Question

Kate, a bungee jumper, wants to jump off the edge of a bridge that spans a river below. Kate has a mass m, and the surface of the bridge is a height h above the water. The bungee cord, which has length L when unstretched, will first straighten and then stretch as Kate falls.

Assume the following:
A) The bungee cord behaves as an ideal spring once it begins to stretch, with spring constant .
B) Kate doesn't actually jump but simply steps off the edge of the bridge and falls straight downward.
C) Kate's height is negligible compared to the length of the bungee cord. Hence, she can be treated as a point particle.
Use g for the magnitude of the acceleration due to gravity.
1) How far below the bridge will Kate eventually be hanging, once she stops oscillating and comes finally to rest? Assume that she doesn't touch the water.
Express the distance in terms of quantities given in the problem introduction.
Update:
I already found the answer to 1) to be:
d = L + (mg / k)
2) If Kate just touches the surface of the river on her first downward trip (i.e., before the first bounce), what is the spring constant k? Ignore all dissipative forces.
Express k in terms of L, h, m, and g.

Answer 1

The spring constant k can be found using the formula k = (m * g) / (h + L), where m is the mass of Kate, g is the acceleration due to gravity, h is the height of the bridge, and L is the unstretched length of the bungee cord.

Step-by-step explanation:

To calculate the spring constant k, we can use the equation:

k = (mg - T) / x

where m is the mass of Kate, g is the magnitude of the acceleration due to gravity, T is the tension in the cord, and x is the stretch of the cord (h + L).

Since Kate just touches the surface of the river on her first downward trip, the tension in the cord will be equal to her weight mg.

Substituting the given values, we have:

k = (m * g) / (h + L)

Therefore, the spring constant k is (m * g) / (h + L).