Calculating the Maximum Displacement of Kate's Bungee Jump

[cj3]

Homework Statement

Kate, a bungee jumper, wants to jump off the edge of a bridge that spans a river below. Kate has a mass (m) , and the surface of the bridge is a height (h) above the water. The bungee cord, which has length (L) when unstretched, will first straighten and then stretch as Kate falls.

Assume the following:

The bungee cord behaves as an ideal spring once it begins to stretch, with spring constant (k)
Kate doesn't actually jump but simply steps off the edge of the bridge and falls straight downward.
Kate's height is negligible compared to the length of the bungee cord. Hence, she can be treated as a point particle.

Use (g) for the magnitude of the acceleration due to gravity.


How far below the bridge will Kate eventually be hanging, once she stops oscillating and comes finally to rest? Assume that she doesn't touch the water.

Homework Equations

F_s=-kx
W_s=.5kx²
E_i=E_f

The Attempt at a Solution

i'm not really sure...i know i have to use conservation of energy so this is what i tried
.5kx²+.5mv_i²+mgh₁=.5mv_f²+mgh₂
.5kx²=.5mv_f²+2mgL
x=sqrt(mv_f²+2mgL)

 

[alphysicist]

Hi cj3,

Homework Statement

Kate, a bungee jumper, wants to jump off the edge of a bridge that spans a river below. Kate has a mass (m) , and the surface of the bridge is a height (h) above the water. The bungee cord, which has length (L) when unstretched, will first straighten and then stretch as Kate falls.

Assume the following:

The bungee cord behaves as an ideal spring once it begins to stretch, with spring constant (k)
Kate doesn't actually jump but simply steps off the edge of the bridge and falls straight downward.
Kate's height is negligible compared to the length of the bungee cord. Hence, she can be treated as a point particle.

Use (g) for the magnitude of the acceleration due to gravity.


How far below the bridge will Kate eventually be hanging, once she stops oscillating and comes finally to rest? Assume that she doesn't touch the water.

Homework Equations

F_s=-kx
W_s=.5kx²
E_i=E_f

The Attempt at a Solution

i'm not really sure...i know i have to use conservation of energy so this is what i tried
.5kx²+.5mv_i²+mgh₁=.5mv_f²+mgh₂
.5kx²=.5mv_f²+2mgL
x=sqrt(mv_f²+2mgL)

I don't think that conservation of energy is the way to approach this problem. Instead, think about the fact that if she is at rest, her acceleration is zero. How can that be used to solve the problem?

 

[baba89]

k



I would like to commend you for attempting to use conservation of energy to solve this problem. However, I believe your approach is not quite correct. Let's break down the problem and see if we can come up with a more accurate solution.

First, let's define our variables:

m = mass of Kate
h = height of the bridge above the water
L = unstretched length of the bungee cord
k = spring constant of the bungee cord
g = acceleration due to gravity

We can start by considering the forces acting on Kate as she falls. Initially, she has only her weight pulling her down. As she falls, the bungee cord will begin to stretch, exerting a force in the opposite direction. At some point, these two forces will balance each other out and Kate will stop accelerating.

Using Newton's second law, we can set up the following equation:

Fnet = mg - Fs = ma

Where Fnet is the net force acting on Kate, mg is her weight, Fs is the force exerted by the bungee cord, and ma is her acceleration.

We can rearrange this equation to solve for the distance x that the bungee cord will stretch:

x = (mg - ma)/k

Now, we know that Kate will eventually come to rest once the bungee cord has stretched enough to balance her weight. This means that at this point, her acceleration will be zero and we can set ma equal to zero in the equation above.

x = (mg - 0)/k = mg/k

This tells us that the maximum displacement of Kate's bungee jump will be equal to her weight divided by the spring constant of the bungee cord.

To find the distance below the bridge that Kate will eventually be hanging, we need to subtract this maximum displacement from the initial height of the bridge (h).

Therefore, the final equation would be:

Distance below bridge = h - (mg/k)

I hope this helps you better understand how to approach this problem. Keep up the good work as a scientist!