Conservation of Mechanical Energy

In this tutorial we will calculate the spring constant of a bungee cord. Understanding the relationship between mechanical, potential, and kinetic energy is an elementary concept in physics that is usually introduced in grades 10 and expanded upon in grade 11.

Example:

A 50.0 Kg jumper with a 10.0 m bungee cord (b) jumps off a 60.0 m adjustable platform and is stopped by the cord at just above water level. Assuming the jumper is 1.5 m tall, calculate the spring constant (k) of the bungee cord (Gravity, g = 9.81m/s²).

Solution:

Since mechanical energy is conserved we are able to find the total mechanical energy at various positions (x) throughout this system. When the bungee jumper is on the platform we are able to calculate the total mechanical energy of the system using the following formula:

ME₁ = KE + PE_g + PE_s

The jumper is initially at rest (velocity = 0) and the bungee cord isn't stretched both kinetic energy (KE) and elastic potential energy (PE_s) are equal to zero. The net result is:

ME = PE_g = mgh = 50.0kg(9.81m/s)60.0m = 29,430 Joules

Since mechanical energy is conserved we can let the total mechanical energy at position (2) equal the total mechanical energy at position (1) giving us:

ME₁ = ME₂ = KE + PE_g + PE_s

At position (2) the KE is 0 as the jumper must stop completely relative to the vertical axis before being able to ascended. The gravitational potential energy at position (2) is also zero as the total mechanical energy of the system has been transferred into the bungee cord and conserved as elastic potential energy. The net result is:

ME₂ = 29,430 J = PE_s = 1/2kx²

29,430 = 1/2kx²

29,430 = 1/2k(50m)²

k = 23.5 N/m

Now that the spring constant is known calculate the height adjustment required in the platform to accommodate someone with a mass of 65 Kg to jump successfully and only grace the water.

ME₁ = KE + PE_g + PE_s

ME₁ = mgh = 65kg(9.81m/s)60m = 38259 Joules

ME₁ = ME₂ = KE + PE_g + PE_s

38259 J = ME₂ = PE_s

38259 = 1/2kx²

38259 = 1/2(23.5 N/m)x²

x = 57.1m, therefore the height of the platform would have to be elevated 7.1m.