A daredevil plans to bungee jump from a balloon 53.0 m above a carnival midway. 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 and obeying 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 1.50 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.

Question

A daredevil plans to bungee jump from a balloon 53.0 m above a carnival midway. 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 and obeying 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 1.50 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.

(a) What length of cord should he use?
_____m

(b) What maximum acceleration will he experience?

____m/s2

Answer 1

To solve this problem, we need to use Hooke's law for the elastic cord. Hooke's law states that the force exerted by a spring or elastic material is directly proportional to the displacement from its equilibrium position. The formula for Hooke's law is:

F = -kx

where F is the force exerted by the cord, k is the spring constant, and x is the displacement from the equilibrium position.

In this scenario, the displacement is given by the difference in heights between the top attachment point and the stopping point. Let's call this displacement H.

H = 53.0 m - 10.0 m
H = 43.0 m

We are also given that a 5.00 m length of the cord stretches by 1.50 m under his body weight. This means that when the displacement is 1.50 m, the force exerted by the cord is equal to the weight of the daredevil.

Now, we need to find the spring constant k. We can rearrange Hooke's law to solve for k:

k = -F / x

Since we want to find k when the displacement is 1.50 m, we substitute the force F with the daredevil's weight (mg), and the displacement x with 1.50 m.

k = -mg / 1.50 m

We can use the equation for gravitational force to find the weight of the daredevil:

F = mg

So, the equation for k becomes:

k = -F / 1.50 m

Now, we have all the information we need to find the length of the cord. The maximum displacement of the cord will occur when the entire length is stretched from the equilibrium position to the stopping point. Let's call this length L.

L = H + x
L = 43.0 m + 1.50 m

Now we can calculate the length of the cord:

(a) L = 44.5 m

Next, we need to find the maximum acceleration experienced by the daredevil. The maximum acceleration occurs at the point where the cord is stretched the most, which is at the beginning of the fall when the length of the cord is L.

To find the maximum acceleration, we can use the equation:

a = -kx / m

Substituting in the values we know:

a = -kL / m

Now we can calculate the maximum acceleration:

(b) a = (-k * 44.5 m) / m = -k * 44.5

Since we don't have the specific value of k, we cannot calculate the exact maximum acceleration. However, we have all the information needed to calculate it once the value of k is known.