A 63.0-kg bungee jumper is standing on a tall platform (h0 = 45.8 m). The bungee cord has an unstrained length of L0 = 9.18 m and, when stretched, behaves like an ideal spring with a spring constant of k = 67.2 N/m. The jumper falls from rest, and it is assumed that the only forces acting on him are his weight and, for the latter part of the descent, the elastic force of the bungee cord. Determine how far the bungee jumper is from the water when he reaches the lowest point in his fall.
To determine how far the bungee jumper is from the water when reaching the lowest point, we need to calculate the maximum displacement of the bungee cord.
We can break down the problem into two parts:
1. Free-fall: In this part, the only force acting on the bungee jumper is gravity. The acceleration due to gravity is approximately 9.8 m/s². We can calculate the time it takes for the jumper to reach the lowest point using the equation:
h(t) = h0 + v0t + (1/2)at²
Where:
h(t) = height at time t
h0 = initial height (45.8 m)
v0 = initial velocity (0 m/s, as the jumper starts from rest)
a = acceleration due to gravity (-9.8 m/s²)
t = time
By rearranging the equation and solving for t, we get:
t = sqrt((2(h - h0))/a)
Substituting the values, we find:
t = sqrt((2(0 - 45.8))/(-9.8))
2. Elastic part: Once the bungee cord starts stretching, it behaves like an ideal spring. The displacement of the spring can be calculated using Hooke's Law:
F = kx
Where:
F = force
k = spring constant (67.2 N/m)
x = displacement of the bungee cord
The force due to the spring is equal to the force due to gravity, so we can set F equal to the weight of the bungee jumper:
mg = kx
Solving for x, we get:
x = mg/k = (63.0 kg)(9.8 m/s²)/(67.2 N/m)
Now, to find the total displacement, we add the displacement from the free-fall part and the elastic part:
displacement = x + h0 + L0
Substituting the values we have:
displacement = (63.0 kg)(9.8 m/s²)/(67.2 N/m) + 45.8 m + 9.18 m
Evaluating this equation will give us the answer.