How to find position along sine curve at given time?

Question

I'm having trouble figuring out the following, it would be great if someone could help me out!

If Person A is moving along the curve $y=sin(x)$ at the constant speed $S$, then what is their position in the form (x, y) at time $T$?

Edit for clarification:

$S$ is constant speed along the curve. If Person A were moving along a straight line, it's pretty clear that their x-coordinate would be just be $ST$ at time $T$. But since the arc of the sine function is longer than a straight line, this isn't the case. Thus I'm trying to find a way to express Person A's x and y coordinates at $T$, given that they're moving at a constant speed along $y=sin(x)$

Answer 1

The velocity magnitude is constant. Given that the velocity magnitude is $$v = \sqrt{\dot x ^2 + \dot y ^2 },$$ then

$$v = \sqrt{\dot{x}^2 + \dot{y}^2} = S \Rightarrow \\ \sqrt{\dot{x}^2 + \left(\dot{x} \cos(x)\right)^2} = S \Rightarrow \\\dot x = \frac{S}{\sqrt{1+\cos^2(x)}}.$$

The last equation is a separable-variables ODE which is not solvable with standard functions.

Answer 2

For constant speed, you can do $x=ST$.

Thus $(x,y) = (ST,sin(ST))$.

This keeps the velocity in x-direction constant and ignores the y-direction changes.