A bungee jumper jumps (with no initial speed) from a tall bridge attached to a light elastic cord (bungee cord) of unstretched length L. The cord first straightens and then extends as the jumper falls. This prevents her from hitting the water! Suppose that the bungee cord behaves like a spring with spring constant k = 80 N/m. The bridge is h = 110 m high and the jumper's mass is m = 40 kg. Use g = 10 m/s2.

(a) What is the maximum allowed length L of the unstretched bungee cord (in m) to keep the jumper alive?
(b) Before jumping, our jumper verified the spring constant of the cord. She lowered herself very slowly from the bridge to the full extent of the cord and when she is at rest she measured the distance to the water surface. What was the measured distance (in m)?

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To find the maximum allowed length of the unstretched bungee cord, we need to consider the forces acting on the jumper.

(a) The forces acting on the jumper when the cord is fully extended are gravity (mg) pulling the jumper downwards and the elastic force of the cord pulling the jumper upwards. At the maximum extension, the elastic force is equal to the weight of the jumper.

Setting up the equation:

mg = kΔL

Where ΔL is the change in length of the cord from its unstretched length L.

We can rearrange the equation to solve for ΔL:

ΔL = mg/k

Substituting the given values:

ΔL = (40 kg * 10 m/s^2) / 80 N/m
ΔL = 5 m

The maximum allowed length of the unstretched bungee cord is the sum of the jumper's height (h) and the change in length of the cord (ΔL):

Lmax = h + ΔL
Lmax = 110 m + 5 m
Lmax = 115 m

Therefore, the maximum allowed length of the unstretched bungee cord is 115 meters.

(b) When the jumper lower herself slowly from the bridge to the full extent of the cord and comes to rest, the distance between the jumper and the water surface is the change in length of the cord, ΔL.

Using the equation:

ΔL = mg/k

Substituting the given values:

ΔL = (40 kg * 10 m/s^2) / 80 N/m
ΔL = 5 m

Therefore, the measured distance from the jumper to the water surface is 5 meters.