A bungee jumper of mass m uses a massless bungee cord of (unstretched) length L

Question

A bungee jumper of mass m uses a massless bungee cord of (unstretched) length L and spring constant k to jump off a bridge. At the bottom of jump, the bungee cord will be stretched to a total length L + L. Assume the jumper hauls the bungee cord up onto the bridge and ties the end to her waist before jumping off. Also assume no air resistance.

[a] What is the speed of the jumper after she has fallen a distance L? (check units)

[b] Determine how far the bungee cord stretches at maximum extension (i.e. L) in terms of

m, g, k, and L. Check units.

[c] What speed will the bungee jumper have when she bounces back up to a distance L below

the bridge (such that bungee cord is unstretched)?

[d] What is the maximum height that the bungee jumper reaches after her first bounce?

Explanation / Answer

a. P.E lost = mgL
K.E. gained = 0.5mv^2
from conservation of energy , 0.5mv^2 = mgL
v = sqroot(2gL)

b. let max extension be dx
then energy stored in chord = 0.5k(dx)^2
PE lost = mg(L + dx)
KE = 0
so from conservation of energy
0.5k(dx)^2 = mg(L+dx)
0.5k(dx)^2 - mgdx - mgL = 0
dx = (mg +- sqroot(m^2g^2 + 2kmgL))/k

c. v = sqroot(2gL) [ from energy conservation, same as part 1]
d. max hieght, h
=> 0.5k(dx)^2 = mg(dx) + mgh
h = [0.5k(dx)^2 - mg(dx)]/mg
where dx can be found from part b

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