Bungee Jumping Spring Constat/Work-Energy

[bang]

Homework Statement

You have persuaded your friend Astrid to attempt an illegal bungee jump from the New River Gorge Bridge. You will provide the bungee cord which has a total length of 40 m and a spring constant of k = 16 N/m. During the jump, Astrid will first fall freely for a distance equal to the length of the cord, after which the cord will begin to stretch, obeying Hooke’s law. Astrid’s mass is 52 kg. The lowest point she reaches before rebounding is _____________ below the bridge.

Homework Equations

No idea really, but I assume:
f=kx
PE=1/2kx^2
PE=mgh
KE=1/2mv^2

The Attempt at a Solution

So I found the velocity she has right when the bungee is at it's normal length of 40m, but I have no idea where to go from there. The velocity I calculated was 28m/s, and that might not be right.

 

[Ibix]

There are two phases to Astrid's motion, so three interesting times. Can you talk us through what's happening with forces and energies at each phase and the three times?

Agree 28m/s, by the way.

 

[bang]

So for the first phase I just used her potential energy from the bridge to the extension of the bungee, the next interesting time is when the bungee starts affecting her fall, and the third interesting time is when she is completely stopped at her lowest point. I've found the important information about the first half of her fall, but I have no idea how to approach the second half since gravity has an effect as well.

 

[Averki]

What forces are acting on Astrid as she falls? Specifically, what forces are acting on her just before she rebounds and how are these forces related?

 

[bang]

The elastic force pulling up, and the gravitational force pulling down. How would I relate the two though?

 

[Averki]

Start by drawing a free body diagram. This should help you visualize the next step.

 

[bang]

So I have to figure out the point at which the bungee would completely counteract the force of gravity pulling down. I get 31.85m, which brings the lowest point to basically 70m but that isn't the correct answer.

 

[Ibix]

No, because no net force means no acceleration, not no velocity. If I were you, I'd stick to energies. What types of energy are in play at each of the three interesting times?

 

[bang]

Gravitational potential at the top, right before the spring both kinetic and still gravitational potential, and then at the bottom kinetic would be 0 and the spring potential would be at it's maximum?

 

[Ibix]

Right. So can you write an expression for the energies at the different times? Do you know the relationship between them?