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My physics class is doing the classic egg drop experiment, but with a different twist.

We are asked to find the length of the bungee cord needed given a mass and a height to drop it from. The success of the lab is determined by how close we can get to the ground without touching.

We are using a single bungee cord, and not multiple bungees.

This is a common question here, but I havent been able to determine the equations to use to determine what I need.

I have tried using U1 + K1 = U2 + K2 +Uspring

Where U1 is the initial potential energy (mgh1)

K1 is the initial kinetic energy (K1 = 0)

U2 is the final kinetic energy (mgh2 with h2 being the closest distance off the ground)

K2 is the final kinetic energy (K2 = 0)

And Uspring is the integral of Force with respect to distance F(x) from 0 to the unstretched bungee length - h2

Our F(x) function is 2.823 - 6.696x + 31.64x^2 - 110.9x^3 + 177.3x^4 - 103.3x^5

This is the characteristic that our bungee follows.

Using this, I get the correct value, if the egg wasn't dropped, but rather if the egg was just hanging there.

My question is, what am I doing wrong? and what equations/principles could I use to determine the length of the string that I need?