World Record in Pole Vaulting: Math and Physics applied to Olympic Games

Question

For more than two decades the world record in pole vaulting had been at 6.14 m (Sergej Bubka, 07/31/1994). Today there are a few athletes with even more impressive results.

1. Assume the maximal velocity of an athlete is approximately 9.5 m/s. What is his kinetic energy at the beginning of the jump (assume an appropriate mass)?
2. Through elaborated technique almost all of the kinetic energy can be transformed into potential energy. To what height can he lift his center of mass (initially at the height of his belly) with the calculated kinetic energy?
3. Today the most professional vaulters increase the maximum height by practically performing a handstand on the pole. Considering this additional effect, what would be the theoretical world record?

Help:

1. Assume the height of the belly at 1.1 m
2. Assume the height of the hand of the jumper reaching up at 2.1 m.

Answer 1

At first, I recommend watching a brief video of the actual event. Usually, an exercise like this will also supply an illustration of the situation.

1. For the first question it is important to realize that we are only interested in that one moment when the jumper reaches his maximal velocity. That is still before he takes off as a large portion of his energy will be stored in form of deformation of the pole. So leave everything aside, just focus on that moment.
2. Kinetic energy can be directly calculated when we have mass and velocity. Let's assume 80 kg (also given in the help section).
3. E_kin = 1/2 * m * (v^2) = 1/2 * 80 * (9.5^2) J = 3600 J
4. Since we know that the SI unit of Energy is Joule, we use that simple fact to do the calculation without too many units. That works since all of the input values are also in SI.
5. The Belly of the jumper is at 1.1 m. Kinetic Energy is transformed into potential energy. m * g * h = 1/2 * m * (v^2).
6. Solve the equation for h. h= (v^2) / (2 * g)
7. Plug in the values. h = (9.5^2) / (2 * 9.8) m = 4.6 m
8. Again use all SI units and thereby simplify the calculation.
9. For the height of the center of mass don't forget that it started at an initial height of h_0 =1.1 m
10. The final theoretical height is the sum of these two: h_final = h + h_0 = 4.6 m + 1.1. m = 5.7 m
11. This one is slightly more complicated. A drawing helps (that I unfortunately can't supply with this text-based editor).
12. We need the distance belly to hand: d = 2.1 m - 1.1 m = 1 m
13. The jumper first pulls and then pushes his belly up so the distance from beginning to end is covered twice.
14. h_additional = 2 * d = 2 m
15. The maximal theoretical record is therefore the combination of the height that can be reached through the conversion of kinetic to potential energy and the additional height: 
16. h_record = h_final + h_additional = 5.7 m + 2 m = 7.7 m

Thus, a new record could still be quite a bit higher....

I hope this helps ;-)