1. three conserved quantities-energy, momentum, and angular momentum,
2. how those conserved quantities are transferred between objects,
3. the relationships between kinetic and potential energies,
4. equilibrium in general and stable equilibrium in particular,
5. restoring forces in general and spring forces in particular.

We are going to examine your motion in this half pipe as you zip up and down the walls and across the bottom. For simplicity, we will neglect both friction and air resistance in the questions that follow. We will also assume that, apart from the curved regions that connect each wall to the bottom, the half pipe's walls are perfectly vertical and its bottom is perfectly horizontal. We will also neglect any details associated with your size and shape-in effect,you're just a single, small object moving around in a fancy bowl.

1. After a minute or two of effort, you let yourself coast back and forth between the two sides of the half pipe. You keeping rising to the same height on each wall as you shuttle from one side to the other. At one moment, (a), you are coasting toward the left across the flat, horizontal bottom of the pipe and at another moment, (b), you are coasting toward the right across the bottom. Compare

1. the net force on you,
2. your acceleration,
3. your speed,
4. your velocity,
5. your momentum,
6. and your total energy

at those two times, (a) and (b). For example, you might answer two of the six lettered parts:

X. Equal in amount (or magnitude) but opposite in direction.
Y. This quantity is zero at both times.

2. You get back to business and, after some effort, you find yourself rising twice as high up each vertical wall as you did during question 1. How does your new speed along the half pipe's horizontal bottom, as you head toward the left, compare with your speed at that point during question 1?

3. A friend joins you in the half pipe. The two of you are exactly the same size and weight. Your friend is motionless in the middle of the pipe and you are coasting leftward when the two of you accidently collide. You push against one another with your arms for 1 second and avoid injury. As the result of this pushing, you come to a complete stop and your friend is now moving with exactly the direction and speed of your motion before the collision.

But suppose that the two of you had pushed against one another for only 0.5 seconds, with the same resulting motions (your friend assumes your motion and you stop). How would this shorter time of pushing affected (A) the forces the two of you exerted on one another and (B) the impulse you give to your friend?

4. After the collision in question 3, your friend travels up and down the opposite wall and then bumps into you again. You were still motionless when your friend reached you. This time, however, you hold onto one another when you collide and begin moving together instead of separately. (A) How fast do the two of you move and (B) why?

5. Your friend heads home and you are alone again in the half pipe. You get yourself going so hard that you begin to pop up above the top of each vertical wall and are briefly in free fall. While you are airborne above one of the walls, which of the following quantities remain constant?

1. momentum,
2. angular momentum,
3. total energy,
4. kinetic energy,
5. potential energy
   

6. The half pipe becomes crowded again, so you move along to a somewhat different structure. It is also U-shaped, but its bottom never flattens out completely: it curves continuously from its left vertical wall to its right vertical wall. As a result, you find yourself tending to settle into the low point in the middle of the structure. (A) Why is that lowest point a stable equilibrium and what are (B) the net forces and (C) the accelerations that you experience like when you are in or near that lowest point?

7. Ever a generous soul, you decide to use your tremendous skills and experience to teach local school children how to skateboard. Although you rarely fall anymore, you remember how unpleasant it is to bump your knees on the bottom of the half pipe. To soften the blow for one of your tiny students, you rig up a giant overhead spring to act as a safety system.

The top end of the spring connects to a tower high above the half pipe and the bottom end of the spring connects to a harness worn by the pupil. You have selected the spring length and stiffness so that it exerts zero force on your pupil when your pupil is even with the top of the half pipe's vertical walls and exerts an upward force equal to your pupil's weight when your pupil is standing on the horizontal bottom of the half pipe.

One of the first things that your pupil does is to dangle about from the spring and find equilibrium. After a minute or two, you find your pupil hanging motionlessly from the spring, without touching the half pipe at all. (A) At what height is your pupil located (for example: one-third of the way up the half pipe from its bottom or two feet above the top of the half pipe) and (B) why?

8. While this spring arrangement certainly lessens the pain of ordinary falls, your pupil eventually makes the mistake of jumping off the top of a vertical wall toward the middle of the half pipe and hits the bottom surface hard. Use the concepts of (A) net force, (B) acceleration, (C) momentum, and (D) energy to explain why this impact occurs and does not conflict with your answer to question 7.