Find the height of an object when a bungee cord stops it

[Amber_]

Homework Statement

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 = 34 m when
unstretched; when it’s stretched to L > L0,
the cord’s tension obeys Hooke’s law with
“spring” constant 53 N/m. To test the cord’s
reliability, one end is tied to a high bridge of
height 94 m above the surface of a river) and
the other end is tied to a steel ball of mass
98 kg. The ball is dropped off 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 trajec-
tory.
Answer in units of m.

2. The attempt at a solution

At first I thought maybe this could be solved using Fg= ma and then dividing Fg by the spring constant because at the bottom point Ft should be equivalent to Fg. That didn't work, so I tried setting it up like a conservation of energy problem.

mg(94)= 1/2 k (60-h)^2 + mgh

I solved for h and got 2.3 m. This is also not right. What am I doing wrong here?

 

[anathema]

I would like to offer some clarification and guidance on your approach to solving this problem. Firstly, it is important to recognize that the bungee cord is not a simple spring and cannot be treated as such. While it may obey Hooke's law for small deformations, it also has a maximum length that it can stretch to before it becomes taut and the tension becomes zero.

In order to accurately solve this problem, we need to take into account the non-linear nature of the bungee cord's behavior. This can be done using a combination of energy conservation and the concept of work done by a variable force.

Firstly, let's define some variables:
L0 = unstretched length of the bungee cord (34 m)
L = length of the bungee cord at any given point (in this case, it is equal to the height of the ball above the water's surface)
k = spring constant of the bungee cord (53 N/m)
m = mass of the ball (98 kg)
g = acceleration due to gravity (9.8 m/s^2)

Now, let's consider the different stages of the ball's motion:
1. As the ball is dropped from the bridge, it accelerates downwards due to the force of gravity.
2. As it reaches the point where the bungee cord starts to stretch, the tension in the cord begins to increase and opposes the force of gravity.
3. At some point, the tension in the cord becomes equal to the weight of the ball and the ball stops accelerating downwards.
4. As the ball continues to fall, the cord stretches further and the tension increases, causing the ball to decelerate and eventually come to a stop.
5. The cord then starts to pull the ball back up, causing it to accelerate upwards until it reaches its maximum stretched length.
6. Finally, the ball starts to fall again and the process repeats until it comes to a stop at the lowest point of its trajectory.

Using energy conservation, we can equate the potential energy at the top of the bridge (mgh) with the maximum potential energy at the lowest point of the trajectory (1/2kL^2). This gives us the following equation:

mgh = 1/2kL^2

Using this equation, we can solve for L, which represents the height of the ball above the water's surface at the lowest point of its trajectory