Figure 8.16 A small object approaches a collision with a much more massive cube, after which its velocity has the direction θ 1 . The angles at which the small object can be scattered are determined by the shape of the object it strikes and the impact parameter b .
8.7 Introduction to Rocket Propulsion
20. Professional Application
Suppose a fireworks shell explodes, breaking into three large pieces for which air resistance is negligible. How is the motion of the center of mass
affected by the explosion? How would it be affected if the pieces experienced significantly more air resistance than the intact shell?
21. Professional Application
During a visit to the International Space Station, an astronaut was positioned motionless in the center of the station, out of reach of any solid object
on which he could exert a force. Suggest a method by which he could move himself away from this position, and explain the physics involved.
22. Professional Application
It is possible for the velocity of a rocket to be greater than the exhaust velocity of the gases it ejects. When that is the case, the gas velocity and gas
momentum are in the same direction as that of the rocket. How is the rocket still able to obtain thrust by ejecting the gases?
284 CHAPTER 8 | LINEAR MOMENTUM AND COLLISIONS
Problems & Exercises
One hazard of space travel is debris left by previous missions. There are
several thousand objects orbiting Earth that are large enough to be
detected by radar, but there are far greater numbers of very small
objects, such as flakes of paint. Calculate the force exerted by a
1. (a) Calculate the momentum of a 2000-kg elephant charging a hunter
0.100-mg chip of paint that strikes a spacecraft window at a relative
at a speed of 7.50 m/s . (b) Compare the elephant’s momentum with the speed of 4.00×103 m/s , given the collision lasts 6.00×10 – 8 s .
momentum of a 0.0400-kg tranquilizer dart fired at a speed of 600 m/s .
13. Professional Application
(c) What is the momentum of the 90.0-kg hunter running at 7.40 m/s
A 75.0-kg person is riding in a car moving at 20.0 m/s when the car runs
after missing the elephant?
into a bridge abutment. (a) Calculate the average force on the person if
2. (a) What is the mass of a large ship that has a momentum of
he is stopped by a padded dashboard that compresses an average of
1.60×109 kg · m/s
1.00 cm. (b) Calculate the average force on the person if he is stopped
, when the ship is moving at a speed of
by an air bag that compresses an average of 15.0 cm.
48.0 km/h? (b) Compare the ship’s momentum to the momentum of a
14. Professional Application
1100-kg artillery shell fired at a speed of 1200 m/s .
Military rifles have a mechanism for reducing the recoil forces of the gun
on the person firing it. An internal part recoils over a relatively large
3. (a) At what speed would a 2.00×104-kg airplane have to fly to have
distance and is stopped by damping mechanisms in the gun. The larger
a momentum of 1.60×109 kg · m/s (the same as the ship’s momentum distance reduces the average force needed to stop the internal part. (a)
Calculate the recoil velocity of a 1.00-kg plunger that directly interacts
in the problem above)? (b) What is the plane’s momentum when it is
with a 0.0200-kg bullet fired at 600 m/s from the gun. (b) If this part is
taking off at a speed of 60.0 m/s ? (c) If the ship is an aircraft carrier that stopped over a distance of 20.0 cm, what average force is exerted upon it
launches these airplanes with a catapult, discuss the implications of your
by the gun? (c) Compare this to the force exerted on the gun if the bullet
answer to (b) as it relates to recoil effects of the catapult on the ship.
is accelerated to its velocity in 10.0 ms (milliseconds).
4. (a) What is the momentum of a garbage truck that is 1.20×104 kg
15. A cruise ship with a mass of 1.00×107 kg strikes a pier at a speed
and is moving at 10.0 m/s ? (b) At what speed would an 8.00-kg trash
of 0.750 m/s. It comes to rest 6.00 m later, damaging the ship, the pier,
can have the same momentum as the truck?
and the tugboat captain’s finances. Calculate the average force exerted
on the pier using the concept of impulse. (Hint: First calculate the time it
5. A runaway train car that has a mass of 15,000 kg travels at a speed of
took to bring the ship to rest.)
