Problem Set 5, Linear Momentum
Topics 4.1 through 4.4. Out of 100 points.
Problem 117 points
Object A has mass 2.00 kg and object B has mass 8.00 kg. Each one carries a momentum of magnitude 12.0 kg· m/s along the same straight line. (4.1)
(a)Find the speed of each object.4 pts
(b)Find the kinetic energy of each object, working from the momentum rather than from the speed.4 pts
(c)On the grid, to a stated scale, sketch the kinetic energy of a 2.00 kg object against its momentum for momenta from 0 to 20.0 kg· m/s.4 pts
(d)A student says two objects with the same momentum must be equally dangerous to stand in front of. Write a short argument using K=p2/2m, and say what the student has confused with what.5 pts
Problem 217 points
A net force F(t)=12.0t−3.00t2 newtons acts along a straight line on an object of mass 2.00 kg that is initially at rest. The force acts from t=0 to t=4.00 s. (4.2)
(a)Find the impulse delivered over the whole interval, and the speed of the object at t=4.00 s.4 pts
(b)Find the time at which the force is largest, and the momentum of the object at that instant.4 pts
(c)On the grid, to a stated scale, sketch the momentum of the object against time over the interval.4 pts
(d)State the two times in the interval at which the force is zero, and say what is happening to the momentum at each of them. Then say which feature of your sketch corresponds to the instant of largest force, and why.5 pts
Problem 316 points
Sand falls vertically onto a horizontal conveyor belt at a steady 4.00 kg/s. The belt is driven so that it holds a constant speed of 1.50 m/s. The sand lands with no horizontal velocity. (4.2)
(a)Find the horizontal force that must be applied to the belt to hold its speed.4 pts
(b)Find the power that force delivers.4 pts
(c)Find the rate at which the kinetic energy of the sand on the belt increases.4 pts
(d)Your answers to (b) and (c) differ. Account for the difference, and say why F=ma cannot be used to find the force in part (a).4 pts
Problem 417 points
Cart A has mass 2.00 kg and moves to the right at 6.00 m/s. Cart B has mass 4.00 kg and moves to the left at 1.50 m/s. They are on the same level frictionless track and they collide. (4.3)
(a)Find the total momentum of the two carts and the velocity of their center of mass.4 pts
(b)State the velocity of the center of mass immediately after the collision, and justify your answer.4 pts
(c)On the grid, draw a bar chart of the momentum of each cart and of the system, before the collision and after the carts stick together.4 pts
(d)Show that the total momentum of a system equals its total mass times the velocity of its center of mass, and use that result to say why conservation of momentum and a constant center of mass velocity are the same statement.5 pts
Problem 516 points
A puck of mass 0.500 kg slides east at 4.00 m/s across frictionless ice and strikes a second puck of mass 0.500 kg that is at rest. After the collision the first puck moves off at 30.0° north of east. The collision is elastic. (4.3)
(a)Write the two momentum conservation equations and the kinetic energy condition, taking east as the positive x direction.4 pts
(b)Find the speed of each puck after the collision.4 pts
(c)Find the direction of the second puck, and state the angle between the two outgoing velocities.4 pts
(d)On the grid, to a stated scale, draw the momentum vector before the collision and the two after it, and say what your drawing shows about the total.4 pts
Problem 617 points
A cart of mass 2.00 kg moves at 3.00 m/s along a level track and strikes a cart of mass 4.00 kg that is at rest. (4.4)
(a)The carts stick together. Find their velocity and the kinetic energy lost.4 pts
(b)The collision is instead elastic. Find the velocity of each cart afterward, and verify that both conservation statements hold.4 pts
(c)Find the fraction of the original kinetic energy that survives in each of the two cases.4 pts
(d)Using K=p2/2m for the system, prove that no collision between these two carts can lose more kinetic energy than the case in part (a), and say what would have to happen physically for more to be lost.5 pts