrm{kg} / \mathrm{s} .$ The fuel will last for an actual burn time of $330 \mathrm{kg} /(2.5 \mathrm{kg} / \mathrm{s})=132 \mathrm{s}$
but define a "projected depletion time" as $T_{p}=M_{i} / k=$
$360 \mathrm{kg} /(2.5 \mathrm{kg} / \mathrm{s})=144 \mathrm{s}$ (which would be the burn time
if the rocket could use its payload and 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 $$v(t)=-v_{e} \ln \left(1-\frac{t}{T_{p}}\right)$$ (b) Make a graph of the velocity of the rocket as a function of time for times running from 0 to 132 s. (c) Show that the acceleration of the rocket is $$a(t)=\frac{v_{e}}{T_{p}-t}$$ (d) Graph the acceleration as a function of time. (e) Show that the position of the rocket is $$x(t)=v_{e}\left(T_{p}-t\right) \ln \left(1-\frac{t}{T_{p}}\right)+v_{e} t$$ (f) Graph the position during the burn as a function of time.

Sheh Lit Chang
Sheh Lit Chang
University of Washington
01:15

Problem 45

An orbiting spacecraft is described not as a "zero-g" but rather as a "microgravity" environment for its occupants and for onboard experiments. Astronauts experience slight lurches due to the motions of equipment and other astronauts and as a result of venting of materials from the craft. Assume that a 3500 -kg spacecraft undergoes an acceleration of $2.50 \mu g=2.45 \times 10^{-5} \mathrm{m} / \mathrm{s}^{2}$ due to a leak from one of its hydraulic control systems. The fluid is known to escape with a speed of $70.0 \mathrm{m} / \mathrm{s}$ into the vacuum of space. How much fluid will be lost in $1.00 \mathrm{h}$ if the leak is not stopped?

Sanjeev Kumar
Sanjeev Kumar
Numerade Educator
02:33

Problem 46

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 velocity $\overrightarrow{\mathbf{v}}_{1},$ and the second glider of mass $m_{2}$ moves more slowly, with velocity $\overrightarrow{\mathbf{v}}_{2},$ as shown in Figure $\mathrm{P} 8.46 .$ When $m_{1}$ collides with the spring attached to $m_{2}$ and compresses the spring to its maximum compression $x_{\max },$ the velocity of the gliders is $\overrightarrow{\mathbf{v}}$. In terms of $\overrightarrow{\mathbf{v}}_{1}, \overrightarrow{\mathbf{v}}_{2}, m_{1}, m_{2},$ and $k,$ find (a) the velocity $\overrightarrow{\mathbf{v}}$ at maximum compression, (b) the maximum compression $x_{\max }$ and (c) the velocity of each glider after $m_{1}$ has lost contact with the spring.

Mayukh Banik
Mayukh Banik
Numerade Educator
02:37

Problem 47

Review problem. A 60.0 -kg person running at an initial speed of $4.00 \mathrm{m} / \mathrm{s}$ jumps onto a 120 -kg cart initially at rest (Fig. $\mathrm{P} 8.47$ ). 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 ignored. (a) Find the final velocity of the person and cart relative to the ground. (b) Find the friction force acting on the person while he is sliding across the top surface of the cart. (c) How long does the friction 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 (g) and (h) differ. (What kind of collision is this one, and what accounts for the loss of mechanical energy?)

Mayukh Banik
Mayukh Banik
Numerade Educator
03:19

Problem 48

A bullet of mass $m$ is fired into a block of mass $M$ initially at rest at the edge of a frictionless table of height $h$ (Fig. $P 8.48$ ). 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.

Sheh Lit Chang
Sheh Lit Chang
University of Washington
02:34

Problem 49

When it is threatened, a squid can escape by expelling a jet of water, sometimes colored with camouflaging ink. Consider a squid originally at rest in ocean water of constant density $1030 \mathrm{kg} / \mathrm{m}^{3}$. Its original mass is $90.0 \mathrm{kg}$, of which a significant fraction is water inside its mantle. It expels this water through its siphon, a circular opening of diameter $3.00 \mathrm{cm},$ at a speed of $16.0 \mathrm{m} / \mathrm{s}$. (a) As the squid is just starting to move, the surrounding water exerts no drag force on it. Find the squid's initial acceleration. (b) To estimate the maximum speed of the escaping squid, model the drag force of the surrounding water as described by Equation 5.7 Assume that the squid has a drag coefficient of 0.300 and a cross-sectional area of $800 \mathrm{cm}^{2} .$ Find the speed at which the drag force counterbalances the thrust of its jet.

Sanjeev Kumar
Sanjeev Kumar
Numerade Educator
05:12

Problem 50

Pursued by ferocious wolves, you are in a sleigh with no horses, gliding without friction across an ice-covered lake. You take an action described by these equations:
$$\begin{aligned}(270 \mathrm{kg})(7.50 \mathrm{m} / \mathrm{s}) \hat{\mathbf{i}} &=(15.0 \mathrm{kg})\left(-v_{1 f} \hat{\mathbf{i}}\right)+(255 \mathrm{kg})\left(v_{2 f} \hat{\mathbf{i}}\right) \\
v_{1 f}+v_{2 f} &=8.00 \mathrm{m} / \mathrm{s}\end{aligned}$$ (a) Complete the statement of the problem, giving the data and identifying the unknowns. (b) Find the values of $v_{1 f}$ and $v_{2 f} .$ (c) Find the work you do.

