1. bouncing behavior in springs, balls, and other elastic objects,
2. relative motion and the interactions between objects in relative motion,
3. connections between translational and rotational motions,
4. connections between acceleration and force,
5. the way we perceive acceleration.
   

1. One of the big hits among the braver members of your group is the free fall tower. You and your little charges are strapped into individual seats on a bench that is gradually hauled up to the top of a vertical tower by a cable. After hanging stationary at the top of this tower for a few moments, the bench is suddenly released. Though rails keep the bench from drifting away from the tower, the bench nonetheless drops freely for a terrifying second or two before brakes activate and slow the bench to a stop. When during this trip from the top of the tower to the bottom do you (A) feel the lightest and (B) the heaviest? In each case, briefly explain why you feel that way.

Answer: (A) You feel lightest while you are in free fall (at the beginning of your descent) because that is when you are accelerating downward most rapidly. (B) You feel heaviest while you are braking to a stop (near the end of your descent) because that is when you are accelerating upward most rapidly.

Why: Accelerations cause you to feel fictitious "forces" in the directions opposite those accelerations. When you accelerate downward at the beginning of your descent, you feel a fictitious "force" upward that balances your feelings of weight and you experience an apparent weight of zero. You feel weightless. But as you accelerate upward near the end of your descent, the fictitious "force" you feel is downward and you feel an extremely large apparent weight in the downward direction.

2. You move your gang along to the swinging pirate ship, a huge pretend boat that hangs from a central pivot high overhead. After you are all strapped into seats in the boat, it begins to swing back and forth, higher and higher. Eventually the boat swings so high that it travels over the top of the pivot. The boat and its occupants are upside-down for a few seconds. The motion is slow and during this time, loose change, hats, sunglasses, food, and other items come raining out of the boat onto the ground below. In the normal loop-the-loop of a roller coaster, everyone feels pressed into their seats and nothing falls out of the cars. Why in this case do the boat riders feel that they are hanging upside-down and why is this situation so different from that of the loop-the-loop?

Answer: Although the boat riders are traveling in a huge circle, and are thus accelerating roughly toward its center, they move so slowly that their acceleration is small compared with the acceleration due to gravity. Their apparent weights are thus dominated by their true weights and loose objects simply fall away from the car. In a loop-the-loop, the acceleration is comparable to the acceleration due to gravity and apparent weight is dominated by that acceleration.

Why: Because its motion is so slow, the huge inverted boat is almost moving at constant velocity. The effects of acceleration are so small that the boat and its contents behave as though they were simply hanging upside-down. The riders feel themselves hanging and they lose the contents of their pockets. But in a normal loop-the-loop, the motion is fast enough that there is a rapid downward acceleration as the inverted roller coaster travels over the top of the loop. Since the coaster is accelerating downward faster than the acceleration due to gravity, apparent weights are upward and the plummeting car will actually overtake any loose object that wiggles free of someone's pocket.

3. Sickening though it is, the egg scrambler is still a favorite among 8-year-olds. You sit in your circular cars as this giant mixer throws each car back and forth across a large circle. Although the whole ride gradually pivots, the main motion is very simple: you begin almost at rest near the edge of the circle, then swing at tremendous speed through the center of the circle, and finish almost at rest near the opposite edge of the circle. Even when you close your eyes, you can't ignore the motion; you frequently feel it pressing you fiercely against your seat or against your fellow passengers. Briefly explain why this sideways squishing effect (A) is most severe during the start and stop of each swing and (B) vanishes briefly as your car passes through the center of the circle.

Answer: (A) At the start or stop of each swing, the car is shifting from motionless to high speed and is therefore accelerating rapidly. You feel squished in the direction opposite this acceleration. (B) As the car passes through the center of the circle, it is moving with roughly constant velocity and is not accelerating. You are simply coasting forward and feel no squishing at all.

Why: The fictitious "forces" responsible for the squishing feelings result from rapid accelerations. Each time you accelerate to the right, you feel squished into whatever is on your left. In reality, that stuff on your left is pushing on you to make you accelerate to the right. If that stuff has to push too hard on you and is a person rather than a seat, it may complain loudly.

4. The skydiving ride isn't open to the children, but you take a turn anyway. They cheer you on and laugh as you scream on the way down. The ride resembles a 200-foot-tall playground swing, except that instead of being pushed gradually to greater and greater heights, you are pulled backward and upward just once and then released. You are so high at the start that you begin your descent in near free fall. During those first moments, you feel weightless and terrified, which explains why you are bellowing loudly enough for the kids to hear. But how do you feel as you swing through center and make your closest approach to the ground? Are you still weightless or is there some other apparent weight that you are experiencing?

Answer: As you swing through center, you feel a downward apparent weight that is greater than your actual weight.

Why: As you swing through center, you are essentially in circular motion around the overhead pivot. Your acceleration is therefore centripetal and the fictitious "force" you experience is directed away from the pivot and into the ground. This fictitious "force" combines with your true weight to give you a strong downward apparent weight.

