Formulae for falling

Published on January 11th, 2012
Formulae for falling

We’re sure you’ve heard about the girl who survived the fall from the Victoria Falls bridge? That’s right, the rope snapped and left the 22- year- old Australian plummeting into the Zambezi River. And yes, she survived.

In The Extreme Issue (February 2009) we discovered how Maths allow you to jump off a bridge without dying. Well, when you’re standing on that platform with a rope around your ankles and the ground faaaaar away, it makes sense that your safety-conscious brain might tell you that jumping is totally crazy. But then you go and do it anyway. Why? Because you know that someone has worked through the maths to ensure you can leap to your death … and survive.

The consideration of cord length is vital in a bungee jump – too short, and you won’t get much of a thrill; too long, and ouch! When you jump, the cord stretches to absorb the energy of your fall, and then you fly upwards as the cord snaps back to its original unstretched length. In this way, you’ll oscillate up and down until all the energy is used up. So, a cord producer must design cords to not only support specific weights, but to reach a certain maximum length when fully stretched out.

A mathematician named Robert Hook studied the physical characteristics of springs and rubber bands, which are similar to the cord for a bungee jumper.

Hooke’s Law of Elasticity states that the stretch of a spring is directly proportional to the load that it’s holding, provided the system doesn’t exceed the spring’s elastic limit. This is represented by the following equation:

F = kx, where F = force, k = spring constant and x = the distance the spring can be stretched (or compressed) from its equilibrium length.

In layman’s terms the equation tells us how much tension a spring can endure, and the maximum length it will reach.

Sum it up

Example 1:
A cord is 20 m long when unstretched. If elasticity is 50% and the spring constant is 20 N/m, what is the maximum force the cord can hold?
F = kx
x = 50% of 20 m , ie 10 m.
∴ F = (20 N/m) (10 m)
= 200 N

200 N = 20,4 kg (divide F by 9.8)
Therefore, the maximum mass the cord can support is 20,4 kg.

Example 2.
A cord is 45 m long  when unstretched. It has an elasticity of 75%. The maximum force it can hold is 350 N. What is the spring constant of the cord?
F = kx
∴ 350 N = k (75% x 45 m)
∴ 350 N  = k (33,75 m)
∴      k =  10,37 N/m

Check out this video of how Erin Langworthy survives a fall when the rope breaks…