Bungee jumping is a sport where individuals jump off bridges with a budngee cord

Question

Bungee jumping is a sport where individuals jump off bridges with a budngee cord attached to their feet. It is very important to determine the correct length of the bungee cord as to maximize the length of the fall while avoiding a collision with the water or ground below. We can view the bungee jump in two phases. Phase 1 will be considered the free fall phase and phase 2 will be considered the spring-mass system phase. The spring-mass system phase begins at the first stretch of the bungee cord.

Suppose Joe and Fred are standing on a bridge, which is 195 feet above the water. They both have purchased a bungee cord with a k value of 8 lb/ft. The cord will be tied off so it is 100 feet long when it is hanging from the bridge. Call the position at the bottom of the cord 0, and measure the position of Joe's or Fred's feet below as x(t), where x increases as they go down and is a function of time t.

PART 1 - Assume that Joe is 6 feet tall and weighs 192 lbs.

1.) Phase 1 - the free fall phase. Write the equation that models the free fall phase. (Assume resistance from the air and weight of the bungee cord are both negligible).

2.) How long will Joe free fall?

3.) What is Joe's velocity in ft/sec at the equilibrium position?

4.) Find the equation of motion that models Joe's spring-mass system phase.

5.) Does Joe's head hit the water? If his head does hit the water explain how you know this. If his head does not hit the water explain how close his head comes to the water. In either case you must prove your answer.

PART 2 - Assume that Fred is 6 feet 6 inches tall and weighs 288 lbs.

1.) Phase 1 - the free fall phase. Write the equation that models the free fall phase. (Assume resistance from the air and weight of the bungee cord are both negligible).

2.) How long will Fred free fall?

3.) What is Fred's velocity in ft/sec at the equilibrium position?

4.) Find the equation of motion that models Fred's spring-mass system phase.

5.) Does Fred's head hit the water? If his head does hit the water explain how you know this. If his head does not hit the water explain how close his head comes to the water. In either case you must prove your answer.

Explanation / Answer

PART 1

1) for phase one the equation is x(t) = 1/2 gt^2

2) the free fall will be for 100 feet ; put x =100 in above equation you get t as = 2.47 seconds

3) let us find the equilibrium position spring const = 8lb/ft so 192/8 = 24 feet , so eq position = 124ft,

total energy = 785664

enery in spring = 2304

kinetic energy = 783360

velocity = 90.33 ft/s

4) total enery in spring = 785664 at eq postion

so amplitude = 44.31ft

eq = 44.31sin(wt)

5) no

PART 2

1) for phase one the equation is x(t) = 1/2 gt^2

2) the free fall will be for 100 feet ; put x =100 in above equation you get t as = 2.47 seconds

3) let us find the equilibrium position spring const = 8lb/ft so 288/8 = 36 feet , so eq position = 136ft,

you can do the same calculations again as in part 1

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