5. You now enter the arcade part of the park and watch people try to win prizes by performing seemingly simple feats. Not surprisingly, most of those feats are deceptively difficult.

The first game you encounter is a ring-toss in which you must throw a small, rigid plastic hoop around the neck of a glass soda bottle. An array of motionless upright bottles fills the center of the booth and it looks as though it would be easy to toss a hoop so that it would fall down around one of the bottle necks. But in 10 minutes of watching, not a single hoop stays on a bottle neck. They all bounce up and off. (A) Use the concept of conservation of energy and the fact that both the hoops and the bottles have coefficients of restitution of almost 1.0 to explain briefly why the hoops can't stay on the bottle necks. (B) Why would replacing the rigid hoops with similarly shaped beanbag rings make this game relatively simple to win?

Answer: (A) Each falling hoop has at least enough total energy to lift it to the height from which it fell. Since both of the objects involved in this game, hoop and bottle, are highly elastic, neither one wastes much energy during a collision. The hoop therefore has no easy way to get rid of its total energy and keeps bouncing around among the bottles until it falls onto something softer, like the floor. (B) A beanbag ring wastes energy nicely and would barely bounce at all after colliding with a glass bottle. If it landed properly around a bottle's neck, it would stay there.

Why: The hoops descend on the bottles with so much extra energy that they have trouble settling down on anything. Their elastic nature makes them bounce around until they fall between the bottles or hit the floor. Only by reducing this bounciness, perhaps by replacing the hard elastic plastic hoop with a soft, non-elastic beanbag ring, can the game be made easy.

6. The next arcade game involves tossing quarters onto glass salad plates arranged horizontally in the middle of the booth. Those plates are almost flat and have a shiny, smooth surface. Not surprisingly, the quarters never stop on the plates and all end up on the floor below. Use the concepts of energy and momentum to explain briefly why this game is so nearly impossible to win.

Answer: For the coin to come to rest on the plate, it must get rid of its total energy and its momentum, both of which can only be eliminated by transfer elsewhere or, in the case of energy, by a change in form. The plate doesn't move, so the coin can't transfer energy to it and the plate is slippery so the coin can't transfer horizontal momentum to it. Finally, both the plate and coin are so elastic that they change very little energy into heat during a collision. Unable to get rid of its energy or forward momentum, the coin flies off the plate and is lost.

Why: For the coin to get rid of its initial energy, it has to do work on something or waste that energy as heat. Neither of these is possible because the plate won't move and the two objects are too bouncy to waste energy as heat. For the coin to get rid of its initial horizontal momentum, something must push it backward for a period of time. The plate is horizontal, so it exerts only vertical support forces, and it is slippery, so it exerts very little frictional force. As a result, the coin is unable to get rid of its horizontal momentum and flies off the plate.

7. Another game you enjoy watching involves tossing a basketball into a stiff fruit basket that's mounted with its bottom against a slanted wall. The basket is tipped back just enough that it can hold a basketball, but it doesn't take much to make that basketball roll out and fall to the floor. Over and over, people toss balls into the baskets only to watch them bounce or roll back out. You point out that this feat would be much easier if the baskets were moving when the balls landed in them. Which way should the basket be moving as the ball arrives for this feat to become easier and why that direction?

Answer: The basket should be moving away from the ball when it arrives because then the ball would do work on the basket and would thus get rid of some of its total energy.

Why: The ball normally bounces back out of the basket because it can't get rid of its energy. The basket is too stiff and stationary for the ball to do work on it, so the ball keeps moving until it leaves the basket. But if the basket were to move away from the ball as the ball arrives, then the ball will do work on the basket and the ball will lose energy. That energy loss increases the likelihood that the ball will stay in the basket.

8. A final arcade game requires that you swing a huge mallet over your head and pound downward on the end of a lever. The other end of the lever then flies upward and strikes a block, which immediately slides up a tall vertical track toward a bell. If you hit the lever hard enough, the bell will ring and you'll win a prize.

All the kids are cheering you on as you swing the mallet and drive it violently into the lever. The far end of the level swings upward rapidly, smacks the block hard, and sends that block up the track so that it rings the bell. You're an instant celebrity. Of course, your chances for success were greatly improved when the game operator allowed you to stick whatever material you wanted to the top of the lever and the bottom of the block. You chose very elastic materials (coefficients of restitution of almost 1.0) for both surfaces and that made all the difference. Why is it so helpful to have highly elastic surfaces colliding in this case?

Answer: The lever is batting the block up the track. With elastic materials between the two objects, the block will rebound from the lever at roughly the same relative speed as the two had when they collided. The block will therefore move up and away from the lever at a large relative speed and will rise higher than it would otherwise do.

Why: If the lever and block are not elastic, then the lever will simply push the block up the track and the two will rise at the same speed. But if the two are highly elastic, then the lever will bat the block upward so that the block will effectively "bounce" off the rising lever. As a result of this bounce, the block will travel upward at twice the speed of the rising lever. Since the block is thus traveling twice as fast as it would from a non-elastic lever, the block will have four times as much kinetic energy and will ultimately rise four times as high.