First, we’ll figure out his kinetic energy when he is running at full speed, and then we’ll calculate how high he could vault if he used all of that KE to increase his height and, therefore, his potential energy (PE) without wasting any of it. If he converted all of his KE to PE, then we can solve the equation by setting them equal to each other:
1/2 m v2 = m g h
Since mass is on both sides of the equation, we can eliminate this term. This makes sense because both KE and PE increase with increasing mass, so if the runner is heavier, his PE and KE both increase. So we’ll eliminate the mass term and rearrange things a little to solve for h:
1/2 v2 / g = h
Let’s say our pole vaulter can run as fast as anyone in the world. Right now, the world record for running 100 m is just under 10 seconds. That gives a velocity of 10 m/s. We also know that the acceleration due to gravity is 9.8 m/s2. So now we can solve for the height:
1/2 x (102 / 9.8) = 5.1 m
So 5.1 meters is the height that a pole vaulter could raise his center of mass if he converted all of his KE into PE. But his center of mass is not on the ground; it is in the middle of his body, about 3 ft (1 m) off the ground. So the best height a pole vaulter could achieve is in fact about 20 ft (6.1 m). He may be able to gain a little more height by using special techniques, like pushing off from the top of the pole, or getting a really good jump before takeoff.
The pole vaulter’s energy changes as he makes the vault. When he starts out, both his potential and kinetic energy are zero. As he starts to run, he increases his kinetic energy. Then, as he plants the pole and starts his vault, he trades his kinetic energy for potential energy. As the pole bends, it absorbs a lot of his kinetic energy, just like compressing a spring. He then uses the potential energy stored in the pole to raise his body over the bar. At the top of his vault, he has converted most of his kinetic energy into potential energy.
Our calculation compares pretty well with the current outdoor world record of 20 ft 1-3/4 in (6.15 m), set by Sergey Bubka in 1994.
Now that we’ve done this calculation, let’s look at what a vaulter can do to try to break the record. They have two main ways to increase the height of his vault. One is to increase his running speed, which increases the amount of kinetic energy he can use. The other is to make more efficient use of the energy, perfecting his technique so that absolutely no energy is wasted.