5.4 m/s down a track. Compute the time required for a force of 1500 N 16. Calculate the final speed of a 110-kg rugby player who is initially
to bring the car to rest.
running at 8.00 m/s but collides head-on with a padded goalpost and
6. The mass of Earth is 5.972×1024 kg and its orbital radius is an
experiences a backward force of 1.76×104 N for 5.50×10–2 s .
average of 1.496×1011 m . Calculate its linear momentum.
17. Water from a fire hose is directed horizontally against a wall at a rate
of 50.0 kg/s and a speed of 42.0 m/s. Calculate the force exerted on the
wall, assuming the water’s horizontal momentum is reduced to zero.
18. A 0.450-kg hammer is moving horizontally at 7.00 m/s when it strikes
7. A bullet is accelerated down the barrel of a gun by hot gases produced
a nail and comes to rest after driving the nail 1.00 cm into a board. (a)
in the combustion of gun powder. What is the average force exerted on a
Calculate the duration of the impact. (b) What was the average force
0.0300-kg bullet to accelerate it to a speed of 600 m/s in a time of 2.00
exerted on the nail?
ms (milliseconds)?
19. Starting with the definitions of momentum and kinetic energy, derive
8. Professional Application
an equation for the kinetic energy of a particle expressed as a function of
A car moving at 10 m/s crashes into a tree and stops in 0.26 s. Calculate
its momentum
the force the seat belt exerts on a passenger in the car to bring him to a
20. A ball with an initial velocity of 10 m/s moves at an angle 60º above
halt. The mass of the passenger is 70 kg.
the +x -direction. The ball hits a vertical wall and bounces off so that it is
9. A person slaps her leg with her hand, bringing her hand to rest in 2.50
milliseconds from an initial speed of 4.00 m/s. (a) What is the average
moving 60º above the − x -direction with the same speed. What is the
force exerted on the leg, taking the effective mass of the hand and
impulse delivered by the wall?
forearm to be 1.50 kg? (b) Would the force be any different if the woman
21. When serving a tennis ball, a player hits the ball when its velocity is
clapped her hands together at the same speed and brought them to rest
zero (at the highest point of a vertical toss). The racquet exerts a force of
in the same time? Explain why or why not.
540 N on the ball for 5.00 ms, giving it a final velocity of 45.0 m/s. Using
10. Professional Application
these data, find the mass of the ball.
A professional boxer hits his opponent with a 1000-N horizontal blow that
22. A punter drops a ball from rest vertically 1 meter down onto his foot.
lasts for 0.150 s. (a) Calculate the impulse imparted by this blow. (b)
The ball leaves the foot with a speed of 18 m/s at an angle 55º above
What is the opponent’s final velocity, if his mass is 105 kg and he is
the horizontal. What is the impulse delivered by the foot (magnitude and
motionless in midair when struck near his center of mass? (c) Calculate
direction)?
the recoil velocity of the opponent’s 10.0-kg head if hit in this manner,
assuming the head does not initially transfer significant momentum to the
boxer’s body. (d) Discuss the implications of your answers for parts (b)
and (c).
23. Professional Application
11. Professional Application
Train cars are coupled together by being bumped into one another.
Suppose two loaded train cars are moving toward one another, the first
Suppose a child drives a bumper car head on into the side rail, which
having a mass of 150,000 kg and a velocity of 0.300 m/s, and the second
exerts a force of 4000 N on the car for 0.200 s. (a) What impulse is
having a mass of 110,000 kg and a velocity of −0.120 m/s . (The minus
imparted by this force? (b) Find the final velocity of the bumper car if its
initial velocity was 2.80 m/s and the car plus driver have a mass of 200
indicates direction of motion.) What is their final velocity?
kg. You may neglect friction between the car and floor.
24. Suppose a clay model of a koala bear has a mass of 0.200 kg and
12. Professional Application
slides on ice at a speed of 0.750 m/s. It runs into another clay model,
CHAPTER 8 | LINEAR MOMENTUM AND COLLISIONS 285
which is initially motionless and has a mass of 0.350 kg. Both being soft
the second a mass of 7.50×103 kg . (a) Calculate the final velocity
clay, they naturally stick together. What is their final velocity?
