Find the angle of the parable tangent derived from the theoretical arc traced by a foot

Barkiernan

New member
I am a masters student studying motion analysis in human running.

I need to find the angle of the parable tangent derived from the theoretical arc traced by a foot during a step and the ground (see attached). The arc comprises of a persons step height and step length and I need to find the angle of the arc it creates. No other research regarding this angle has manual calculated it.

I have contacted the researcher and he gave me the following formula:

Stride (step) angle tangent = 4*height / Step length
Therefore, the Stride (step) angle = tan-1(4*height/step length)”

However we are not sure why the height is multiplied by 4 ?

Thank you

MarkFL

Staff member
Hello, and welcome to MHB!

If we let $$\ell$$ be the stride length, and $$h$$ be the max height, and orient our coordinate axes such that the "toe off" is at the origin, then we have:

$$\displaystyle f(x)=kx(x-\ell)$$

Now, we must have:

$$\displaystyle f\left(\frac{\ell}{2}\right)=h$$

$$\displaystyle k\left(\frac{\ell}{2}\right)\left(\frac{\ell}{2}-\ell\right)=h\implies k=-\frac{4h}{\ell^2}$$

And so:

$$\displaystyle f(x)=-\frac{4h}{\ell^2}x(x-\ell)=-\frac{4h}{\ell^2}x^2+\frac{4h}{\ell}x$$

From this we find:

$$\displaystyle f'(x)=-\frac{8h}{\ell^2}x+\frac{4h}{\ell}$$

And then:

$$\displaystyle f'(0)=\frac{4h}{\ell}$$

Thus, we may conclude:

$$\displaystyle \alpha=\arctan\left(\frac{4h}{\ell}\right)$$