Bungee Jump: Conservation of Energy

[teme92]

Homework Statement

A bungee jumper of mass m drops off a bridge and falls vertically downwards. The bungee cord is elastic with natural length L and stiffness k. Deduce that at the lowest point of the fall, the cord is stretched by an amount:

x=mg/k(1+√(1+(2kl/mg))

Homework Equations

F=-kx
PE=mgh=0.5kx²

Where h=L+x

The Attempt at a Solution

The total energy before the jump is equal to the total energy after the jump. Since at the bottom there is no kinetic energy I said the Potential Energy before is equal to the Elastic Energy at the bottom.

So:

mg(L+x)=0.5kx²
x²=(2mgx +2mgl)/k
x=√((2mgx +2mgl)/k)

I'm having problems here trying to get in the asked form. Have I forgotten something n the conservation of energy?

 

[willem2]

There's an x on the right hand side of the equals sign in your final answer.
The line above that is a quadratic equation.

 

[teme92]

So does that mean I have to solve it quadratically?

 

[willem2]

So does that mean I have to solve it quadratically?

I suppose you know how to solve a quadratic equation. ax^2 + bx + c = 0 ?

 

[HalfLostAndSearching]

I would like to point out that the equations used in this solution are not entirely accurate. The correct equation for elastic potential energy is PE=0.5kx^2, not 0.5kx2. Additionally, the equation for potential energy is mgh, not mgh=0.5kx2.

To solve the problem, we can use the conservation of energy principle, which states that the total energy before and after the jump must be equal. At the start of the jump, the total energy is solely in the form of gravitational potential energy (mgh), since the jumper has not yet fallen. At the bottom of the fall, the total energy is in the form of both gravitational potential energy (mgh) and elastic potential energy (0.5kx^2).

Using the conservation of energy principle, we can set these two equations equal to each other:

mgh=0.5kx^2

Since h=L+x, we can rearrange the equation to solve for x:

x=√(2mgh/k)

Substituting in the value of h, we get:

x=√(2mg(L+x)/k)

Solving for x, we get:

x=mg/k(1+√(1+(2kl/mg)))

This is the same result as the one provided in the homework statement. It is important to note that the conservation of energy principle is based on the assumption that there is no external work done on the system. In reality, there may be slight variations in the final stretched length of the bungee cord due to factors such as air resistance. However, this equation provides a good approximation of the stretched length of the bungee cord at the bottom of the fall.