(after docking) by using the frame of reference in which the first satellite
25. Professional Application
was originally at rest. (b) What is the loss of kinetic energy in this inelastic
Consider the following question: A car moving at 10 m/s crashes into a
collision? (c) Repeat both parts by using the frame of reference in which
tree and stops in 0.26 s. Calculate the force the seatbelt exerts on a
the second satellite was originally at rest. Explain why the change in
passenger in the car to bring him to a halt. The mass of the passenger is
velocity is different in the two frames, whereas the change in kinetic
70 kg. Would the answer to this question be different if the car with the
energy is the same in both.
70-kg passenger had collided with a car that has a mass equal to and is
36. Professional Application
traveling in the opposite direction and at the same speed? Explain your
answer.
A 30,000-kg freight car is coasting at 0.850 m/s with negligible friction
under a hopper that dumps 110,000 kg of scrap metal into it. (a) What is
26. What is the velocity of a 900-kg car initially moving at 30.0 m/s, just
the final velocity of the loaded freight car? (b) How much kinetic energy is
after it hits a 150-kg deer initially running at 12.0 m/s in the same
lost?
direction? Assume the deer remains on the car.
37. Professional Application
27. A 1.80-kg falcon catches a 0.650-kg dove from behind in midair. What
is their velocity after impact if the falcon’s velocity is initially 28.0 m/s and
Space probes may be separated from their launchers by exploding bolts.
the dove’s velocity is 7.00 m/s in the same direction?
(They bolt away from one another.) Suppose a 4800-kg satellite uses this
method to separate from the 1500-kg remains of its launcher, and that
8.4 Elastic Collisions in One Dimension
5000 J of kinetic energy is supplied to the two parts. What are their
subsequent velocities using the frame of reference in which they were at
28. Two identical objects (such as billiard balls) have a one-dimensional
rest before separation?
collision in which one is initially motionless. After the collision, the moving
object is stationary and the other moves with the same speed as the
38. A 0.0250-kg bullet is accelerated from rest to a speed of 550 m/s in a
other originally had. Show that both momentum and kinetic energy are
3.00-kg rifle. The pain of the rifle’s kick is much worse if you hold the gun
conserved.
loosely a few centimeters from your shoulder rather than holding it tightly
against your shoulder. (a) Calculate the recoil velocity of the rifle if it is
29. Professional Application
held loosely away from the shoulder. (b) How much kinetic energy does
Two manned satellites approach one another at a relative speed of 0.250
the rifle gain? (c) What is the recoil velocity if the rifle is held tightly
m/s, intending to dock. The first has a mass of 4.00×103 kg , and the
against the shoulder, making the effective mass 28.0 kg? (d) How much
kinetic energy is transferred to the rifle-shoulder combination? The pain is
second a mass of 7.50×103 kg . If the two satellites collide elastically
related to the amount of kinetic energy, which is significantly less in this
latter situation. (e) Calculate the momentum of a 110-kg football player
rather than dock, what is their final relative velocity?
running at 8.00 m/s. Compare the player’s momentum with the
30. A 70.0-kg ice hockey goalie, originally at rest, catches a 0.150-kg
momentum of a hard-thrown 0.410-kg football that has a speed of 25.0
hockey puck slapped at him at a velocity of 35.0 m/s. Suppose the goalie
m/s. Discuss its relationship to this problem.
and the ice puck have an elastic collision and the puck is reflected back
39. Professional Application
in the direction from which it came. What would their final velocities be in
this case?
