Question

7.51

A bungee cord is 30.0 m long and. when stretched a distance x, it exerts a restoring force of magnitude kx. Your father-in-law (mass 100 kg) stands on a platform 45.0 m above the ground, and one end of the cord is tied securely to his ankle and the other end to the platform. You have promised him that when he steps off the platform he will fall a maximum distance of only 41.0 m before the cord stops him. You had several bungee cords to select from, and you tested them by stretching them out. tying one end to a tree, and pulling on the other end with a force of 410 N. When you do this, what distance will the bungee cord that you should select have stretched?

Answer 1

After jumping off platform:
At the point where bungee becomes its full un-stretched length,
the ground is 45.0 - 30.0 = 15.0 m away.
The falling father-in-law is at this point (assuming Free-Fall conditions) moving at:
V = √2g(30.0) = √588.6 = 24.26108 m/s and has a KE = 1/2(100.0)(24.261)² = 29430 J.

The selected bungee must therefore absorb this KE
and
the add'l GPE loss {mgx} in 41.0 - 30.0 = 11.0 m. <= x-value in mgx
29430 + (100.0)(9.81)(11.0) = SPE = 1/2k(11.0)²
29430 + 10791 = (0.5)k(121) = 60.5k
k = 40221/60.5 = 664.8 N/m
when pulled with a force = 410 N, the correct bungee would stretch z: F = kz
z = 410/k = 410/664.8 = 0.617 m = 61.7 cm