What Is the Spring Constant for Kate's Bungee Jump?

[IKonquer]

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.

If Kate just touches the surface of the river on her first downward trip (i.e., before the first bounce), what is the spring constant k? Ignore all dissipative forces.

Homework Equations

The Attempt at a Solution

I started by trying to draw a free-body diagram on Kate. I had kx from the cord that pulled her up and mg down. At the bottom, her acceleration is g upwards so I wrote the equation:
kx - mg = mg.

k = 2mg / x

x is the distance from the bridge to the water - the length of the unstretched cord.
x = h - L

My answer for the spring constant k = 2mg / (h - L)

I'm not sure why my answer is not right.

 

[kjohnson]

I started by trying to draw a free-body diagram on Kate. I had kx from the cord that pulled her up and mg down. At the bottom, her acceleration is g upwards so I wrote the equation:
kx - mg = mg.

If you wanted to write a free body diagram for the problem with down being positive then you would have mg-kx = ma where 'a' is the varying acceleration at any time though out the fall. You can't simply say that the acceleration is that of gravity..

Probably the best way to approach this is using the conservation of energy. What energy forms do you have at the top vs any other point?

 

[IKonquer]

I got the correct answer with the conservation of energy, but I am still confused about the acceleration in this problem. Shouldn't the acceleration at the bottom be g pointing upwards?

 

[kjohnson]

You can't say the acceleration at the bottom is g because the acceleration is related to the motion equation from the FBD of mg-kx=ma...all you can say is that the velocity at the bottom is zero.

What you did was solve for what k "would" be if the acceleration at the bottom was equal to gravity but in the positive direction. This is a possible solution for k, but one that doesn't fit to the problem description.

 

[rwisz]

Your approach and equations are correct, but there may be a mistake in your calculation. Double check your units and make sure they are consistent. The spring constant should have units of force/length, so make sure you are using the correct units for mass, acceleration due to gravity, and distance. Also, keep in mind that the mass m in the equation represents the total mass of Kate and the bungee cord.