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Consider the following example: While running, I do work and I have a certain velocity and acceleration. When I'm in a pool (my feet are touching the bottom), I do the same work but, I have less acceleration. As more of my body is submerged I begin to be slower (doing the same work) . So I require an explanation, by how much water slows me.

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Sorry but there is no relation between the 2 questions . I dont want to run over water. A nice a question though :) –  Abdelrahman Esmat Nov 3 '12 at 22:29

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When running only about 10% of the power you consume is used overcoming air resistance. The rest is used because your muscles are rather inefficient at converting energy to work. But let's ignore this and only consider the air (and water) resistance.

When you're running in air at a constant velocity the power you're expending, i.e. the work per second, is simply your velocity times the drag from air resistance. The drag varies linearly with velocity at low speeds and quadratically with velocity at high speeds. A quick Google should find you figures for the drag on a typical runner.

Exactly the same applies to running in water, except that the viscosity and density of water are much greater than air so the drag is much greater.

There is a second effect in water, unrelated to drag, Archimedes principle tells us that the water you displace exerts an upwards force on you so in effect you weigh less in water. This means the friction between the running shoes and the bottom of the swimming pool is greatly reduced, and this reduces the speed you can run simply because your shoes skid instead of gripping.

If you want a figure for the ratio of top speed in air to top speed in water then I think this would be exceedingly difficult to calculate from first principles. The only way to get this data would be to measure the speeds. I tried Googling for such measurements but couldn't find anything.

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