# How fast would a pole vaulter hit the ground?

## A Fermi mechanics question

10 April 2019

I was recently compiling questions for a quiz for maths teachers, which included a sports round. I thought it would be good to include a question about a more obscure sport, and pole-vaulting popped into my mind.

My first rather dull idea was simply: "What is the highest vault ever achieved?", but then I thought of an amusing if slightly dark twist by wondering what would happen if the landing mat were removed. At what speed would the unfortunate pole-vaulter hit the ground? (You get a point if you're within one metre/second of the right answer.)

This is an example of a so-called Fermi question - you have to come up with an answer without access to any data.

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We’ve all seen pole vaulters in the Olympics. The vaulter runs with the pole stretched in front of them, plants it into a metal lined pit (known as the *box*), and the elastic bend of the pole then catapults them up and over the horizontal bar. But how high is that bar? We can put some limits on it. Even a beginners' level pole-vault bar will be a lot taller than a person, or this would just be the high jump. Could it be as high as a two storey house (which is about four people high)? That looks too much. Just picturing it in your head, I hope you agree that it’s going to be somewhere between the heights of two and three tall people, twelve to twenty feet (roughly four to six metres*).

At the top of the vault, the vaulter releases the pole, and for an instant has no vertical speed. The problem now becomes one of working out the speed of an object dropped from a height and falling under gravity.

We need the maths of acceleration due to gravity. Galileo and Newton both knew that the speed of an object that is falling freely (such as a pole vaulter who has let go of the pole) will follow a ‘square’ law. The formula that links speed, distance and gravitational acceleration should be familiar to any GCSE physics student. It is:

** v ^{2} = 2gh**

Where v is the vertical speed, g is the acceleration due to gravity (about 10 m/s^{2}), and h is the distance fallen. In other words:

** The square of the speed = 20 x height fallen.**

So we can work out the speed of impact on the ground for our two extremes of height:

**Height fallen v ^{2} (=20h) v**

(metres) (metres/second)

4 80 ~9 m/s

6 120 ~11 m/s

Interestingly, despite the wide range of possible heights of the bar, we can be pretty confident that the speed of impact when hitting the ground is around 10m/s, or 25mph. That’s about the same as Usain Bolt running at maximum speed. Ouch!

** It turns out that the pole vault world record for men is 6.12 metres by Renaud Lavillenie in 2014, *