1. One friend measures off 1000 feet of elastic cord and ties it between his ankles and the bridge. He is about to jump when you caution him that he has made mistake and that he won't enjoy trip. What is wrong with his using 1000 feet of cord?

Answer: He will hit the ground.

Why: He won't reach zero velocity until he has accelerated upward long enough to get rid of all his downward speed. That won't happen until the cord has stretch so far that it is pulling upward on him with considerably more force than his own weight. Unfortunately, he'll hit the ground first.

2. Another friend is nervous about breaking the cord, despite weighing less than 100 pounds, and she selects the very stiffest of the elastic cords. Although she chooses the appropriate length for that cord, you caution her that she still won't enjoy the trip. What is wrong with her using the stiffest cord?

Answer: The stiff cord will accelerate her upward very quickly once it pulls taut and will exert an uncomfortably large upward force on her to achieve that upward acceleration.

Why: The stiff cord resists stretching by exerting large tensions even when stretched only a little. Yes, she will bounce once it pulls taut and begins to stretch. However, the time over which she will lose her downward momentum will be short and will involve unpleasantly large forces.

3. Soon, you have everyone set up with cords of appropriate lengths and stiffnesses. It's time to begin jumping. The first jumper steps off the bridge and plummets toward the ground below. The cord pulls taut and begins to pull upward on him. At the exact moment that the cord first pulls taut and has not yet begun to stretch, what is your friend's acceleration?

Answer: The downward acceleration due to gravity.

Why: Just because the cord pulls taut doesn't mean that your friend will stop descending, or even stop accelerating downward. The cord will exert an upward force on him only when it begins to stretch. At the moment when it first pulls taut, the cord hasn't started to stretch yet and therefore doesn't exert any upward force on him. He is still in free fall!

4. Your friend bounces up and down a number of times before coming to rest and you are timing his bounces. During the first two upward bounces, he rises so high that the cord briefly goes slack. After that, the cord remains taut as he travels up and down. You find that the time between those first two upward bounces is longer than between any subsequent bounces. Why?

Answer: The first bounces are not harmonic oscillator bounces and involve a weaker restoring force: gravity.

Why: For your bouncing friend to be a harmonic oscillator, the restoring force he experiences must be proportional to how far he is from equilibrium. As long as the cord remains taut, that restoring force will be proportional to distance from equilibrium. But once the cord goes slack, the force stops changing. It's then just his weight. As a result of this weaker-than-springlike restoring force, he takes longer to complete those first bounces than he does once his bouncing becomes harmonic.

5. The bounces during which the cord remains taut all take exactly the same amount of time to complete, even though they are of ever decreasing amplitude. Why?

Answer: The restoring force he experiences (once the cord stays taut) is proportional to his distance from equilibrium, so he is a harmonic oscillator. Harmonic oscillators have periods and frequencies that are independent of their amplitudes of oscillation.

Why: Your friend on the taut cord is essentially a mass on a spring, the classic harmonic oscillator. As such, he oscillates with a period and frequency that are independent of the amplitude of oscillation.

6. Two of your friends decide to jump simultaneously using identical cords. One friend is considerably heavier than the other. They step off the bridge together and begin to bounce up and down. The cords are remaining taut as they go up and down. Do they take equal times to complete their bounces or is one of them bouncing faster than the other? Why?

Answer: The lighter friend completes her bounces faster because she has less mass to slow the harmonic motion.

Why: Your two friends are essentially two different masses on two identical springs. The springs exert equivalent restoring forces, so your low-mass friend will cycle through her motion faster than your high-mass friend.

7. Finally, it's your turn to jump. You step off the bridge and are soon bouncing up and down on the cord attached to your ankles. The cord stays taut as you bounce and you begin to think about the situation. When during your bounces is (A) your kinetic energy at its maximum, (B) at its minimum, and when during your bounces is (C) your total potential energy at its maximum, (D) at its minimum?

Answer: (A) at equilibrium, (B) at the top or bottom of the bounce, (C) at the top or bottom of the bounce, (D) at equilibrium.

Why: At equilibrium, all of the available potential energy has been converted into kinetic energy so the former is minimal and the latter is maximal. The fact that the total potential energy is at a minimum at equilibrium explains why there is a restoring force: objects accelerate in the direction that reduces their total potential energy as quickly as possible and that direction is toward equilibrium. At the top or bottom of the bounce, you slow to a full stop, so you have zero kinetic energy. Recognizing that your energy is conserved, you know that it must now be in potential form. At the top of a bounce, it is mostly gravitational potential energy. At the bottom of a bounce, it is mostly elastic potential energy in the cord. But otherwise, the two extreme positions have roughly equal total potential energies. In fact, they'd be exactly equal if it weren't for the gradual wasting of total energy through friction in the cord (and in real life, through air resistance).

8. Your friends are always looking for fun, so they grab the upper end of your bungee cord and begin to yank it upward rhythmically. When they get their timing just right, you begin to bounce more and more wildly up and down. Why was it important that they yank rhythmically at just the right frequency and what frequency did they chose to make you bounce so wildly?

Answer: To achieve resonant energy transfer, they must yank the cord upward at the same frequency as the frequency of your bouncing.

Why: Because they must do a little work on you each time you bounce, they must pull upward on the cord in synchrony with your bouncing. More specifically, they must yank upward on the cord as you move upward, so that they do a little work on you each bounce. As a result, you will bounce wildly.