Sheh Lit Chang
Sheh Lit Chang
University of Washington
04:15

Problem 51

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 $\mathrm{P} 8.51$ a. When the block leaves the wedge, its velocity is measured to be $4.00 \mathrm{m} / \mathrm{s}$ to the right as shown in Figure P8.51b. (a) What is the velocity of the wedge after the block reaches the horizontal surface?
(b) What is the height $h$ of the wedge?

Sheh Lit Chang
Sheh Lit Chang
University of Washington
01:14

Problem 52

A jet aircraft is traveling at $500 \mathrm{mi} / \mathrm{h}(223 \mathrm{m} / \mathrm{s})$ 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 $3.00 \mathrm{kg} / \mathrm{s}$. 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 power.

Sanjeev Kumar
Sanjeev Kumar
Numerade Educator
12:08

Problem 53

Review problem. A light spring of force constant $3.85 \mathrm{N} / \mathrm{m}$ is compressed by $8.00 \mathrm{cm}$ and held between a 0.250 -kg block on the left and a 0.500 -kg block on the right. Both blocks are at rest on a horizontal surface. The blocks are released simultaneously so that the spring tends to push them apart. Find the maximum velocity each block attains if the coefficient of kinetic friction between each block and the surface is (a) $0,$ (b) 0.100 , and (c) 0.462 . Assume that the coefficient of static friction is larger than that for kinetic friction.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
00:48

Problem 54

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 workkinetic 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 -kg object has velocity $7.00 \hat{\mathbf{j}} \mathrm{m} / \mathrm{s}$ Then, a total force $12.0 \hat{\mathbf{i}} \mathrm{N}$ acts on the object for 5.00 s. (a) Calculate the object's final velocity,using the impulse-momentum theorem. (b) Calculate its acceleration from $\overrightarrow{\mathbf{a}}=\left(\overrightarrow{\mathbf{v}}_{f}-\overrightarrow{\mathbf{v}}_{i}\right) / \Delta t .$ (c) Calculate its acceleration from $\overrightarrow{\mathbf{a}}=\Sigma \mathbf{F} / m .$ (d) Find the object's vector displacement from $\Delta \overrightarrow{\mathbf{r}}=\overrightarrow{\mathbf{v}}_{i} t+\frac{1}{2} \overrightarrow{\mathbf{a}} t^{2}$. (e) Find the work done
on the object from $W=\overrightarrow{\mathbf{F}} \cdot \Delta \overrightarrow{\mathbf{r}}$. (f) Find the final kinetic energy from $\frac{1}{2} m v_{f}^{2}=\frac{1}{2} m \overrightarrow{\mathbf{v}}_{f} \cdot \overrightarrow{\mathbf{v}}_{f} \cdot(\mathrm{g})$ Find the final kinetic energy from $\frac{1}{2} m v_{i}^{2}+W$.

Mayukh Banik
Mayukh Banik
Numerade Educator
01:46

Problem 55

Two particles with masses $m$ and $3 m$ are moving toward each other along the $x$ axis with the same initial speeds $v_{i}$ The particle with mass $m$ is traveling to the left, and particle $3 m$ 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 particles.

Sanjeev Kumar
Sanjeev Kumar
Numerade Educator
06:40

Problem 56

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

Sheh Lit Chang
Sheh Lit Chang
University of Washington
10:29

Problem 57

George of the Jungle, with mass $m$, swings on a light vine hanging from a stationary tree branch. A second vine of equal length hangs from the same point, and a gorilla of larger mass $M$ swings in the opposite direction on it. Both vines are horizontal when the primates start from rest at the same moment. George and the gorilla meet at the lowest point of their swings. Each is afraid that one vine will break, so they grab each other and hang on. They swing upward together, reaching a point where the vines make an angle of $35.0^{\circ}$ with the vertical. (a) Find the value of the ratio $m / M .$ (b) Try this experiment at home. Tie a small magnet and a steel screw to opposite ends of a string. Hold the center of the string fixed to represent the tree branch and reproduce a model of the motions of George and the gorilla. What changes in your analysis will make it apply to this situation? Assume next that the magnet is strong so that it noticeably attracts the screw over a distance of a few centimeters. Then the screw will be moving faster just before it sticks to the magnet. Does this change make a difference?

Sheh Lit Chang
Sheh Lit Chang
University of Washington
01:57

Problem 58

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 P8.58. The cannon fires a 200 -kg projectile at a velocity of $125 \mathrm{m} / \mathrm{s}$ directed $45.0^{\circ}$ above the horizontal. (a) Assuming that 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 projectile. Is the momentum of this system conserved during the firing? Why or why not?

Sanjeev Kumar
Sanjeev Kumar
Numerade Educator
07:35

Problem 59

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 P8.59. 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 $\overrightarrow{\mathbf{F}}_{\mathrm{ext}}$ 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 $\overrightarrow{\mathbf{F}}_{\mathrm{ext}},(\mathrm{d})$ the work done by $\overrightarrow{\mathbf{F}}_{\mathrm{ext}}$ 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 (d) and (e) different?

Learnmore Shenje
Learnmore Shenje
Numerade Educator
02:55

Problem 60

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 $\mathrm{P} 8.60$ a. Find the force exerted by the table on the chain after the chain has fallen through a distance $x$ as shown in Figure $\mathrm{P} 8.60 \mathrm{b}$. (Assume that each link comes to rest the instant it reaches the table.)

Mayukh Banik
Mayukh Banik
Numerade Educator