A daredevil plans to bungee jump from a hot-air balloon 65.0 m above the ground.
ID: 2276983 • Letter: A
Question
A daredevil plans to bungee jump from a hot-air balloon 65.0 m above the ground. He is going to use a long bungee cord that is supposed to stop his fall when he is 10.0 m above the ground. Take his body to be a point mass and treat the bungee cord as massless, with force as a function of stretching that obeys Hooke's law. This first part of the problem involves getting the cord length correct. It is important if it's too long he may hit the ground (at high speed!). In a preliminary test, hanging at rest from a 5.00-m length of the cord, the daredevil finds that his weight stretches it by 1.50 m. When the final length of cord is cut from a spool, he intends to drop from rest at the point where one end of this cord is tied to the stationary balloon gondola and the other end is attached to his waist.
(a) What length of cord should he use?
(b) What maximum acceleration will he experience?
Explanation / Answer
When the daredevil jumps from height h above his final position, his PE is mgh and his KE is 0.
When the cord is fully stretched as he starts to go up again, he once again has KE = 0, and his initial PE is all stored in the stretched cord.
Let L be the unstretched length of the cord, and x be its maximum extension when the daredevil starts to go up again.
Then:
kx^2 / 2 = mg(L + x) ...(1)
L + x = 55 ...(2)
From the stationary test, the daredevil's weight mg stretches a cord of length 5 by 1.5.
It will therefore stretch a cord of length L by 1.5 L / 5 = 0.3L.
k for a cord of length L therefore satisfies:
mg = 0.3kL
k = mg / (0.3L) ...(3)
Substituting for x from (2) in (1):
k(55 - L)^2 / 2 = 55mg
Using (3) to eliminate k:
mg(55 - L)^2 = 2 * 55mg * 0.3 L
(55 - L)^2 = 33L
3025 + L^2 - 110L - 33L = 0
L^2 - 143L + 3025 = 0
L = (143 +/- sqrt(143^2 - 4 * 3025)) / 2
Discarding the positive sign:
L = (143 - 91.3728) / 2
= 25.8 m.
His maximum acceleration will be g. Until he has descended a distance equal to the length of the cord, there is nothing to slow him down.