How to convert x versus v graph to v versus t

Question

Suppose that I have a V (speed) vs X (distance) graph as follows :

I want to draw V (speed) vs t (time) graph.

But I don't wat to get the graph, I want to get the proper way of transforming graphs. How can I transform V vs X to V vs t?

Regards.

Answer 1

let the equation of $v$ vs $x$ graph be $v=ax+b$

Then $$\dfrac{dx}{dt}=ax+b$$

$$\dfrac{dx}{ax+b}=dt$$

Integrate both sides

$$\int \dfrac{dx}{ax+b}=\int dt$$

$$\dfrac{1}{a}\ln(ax+b) = ct$$

you probably know $a$ and $b$ from the graph, find $c$ from initial condition [should be given]