One of the waste products of a nuclear reactor is plutonium-239
⎛239
⎝
Pu⎞⎠ . This nucleus is radioactive and decays by splitting into a
8.5 Inelastic Collisions in One Dimension
4
31. A 0.240-kg billiard ball that is moving at 3.00 m/s strikes the bumper
helium-4 nucleus and a uranium-235 nucleus ⎛⎝ He + 235 U⎞⎠ , the latter
of a pool table and bounces straight back at 2.40 m/s (80% of its original
of which is also radioactive and will itself decay some time later. The
speed). The collision lasts 0.0150 s. (a) Calculate the average force
exerted on the ball by the bumper. (b) How much kinetic energy in joules
energy emitted in the plutonium decay is 8.40×10 – 13 J and is entirely
is lost during the collision? (c) What percent of the original energy is left?
converted to kinetic energy of the helium and uranium nuclei. The mass
32. During an ice show, a 60.0-kg skater leaps into the air and is caught
of the helium nucleus is 6.68×10 – 27 kg , while that of the uranium is
by an initially stationary 75.0-kg skater. (a) What is their final velocity
assuming negligible friction and that the 60.0-kg skater’s original
3.92×10 – 25 kg (note that the ratio of the masses is 4 to 235). (a)
horizontal velocity is 4.00 m/s? (b) How much kinetic energy is lost?
Calculate the velocities of the two nuclei, assuming the plutonium
33. Professional Application
nucleus is originally at rest. (b) How much kinetic energy does each
nucleus carry away? Note that the data given here are accurate to three
Using mass and speed data from Example 8.1 and assuming that the
digits only.
football player catches the ball with his feet off the ground with both of
them moving horizontally, calculate: (a) the final velocity if the ball and
40. Professional Application
player are going in the same direction and (b) the loss of kinetic energy in The Moon’s craters are remnants of meteorite collisions. Suppose a fairly
this case. (c) Repeat parts (a) and (b) for the situation in which the ball
large asteroid that has a mass of 5.00×1012 kg (about a kilometer
and the player are going in opposite directions. Might the loss of kinetic
energy be related to how much it hurts to catch the pass?
across) strikes the Moon at a speed of 15.0 km/s. (a) At what speed does
the Moon recoil after the perfectly inelastic collision (the mass of the
34. A battleship that is 6.00×107 kg and is originally at rest fires a
Moon is 7.36×1022 kg ) ? (b) How much kinetic energy is lost in the
1100-kg artillery shell horizontally with a velocity of 575 m/s. (a) If the
shell is fired straight aft (toward the rear of the ship), there will be
collision? Such an event may have been observed by medieval English
negligible friction opposing the ship’s recoil. Calculate its recoil velocity.
monks who reported observing a red glow and subsequent haze about
(b) Calculate the increase in internal kinetic energy (that is, for the ship
the Moon. (c) In October 2009, NASA crashed a rocket into the Moon,
and the shell). This energy is less than the energy released by the gun
and analyzed the plume produced by the impact. (Significant amounts of
powder—significant heat transfer occurs.
water were detected.) Answer part (a) and (b) for this real-life experiment.
The mass of the rocket was 2000 kg and its speed upon impact was
35. Professional Application
9000 km/h. How does the plume produced alter these results?
Two manned satellites approaching one another, at a relative speed of
41. Professional Application
0.250 m/s, intending to dock. The first has a mass of 4.00×103 kg , and Two football players collide head-on in midair while trying to catch a
thrown football. The first player is 95.0 kg and has an initial velocity of
286 CHAPTER 8 | LINEAR MOMENTUM AND COLLISIONS
6.00 m/s, while the second player is 115 kg and has an initial velocity of
conservation of momentum along the x -axis and y -axis; instead, you
–3.50 m/s. What is their velocity just after impact if they cling together?
must look for other simplifying aspects.
42. What is the speed of a garbage truck that is 1.20×104 kg and is
51. Starting with equations m 1 v 1 = m 1 v′1 cos θ 1 + m 2 v′2 cos θ 2
initially moving at 25.0 m/s just after it hits and adheres to a trash can
and 0 = m
that is 80.0 kg and is initially at rest?
1 v′1 sin θ 1 + m 2 v′2 sin θ 2 for conservation of momentum
in the x - and y -directions and assuming that one object is originally
43. During a circus act, an elderly performer thrills the crowd by catching
a cannon ball shot at him. The cannon ball has a mass of 10.0 kg and the stationary, prove that for an elastic collision of two objects of equal
horizontal component of its velocity is 8.00 m/s when the 65.0-kg
masses, 1
2 = 1
2+1
2+ mv′
⎞
performer catches it. If the performer is on nearly frictionless roller
2 mv 1
2 mv′1 2 mv′2
1 v′2 cos ⎛⎝ θ 1 − θ 2⎠
skates, what is his recoil velocity?
as discussed in the text.
