7 Momentum and Impulse
Answers to Questions
Q1Yes. Impulse is the product of the average force times the time the force acts.
Q2The first force is the larger force because impulse is equal to the product of force and time. If the second time is twice that of the first, then the first force must be twice that of the second force.
Q3Momentum is mass times velocity, so a fast baseball could possess the same momentum as a slow bowling ball.
Q4No. Impulse and force are not the same thing. Impulse depends on force, but it also depends on the length of the time interval the (average) force was applied.
Q5No. Impulse and momentum are not the same thing. Impulse is equal to the change in momentum, either its magnitude or its direction or both.
Q6a.Yes. Momentum is a vector pointing in the direction of the velocity, so a change in direction implies a change in momentum.
b.Yes. The force exerted by the wall on the ball first slows the ball down and continues to act on the ball as it starts to speed up in the opposite direction until it clears the wall. The impulse is the average force on the ball times the time it acts.
Q7The change in momentum of the baseball depends on the impulse. By following through, the bat can remain in contact with the ball longer, yielding a larger impulse.
Q8A padded dashboard gives a longer impact time; the change of momentum (impulse) takes longer thus decreasing the average force of the dash against the rider’s face/body. Average force is decreased because the force is given a longer time to vary. Force is the change of velocity over time and a longer time gives a smaller force for the same change of velocity.
Q9The airbag lengthens the amount of time and distance to stop the occupant. This reduces the amount of force necessary to stop the occupant.
Q10A hard, rapidly expanding airbag is somewhat equivalent to a hard dashboard and does not significantly lengthen the amount of impact time.
Q11Yes. This will result in a longer impact time and the change of momentum will increase, thus decreasing the average force of the ball against the hand.
Q12The truck will require the larger impulse to stop it. Impulse is the change of momentum and though both the bicycle and the truck are going from the same velocity to zero, the truck’s larger mass gives it the greater momentum change.
Q13Momentum is conserved for any system when the net force acting on the system is equal to zero.
Q14No. Momentum is not conserved since there is a net force on the ball, which is the component of weight along the incline.
Q15No, but the total momentum of the two-body system remains constant. An object exerts an impulse on a second object changing the momentum of the second object. The reaction of the second object provides an impulse on the first object changing its momentum. Overall, since the action and reaction forces are equal and opposite and the time each force acts is the same for both, the change in momentum of the second will be equal and opposite to the change in momentum of the first leaving the total momentum of the system unchanged.
Q16Conservation of momentum for a single body follows from Newton's first law. For a system of interacting bodies, Newton's third law justifies the fact that there is no net force on the system as a whole, which is the condition for the total momentum to be conserved.
Q17a.The magnitude of the average force on each vehicle will be the same.
b.Since the force on each vehicle and the time the force acts are the same, each has the same impulse in magnitude.
c.Both vehicles experience the same magnitude of change of momentum, since each experiences the same impulse.
d.The compact car will suffer the greater acceleration. The acceleration is force divided by mass. Both vehicles will be acted upon by the same magnitude of force, but the mass of the compact car is much less than that of the truck.
Q18It is possible if the lighter defensive’s momentum is greater than the heavier fullback’s.
Q19Zero. Because of the third law, the force on each skater is equal and opposite in direction. The momentum gained by one skater will be equal and opposite that gained by the other.
Q20If free to move, both shotguns will possess the same momentum upon recoiling. The lighter one will recoil faster. The impulse delivered to the shooter's shoulder will be the same for both.
Q21Yes. The total momentum of the boat + gun before firing is zero. After firing the shell, the forward momentum of the shell plus the backward momentum of the boat (with gun) must still be zero; hence, the recoil of the boat. Since the boat can be very massive compared to the shell, the recoil speed of the boat may be quite small.
Q22The rocket moves forward because the exhaust material ejected backwards relative to the motion of the rocket exerts a reaction force on the rocket.
Q23Throw the bag of oranges away thus propelling me in the opposite direction. This would be equivalent to “pushing off’’ against the bag of oranges. One might also toss away the oranges singly for a more controlled mode of travel.
Q24She should remove a wrench from her tool belt and throw it in the opposite direction to the location of the space shuttle. By conservation of momentum, she will then gain an equal amount of recoil momentum in the direction of the shuttle.
