Leonard Kelley holds a bachelor's in physics with a minor in mathematics. He loves the academic world and strives to constantly explore it.
Gravity and athletics? How much could there be to talk about here besides the obvious? Well, as it turns out, its really a pressing matter…
Strength and Weight
Looking at athletes, many wonder what the limit to their capabilities is. Can a person only grow so much muscle mass? To figure this out, we need to look at proportions. The strength of any object is proportional to the cross-sectional area of it. The example Barrows gives is a breadstick. The thinner a breadstick is, the easier it is to break it but the thicker is the harder it would be to snap it in half (Barrow 16).
Now all objects have density, or the amount of mass per a given amount of volume. That is, p = m/V. Mass is also related to weight, or the amount of gravitational force a person experiences on an object. That is, weight = mg. So since density is proportional to mass, it is also proportional to weight. Thus, weight is proportional to volume. Because area is square units and volume is cubic units, area cubed is proportional to volume squared, or A3 is proportional to V2 (to get unit agreement). Area is related to strength and volume is related to weight, so strength cubed is proportional to weight squared. Please note that we do not say they are equal but only that they are proportional, so that if one increases then the other increases and vice versa. Thus as you get larger, you do not necessarily get stronger, for proportionally strength does not grow as fast as weight does. The more of you there is, the more your body has to support before breaking like that breadstick. This relation has governed the possible life forms that exist on Earth . So a limit does exist, it all depends on your body geometry (17).
A favorite of the Olympics, this event used to be straight forward. One would get a running start, hit the pole into the ground, then holding onto the top launch themselves feet-first over a bar high up in the air.
That changes in 1968 when Dick Fosbury leaps head-first over the bar and arching the back, totally clearing it. This became known as the Fosbury Flop and is the preferred method for pole vaulting (44). So why does this work better than the feet-first method?
It is all about mass being launched to a certain height, or the conversion of kinetic energy to potential energy. Kinetic energy is related to the speed launched and is expressed as KE = ½*m*v2, or one-half mass times the velocity squared. Potential energy is related to the height from the ground and is expressed as PE = mgh, or mass times gravitational acceleration times height. Because PE is converted to KE during a jump, ½*m*v2 = mgh or ½*v2 = gh so v2 = 2gh. Note that this height is not the height of the body but the height of the center of gravity. By curving the body, the center of gravity extends to outside the body and thus gives a jumper a boost they normally would not have. The more you curve, the lower the center of gravity is and thus the higher you can jump (43-4).
How high can you jump? Using the earlier relation ½*v2 = gh, this gives us h = v2 / 2g. So the faster you run the greater the height you can achieve (45). Combine this with moving the center of gravity from inside your body to the outside and you have the ideal formula for pole vaulting.
Running vs. Walking
According to official rules, walking is different from running by always maintaining at least one foot on the ground at all times and also keeping your leg straight as you push off the ground (146). Definitely not the same, and definitely not as fast. We constantly see runners breaking new records for speed, but is there a limit to how fast a person can walk?
For a person with leg length L, from sole of foot to the hip, that leg moves in a circular fashion with the pivot point being the hip. Using the circular acceleration equation, a = (v2)/L. Because we never conquer gravity as we walk, the acceleration of walking is less than the acceleration of gravity, or a < g so (v2)/L < g. Solving for v gives us v < (Lg)1/2. This means that the top speed a person can reach is dependent on the leg size. The average leg size is 0.9 meters, and using a value of g = 10 m/s2, we get a v max of about 3 m/s (146).
Barrow, John D. 100 Essential Things You Didn't Know You Didn't Know: Math Explains Your World. New York: W.W. Norton &, 2009. Print. 16-7, 43-5, 146.
This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.
© 2022 Leonard Kelley