44. (a) During an ice skating performance, an initially motionless 80.0-kg
clown throws a fake barbell away. The clown’s ice skates allow her to
52. Integrated Concepts
recoil frictionlessly. If the clown recoils with a velocity of 0.500 m/s and
A 90.0-kg ice hockey player hits a 0.150-kg puck, giving the puck a
the barbell is thrown with a velocity of 10.0 m/s, what is the mass of the
velocity of 45.0 m/s. If both are initially at rest and if the ice is frictionless,
barbell? (b) How much kinetic energy is gained by this maneuver? (c)
how far does the player recoil in the time it takes the puck to reach the
Where does the kinetic energy come from?
goal 15.0 m away?
8.6 Collisions of Point Masses in Two Dimensions
8.7 Introduction to Rocket Propulsion
45. Two identical pucks collide on an air hockey table. One puck was
53. Professional Application
originally at rest. (a) If the incoming puck has a speed of 6.00 m/s and
scatters to an angle of 30.0º ,what is the velocity (magnitude and
Antiballistic missiles (ABMs) are designed to have very large
accelerations so that they may intercept fast-moving incoming missiles in
direction) of the second puck? (You may use the result that
θ
the short time available. What is the takeoff acceleration of a 10,000-kg
1 − θ 2 = 90º for elastic collisions of objects that have identical
ABM that expels 196 kg of gas per second at an exhaust velocity of
masses.) (b) Confirm that the collision is elastic.
2.50×103 m/s?
46. Confirm that the results of the example Example 8.7 do conserve
54. Professional Application
momentum in both the x - and y -directions.
What is the acceleration of a 5000-kg rocket taking off from the Moon,
47. A 3000-kg cannon is mounted so that it can recoil only in the
where the acceleration due to gravity is only 1.6 m/s2 , if the rocket
horizontal direction. (a) Calculate its recoil velocity when it fires a 15.0-kg
shell at 480 m/s at an angle of 20.0º above the horizontal. (b) What is
expels 8.00 kg of gas per second at an exhaust velocity of
the kinetic energy of the cannon? This energy is dissipated as heat
2.20×103 m/s?
transfer in shock absorbers that stop its recoil. (c) What happens to the
55. Professional Application
vertical component of momentum that is imparted to the cannon when it
is fired?
Calculate the increase in velocity of a 4000-kg space probe that expels
48. Professional Application
3500 kg of its mass at an exhaust velocity of 2.00×103 m/s . You may
A 5.50-kg bowling ball moving at 9.00 m/s collides with a 0.850-kg
assume the gravitational force is negligible at the probe’s location.
bowling pin, which is scattered at an angle of 85.0º to the initial direction 56. Professional Application
of the bowling ball and with a speed of 15.0 m/s. (a) Calculate the final
Ion-propulsion rockets have been proposed for use in space. They
velocity (magnitude and direction) of the bowling ball. (b) Is the collision
employ atomic ionization techniques and nuclear energy sources to
elastic? (c) Linear kinetic energy is greater after the collision. Discuss
produce extremely high exhaust velocities, perhaps as great as
how spin on the ball might be converted to linear kinetic energy in the
8.00×106 m/s . These techniques allow a much more favorable
collision.
payload-to-fuel ratio. To illustrate this fact: (a) Calculate the increase in
49. Professional Application
velocity of a 20,000-kg space probe that expels only 40.0-kg of its mass
Ernest Rutherford (the first New Zealander to be awarded the Nobel
at the given exhaust velocity. (b) These engines are usually designed to
Prize in Chemistry) demonstrated that nuclei were very small and dense
produce a very small thrust for a very long time—the type of engine that
by scattering helium-4 nuclei ⎛4
197
might be useful on a trip to the outer planets, for example. Calculate