Constant Moving Speed

Question

I've made this graph:

https://www.desmos.com/calculator/czk3ylyokj

As you can see, the purple point is slowing down as it approaces the extreme point. How can I make this point move with constant speed along the f function graph?

Answer 1

Your point is moving at a constant speed with respect to the $x$ axis since that's how you parametrized it. Since the function you are using $x^2$ grows much faster then $x$ the point travels a further distance along the curve when it's further from the origin.

To fix this you would need to parametrize the curve in such a way that the point travels a constant distance at any given time. You should consider computing the arclength traveled and making that linear in the parameter $t$.

To be honest I would have to write that down before I can give you a formula.

Actually an easy partial solution is to plug in $\sqrt{|t|}$ instead of $t$ everywhere. You need to fix the fact that now you're only going up down one side of the parabola though. Which can be done by clever usage of the sign function.

Answer 2

Using your parametrization (time t), the goal is to get the coordinate $(v(t), f(v(t)))$ that has constant velocity as follows, \begin{align} v'(t)^2 + f'(v(t))^2 v'(t)^2 = c^2. \end{align}

This is a differential equation, and you can solve it in terms of $v(t)$ with $f(x) = x^2 - 5$. But it needs some classification by the sign of $t$ and might be difficult to visualize in the web application you used.