01 September, 2014

Pole vault: a sprinkle of physics

How high can an athlete jump with a pole? Let us make the simplest possible calculation and consider a vaulter approaching the pit at a speed of v. If his kinetic energy is converted into potential energy with 100 % efficiency we find that the jump height is $$h={v^2\over 2g}$$ Assuming a reasonable velocity of 10 m/s leads to a jump height of slightly over 5 m. But wait, this cannot be true. Today’s records exceed and by far the 5 m barrier. A look at the evolution of the world record over the years is quite instructive.

The 5 m barrier appeared realistic as long as the athletes were using a bamboo (or, later, steel) pole. But all this changed with the introduction of fiberglass poles. The world record stands now at over 6 m for men and even the women’s record exceeds 5 m. How is this possible. The standard answer is that the extra energy comes from the upper-body athlete’s muscles which bend the pole storing energy in it, energy which is later returned to the vaulter when the pole straightens. While there is some truth in this argument the situation is far more complicated. 

First, we must not neglect the fact that the centre of mass of the athlete is already elevated with respect to the soil by the order of a metre. Factoring this in would lift the barrier at 6 m in agreement with the current world record. 

But has anyone seen a 100 % efficiency necessary for the 5 (or 6) m jump? Linthorne discusses this matter in detail. According to his findings two important factors entrer into play. First, a flexible pole reduces the energy dissipated in the vaulter's body when the pole is planted into the takeoff box. Second, a flexible pole lowers the optimum takeoff angle, and so the athlete loses less energy in jumping off the ground. While a rigid pole would lead to an optimum takeoff angle of around 30°, for a flexible pole the optimum would be less than 20° and an approximate 25 % energy gain. 

Ekevad and Lundberg have made more detailed calculations trying to determine the optimal length and stiffness of the pole. In a first study they consider a passive vaulter (represented by a point mass at the top of the pole)

and find that an optimal pole length of 5.5 m (slightly longer that the 5.2 poles used by elite vaulters). The interesting result is that they obtain a 87 % efficiency in the conversion of kinetic energy to potential. However in another work of theirs they allow for  what they call ‘smart’ vaulters, meaning that they model the vaulter as a number of connected segments which can execute a given sequence of movements, and they find that the potential energy can attain 1.27 times the value of the initial kinetic energy of the athlete and the pole. This means that the contribution of the muscle work to the increase of the potential energy is quite significant. I would take this calculation with a grain of salt, since there exist neglected effects and simplifications which may modify the precise value. But if we may draw one conclusion from these calculations this is that we haven’t reached yet the limit of human capabilities in pole vaulting 

as the recent improvement upon a world record standing for 20 years has shown.

1 comment:

  1. I find your pole vault writings interesting as they appear well-intended. For the fiberglass pole origination facts (firsthand from the athlete himself) visit this YouTube link: https://www.youtube.com/watch?v=AV6rYC7n0NQ