# Homework Help: Work done during running

1. Aug 14, 2010

### spotmeter

I train runners to run more efficiently, in part by reducing their vertical oscillation (bounce), but I don't know how to figure out how much work they do during a marathon based on the size of their oscillation and their weight. I also want to be able to tell them how much falling force they have to cope with.

I have a runner who weighs 130 lbs. He moves up 3" and down 3" with each stride he takes. During a marathon, he takes 26,200 strides. How do I compute how much work he does during a marathon pushing his body weight up? How much work would he do if he cut his bounce to just one inch? To 1/2"?

At the height of his oscillation (usually when he toes off his rear leg) he then falls back down to earth. How much force does he land with? Years ago I read that with a 2" bounce, a runner lands with 4X his body weight, 3" bounce = 6X body weight and 4" bounce = 8X body wieght. Are these numbers correct?

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Aug 15, 2010

### rock.freak667

If the person's center of gravity is moving 3" up and down 3" then upwards, the energy is E1=mgh1 and down is E2=mgh2. So the total energy would be E=E1+E2

As for the force on the runner, I cannot say if your numbers are correct. But from my classes on strength of materials, I have examples where dropping a mass little as 4 centimeters showed to double the stress experienced on an object. If you want I could try to find the formulas for the stress due to impact/shock loading.

3. Aug 15, 2010

### spotmeter

Thanks for helping me figure out the energy used going up. I would still like to calculate the force coming down as a percentage of bodyweight. Years ago I read that a runner with a 3" bounce lands with 6 times his bodyweight, but I don't know for sure if this is true, or how to figure it out. I recall the figure for a 2" bounce was 4 times bodyweight.

Appreciate any help in this matter.

Thanks.

4. Aug 15, 2010

### Gyro

I think the runner only exerts force during push off from foot. So I'm not sure if E2 counts since gravity is doing work on the runner during the 3" downfall.

5. Aug 15, 2010

### spotmeter

It seems to me that the runner has to do some work to prevent his body falling all the way to the ground. He mainly does this will his quads, preventing the landing knee from collapsing completely. He also uses his back muscles to prevent his upper body from collapsing.

I would like to figure out just how much work is involved in this, or how much energy is expended.

In addition, I would also like to know how much force the runner strikes the ground with as this is something that leads to injuries.

6. Aug 15, 2010

### Gyro

I think you're mixing up what you're choosing as your "system". If the "system" is the runner, then he is doing work on his surroundings when he launches. But when he lands, I don't think the runner does the work, I think surroundings are doing work on him and his body "deforms" semi-elastically to compensate. So technically he is not doing the work in the fall, it is work being done on him, which relates to your injury concept.

I think what you're talking about is modeling a pretty complex motion mathematically. Doing that, you usually make some assumptions to simplify the model. The model detail you want requires understanding the exact force curve when the runner launches and falls, and integrating it between certain limits to find area under the curve, which is the work done by him and on him.... I think, but I'm not a homework helper or expert, just trying to offer my two cents -- which may be worth more or less depending what a PF helper would say.

I think I'll put my muzzle back on now.

Last edited: Aug 15, 2010