Power curve calculation

Question

I have a power curve relating to a turbo trainer I use on my bike.

I save my sessions on a website called strava using speed, cadence, heart rate and time.

Using the power curve I've been given I wish to calculate the gradient of the curve, providing me with a power calculation for a given speed.

$\qquad\qquad\qquad$

I've got a rough table of figures that can be used to plot a graph, but I can't remember the maths I need to work out the formula.

$$ \begin{array}{c|r} \text{speed} & \text{power} \\ \hline 5 & 25 \\ 9 & 50 \\ 12.5 & 100 \\ 17 & 200 \\ 20.5 & 300 \\ 23 & 400 \\ 25 & 500 \\ 27 & 600 \\ \end{array} $$

Thanks for any advice you can give me.. even better if you can provide the completed formula.

Answer 1

The power curve seems to be approximately of the form $P=cv^{2}$, where $P$ is the power, $v$ is the speed and $c$ is a constant that we need to find. To reduce the error I computed the following mean based on your figures $$\begin{equation*} c=\sqrt[8]{\frac{25}{5^{2}}\frac{50}{9^{2}}\frac{100}{12.5^{2}}\frac{200}{ 17^{2}}\frac{300}{20.5^{2}}\frac{400}{23^{2}}\frac{500}{25^{2}}\frac{600}{ 27^{2}}}\approx 0.75. \end{equation*}$$

So the approximate equation of the form $P=cv^{2}$ is $$\begin{equation*} P\approx 0.75v^{2}\qquad \text{(}P\text{ in Watt and }v\text{ in mph),} \end{equation*}$$

or $$\begin{equation*} v\approx 1.16\sqrt{P}. \end{equation*}$$

This approximation gives

$$\begin{eqnarray*} v(600) &\approx &28.4 \\ v(300) &\approx &20.1 \\ v(100) &\approx &11.6. \end{eqnarray*}$$

Answer 2

"Using the power curve I've been given I wish to calculate the gradient of the curve, providing me with a power calculation for a given speed."

If I understand right, the curve itself already gives you the power as a function of speed -- you don't need to do anything with the gradient. And I think that the curve itself was derived empirically by the company so there is not necessarily a formula to "work out" other than just interpolating between the points.