# Power of a runner

How can someone calculate the power in watt that a runner produces, when he runs uphill and downhill?

Is there any algorithm? It is important to take under consideration the uphill and downhill elements of the run.

Thank you for your answers but i am confused since i don't have a background in physics. To make thinks simple, i have some values, and using these values and maybe some constants like gravity, i want to create a calculator for real time watt production. These values are:

INSTANT VALUES: speed, distance, hr, kcal, ascent, descent, duration. OTHER VALUES: body mass, vertical speed (average), ascent time, descent time, max hr, rest hr, vo2max.

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This is in fact fully answered by the answers on 5882, but I'm a little hesitant to close this as a duplicate. Opinions are solicited from passers by. – dmckee Jan 13 '13 at 18:02

Assuming the runner's velocity $v$ is constant, his kinetic energy $\frac{1}{2} m v^2$ will be constant, so all we have to do is worry about his potential energy $U=mgh$. Power is rate of change in energy, so we take $P=\frac{dU}{dt}=m g \frac{dh}{dt}$. That's the power he outputs. If he weighs 70kg, and goes up a hill at 1 meter per second (So... that might be similar to sprinting up stairs?) Then we have $P=70 * 9.8 * 1 \text{kg} \frac{\text{m}}{\text{s}^2} \frac{\text{m}}{\text{s}}=686 \frac{\text{J}}{\text{s}}$

Note that if $\frac{dh}{dt}$ is negative, $P$ will be negative. It's also important to note that it's important that we took the runner's kinetic energy to be constant. If we had $E=U+\text{Ke}$, $\frac{dE}{dT}=\frac{dU}{dt}+\frac{d\text{Ke}}{dt}$, but since $\frac{d\text{Ke}}{dt}$ is zero we can ignore it.

This assumes simplifications like no friction, no heat generation, etc., in order to reduce it to a simple kinematics problem.

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The question is not easy as it involves a complex system as a human body. There is no algorithm as far as I know. It depends on what level of correctness you need.

First, you'll have to take into account the mechanical work. This is always $mgh$ where $m$ is the man's mass, $g$ is gravity acceleration, $h$ is the difference in height (can be negative).

The problem is that a runner will spend energy even when running on a flat surface (this doesn't happen in the ideal case). For this I'll suggest to take into account: 1. the heat: the runner warms up and dissipates energy; 2. the friction. I think they are very difficult to estimate, though.

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It's easy to find estimates of the amount of energy you use for running, on training-sites etc. The figure is given as a power as well (J/s). Then you just need to figure out the power used for running uphill. That is explained well by neurofuzzy.

That should give you a decent estimate.

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