A stuntman wants to bungee jump from a hot air balloon 60.0 m above a the ground

Question

A stuntman wants to bungee jump from a hot air balloon 60.0 m above a the ground. He will use a uniform elastic cord, tied to a harness around his body, to stop his fall at a point 10.0 m above the ground. Model his body as a particle and the cord as having negligible mass that obeys Hooke's law. In a preliminary test, hanging at rest from a 5.00 m length of the cord, he finds that his body weight stretches it by 1.40 m. He will drop from rest at the point where the top end of a longer section of the cord is attached to the stationary balloon. What length of cord should he use? m What maximum acceleration will he experience? m/s^2

Explanation / Answer

Here, 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 = 50 ...(2) (60-10)

From the stationary test, the daredevil's weight mg stretches a cord of length 5 by 1.4.

It will therefore stretch a cord of length L by 1.4 L / 5 = 0.28L.

k for a cord of length L therefore satisfies:

mg = 0.28kL

k = mg / (0.28L) ...(3)

Substituting for x from (2) in (1):

k(50 - L)^2 / 2 = 50mg

Using (3) to eliminate k:

mg(50 - L)^2 = 2 * 50mg * 0.28 L

(50 - L)^2 = 28L

2500 + L^2 - 100L - 28L = 0

L^2 - 128L + 2500 = 0

L = (128 +/- sqrt(128^2 - 4 * 2500)) / 2

Discarding the positive sign:

L = (128 - 79.90) / 2

= 24.05 m.

(a) So, the length of the cord = 24.05 m

(b)At the lowest point, restoring force in the cord in the upward direction
= k*elongation of the cord
= (mg/1.5)*(50-24.05)
= mg * (25.95/1.4)
Downward force = mg
Net force upwards = mg* (25.95/1.4) - mg = mg(24.55/1.4)
Maxm. acceleration upwards
= Net Force/mass
=mg(24.55/1.4) /m
= 17.53g m/s^2
= 17.53 * 9.8 m/s^2
=171.8 m/s^2.

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