A stuntman wants to bungee jump from a hot air balloon 58.0 m above a the ground

Question

A stuntman wants to bungee jump from a hot air balloon 58.0 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 1.25 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. Hint: the spring constant changes with a longer piece of cord.

(a) What length of cord should he use?

(b) What maximum acceleration will he experience?

Explanation / Answer

(a) When the stunt man 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 = 48m ...(2)

From the stationary test, the stuntman weight mg stretches a cord of length 5m by 1.25m
It will therefore stretch a cord of length L by 1.25 L / 5 = 0.25L.

k for a cord of length L therefore satisfies:
mg = 0.25kL
k = mg / (0.25L) ...(3)

Substituting for x from (2) in (1):
k(48 - L)^2 / 2 = 48mg

Using (3) to eliminate k:
mg(48 - L)^2 = 2 * 48mg * 0.25 L
(48 - L)^2 = 24L
2304 + L^2 - 96L - 24L = 0
L^2 - 120L + 2304 = 0
L = (120 +/- sqrt(120^2 - 4 * 2304)) / 2

Discarding the positive sign:
L = (120 - 72) / 2
= 24 m.

(b)

Untill the cord begins stretching, at L+x = L , the person falls with accel. "g" , after the cord begins to stretch the net force is;

mg - Kx = ma

Since L+x > L or = L , the 2nd term is always positive or zero and it is seen that "a" will be a max at L+x = L and a(max) = g

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

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