Q25a.At fan: to the right. At sail: to the left.
b.On fan: to the left. On sail: to the right.
c.If he doesn't furl the sail, he won't go anywhere. The fan blowing air against the sail is an example of internal force. Internal forces do not impart acceleration to a body. With the sail furled he can move opposite the way the fan is blowing. The fan will then be blowing against an external body (the atmosphere).
Q26The skateboard will slow down. By adding the mass of the skateboarder to that of the skateboard, the total mass of the system will increase. By conservation of momentum, if the mass increases, the velocity must decrease in order for the momentum mv to remain constant.
Q27Momentum is conserved, therefore momentum after the collision must equal momentum before the collision (due to the lack of external forces in this problem). Because the mass of the moving system is increased by the addition of the second car, the velocity will decrease.
Q28The coupling of the two cars after collision satisfies the conditions for a perfectly inelastic collision.
Q29No. Momentum is conserved in all collisions, elastic or inelastic, as long as there is no net external force.
Q30No. There is a loss of kinetic energy in the collision, so it is inelastic.
Q31a.No. The direction of the ball’s momentum vector was changed by an external force (the wall).
b.Define the system as the ball, the wall anchored to Earth; then yes, momentum of the system is conserved.
Q32Yes. The collision is elastic. There has been no change in the kinetic energy of the system.
Q33The resultant momentum will point in the direction of the larger momentum, p1, and will have a magnitude (p1 - p2).
Q34No. In this problem the final momentum will be in a direction making a 45o degree angle with respect to each of the initial momentum vectors.
Q35If the velocities had been equal, the direction of motion after the perfectly inelastic collision would have been at a 45o angle with respect to the direction of A. Since the angle is smaller, A must have had a larger momentum and thus was traveling faster.
Q36
The momentum of the truck is twice that of the car, so the final direction of the coupled vehicles is closer to the original direction of the truck than that of the car.
Q37
Answers to Exercises
E1a.12.0 Ns
b.12.0 kg m/s
E2P = 32.4x103 Ns
E3The bowling ball has the larger momentum.
E4a.9 Ns
b.9 kg m/s
E54.8 Ns
E6a.24 Ns
b.160 N
E7a.1080 Ns
b.2700 N
E8a.5.0 Ns
b.5.0 Ns
E9a.-7.5 kg m/s
b.-7.5 Ns
E10a.Pfb = 350 kg m/s; Pdb = -480 kg m/s
b.130 kg m/s (east)
c.In the initial direction of the defensive back, east.
E11a.Zero
b. -7.5 m/s
E12a.3.1 kg m/s
b.2.58 m/s
E132.0 x 104 kg m/s in the forward direction.
E14a.1.44 x 105 kg m/s
b.4.8 m/s
E15a.2.25 x 106 J
b..45 x 106 J
c.Kinetic energy is not conserved; some is lost to
frictional energy during coupling.
E16a.Pt = 4.0 x 104 kg m/s north; Pc = 2.4 x 104 kg m/s south.
b.1.6 x 104 kg m/s, north
E17
E18a.
Scale: 1 cm = 104 kg m/s
b.5.0 x 104 kg m/s
Answers to Synthesis Problems
SP1a.12 kg m/s
b.Yes. Momentum changed direction.
c.12 Ns
d.300 N
SP2a.2.07 m/s
b.2.49 kg m/s
c.Essentially, yes. The mass of the block is large compared to the mass of the bullet so the block carries most of the final momentum.
SP3a.Case B
b.Case B
c.For the ball alone, momentum is not conserved because direction changes. For the system of ball plus wall attached to the earth, momentum is conserved in both cases.
SP4a.-1620 kg m/s
b.Impulse = 1620 kg m/s
c.Wearing a seatbelt distributes the stopping force along the portions on the body restrained by the belt, usually muscular areas of the body. Hitting the dashboard places all the force at the small contact point where the head hits the windshield. The time of action will be slightly longer in the seatbelt scenario because of the action of the seatbelt stop action and the possible stretching of the belt.
SP5a.30,000 kg m/s, South
b.5 m/s South
c.9.75 x 105 J
d.75 x 103 J
e.No. The collision is inelastic; the vehicles stuck together and kinetic energy was not conserved.
1