The Physics of the Pole Vault Revisited

In yesterday’s Mechanics lecture I decided to illustrate the use of energy conservation arguments with an application to the pole vault. I have done this a few times and indeed wrote a blog post about it some years ago. At the time I wrote that post however the world record for the pole vault was held by the legendary Ukrainian athlete Sergey Bubka at a height of 6.14m which he achieved in 1994. That record stood for almost 20 years but has since been broken several times, and is now held by Armand Duplantis at a height of 6.21m.

Here he is breaking the record on July 24th 2022 in Eugene, Oregon:

He seemed to clear that height quite comfortably, actually, and he’s only 23 years old, so I dare say he’ll break quite a few more records in his time but the fact that world record has only increased by 7cm in almost 30 years tells you that the elite pole vaulters are working at the limits of what the human body can achieve. A little bit of first-year physics will convince you why.

Basically, the pole is a device that converts the horizontal kinetic energy of the vaulter \frac{1}{2} m v^2,  as he/she runs in, to the gravitational potential energy m g h acquired at the apex of his/her  vertical motion, i.e. at the top of the vault.

Now assume that the approach is at the speed of a sprinter, i.e. about 10 ms^{-1}, and work out the height h = v^2/2g that the vaulter can gain if the kinetic energy is converted with 100% efficiency. Since g = 9.8 \, ms^{-2} the answer to that little sum turns out to be about 5 metres.

This suggests that  6.21 metres should not just be at, but beyond, the limit of a human vaulter,  unless the pole were super-elastic. However, there are two things that help. The first is that the centre of mass of the combined vaulter-plus-pole does not start at ground level; it is at a height of a bit less than 1m for an an average-sized person.  Nor does the centre of mass of the vaulter-pole combination reach 6.21 metres.

The pole does not go over the bar, but it’s pretty light so that probably doesn’t make much difference. However, the centre of mass of the vaulter actually does not actually pass over the bar.  That  doesn’t happen in the high jump, either. Owing to the flexibility of the jumper’s back the arc is such that the centre of mass remains under the bar while the different parts of the jumper’s body go over it.

Moreover, it’s not just the kinetic energy related to the horizontal motion of the vaulter that’s involved. A human can in fact jump vertically from a standing position using elastic energy stored in muscles. In fact the world record for the standing high jump is an astonishing 1.9m. In the context of the pole vault it seems likely to me that this accounts for at least a few tens of centimetres.

Despite these complications, it is clear that pole vaulters are remarkably efficient athletes. And not a little brave either – as someone who is scared of heights I can tell you that I’d be absolutely terrified being shot up to 6.21 metres on the end of  a bendy stick, even with something soft to land on!

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5 Responses to “The Physics of the Pole Vault Revisited”

  1. John Peacock Says:

    Isn’t part of it that the vaulter continues to push as the pole flexes, so you’re adding the work from that effort to the kinetic energy to give the total stored elastic energy? You could imagine approaching very slowly, but wearing a lead backpack, and flexing the pole at leisure. Then you shrug off the backpack and get whisked up. In that model, your height is limited only by how much elastic energy the pole can store before it shatters (which it does about 1% of the time – that’s what would really frighten me if I was a vaulter).

    • telescoper Says:

      Yes, elastic energy from the arms of the vaulter definitely plays a role. You could push from a static position but it might be hard to arrange it to propel you in the correct direction. I think modern poles have gone the way of cricket bats – they’re very elastic but also prone to break. There’s definitely a risk of being stabbed by a shard, and that unfortunately has happened in competitions.

  2. Anton Garrett Says:

    Non-physicists tend to assume that with a long enough pole you could reach any height, and the limit is merely how fast you can run with a very long and consequently heavy pole. Of course that’s not true.

    Anybody who can do the Indian rope trick would win the world recdord…

    • telescoper Says:

      I remember at school we had to do the pole vault with cylindrical metal poles that didn’t bend at all.

      • Anton Garrett Says:

        That’s handy for getting over a pond (provided you have confidence in the end of the pole ‘sticking’), ie it might help you with distance rather than height. So, if you view the pole vault as an assisted high jump, how about an assisted long jump event?

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