Bringing it Together
Now that we know about potential energy and kinetic energy, we can do some interesting calculations. Let's figure out how high a pole-vaulter could jump if he had perfect technique. First we'll figure out his KE, and then we'll calculate how high he could vault if he used all of that KE to increase his height (and therefore his 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*102 / 9.8 = 5.1 meters
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 1 meter off the ground. So the best height a pole-vaulter could achieve is in fact about 6.1 meters, or 20 feet. 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.
Figure 4. Animation of pole-vault
In Figure 4 you can see how 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 world record of 6.15 meters, set by Sergey Bubka in 1993.
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