Speeding

physics trajectory problem?

Note: A bungee cord can stretch, but it is never compressed. When the distance be- tween the two ends of the cord is less than
its unstretched length L0, the cord folds and its tension is zero. For simplicity, neglect the cord’s own weight and inertia as well as the air drag on the ball and the cord. A bungee cord has length L0 = 24 m when unstretched; when it’s stretched to L > L0, the cord’s tension obeys Hooke’s law with spring” constant 66 N=m. To test the cord’s reliability, one end is tied to a high bridge of
height 67 m above the surface of a river) and the other end is tied to a steel ball of mass 86 kg. The ball is dropped o® the bridge with zero initial speed. Fortunately, the cord works and the ball
stops in the air a few meters before it hits the water | and then the cord pulls it back up.The acceleration of gravity is 9:8 m=s2 :
Calculate the ball’s height above the wa- ter’s surface at this lowest point of its trajectory. Answer in units of m.

This is an energy problem rather than a trajectory problem.
First, let’s go over the givens:
(mass) m = 86kg
(spring constant) k = 66 N*m
(length of cord) L = 24m
(length of cord’s expansion) x = ?

So, due to conservation of energy
Ei = Ef
Ki + U[g]i + U[s]i = Kf + U[g]f + U[s]f
There is no kinetic energy at the beginning or end, nor is there spring potential energy at the beginning, nor gravitational potential energy at the end, so all of these equal zero, and we have
U[g]i = U[s]f
L+x is the distance it will fall, so this is what will be used for height in U[g]i. x is the distance the cord will be stretched, so this will be used for x in U[s]f.

mg(L+x) = k(x^2)/2
mgL + mgx = k(x^2)/2

Plugging in the values we know gives
20227.747 + 842.8x = 33x^2

Get this equal to zero:
33x^2 – 842.8x – 20227.747 = 0

And use the quadratic formual to find
x = 40.627m
The stretched length of the cord is L + x = 24 + 40.627 = 64.627

The height above the water equals the height of the bridge minus the stretched length of the cord, 67 – 64.627 = 2.373m

And there is your answer.

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