I have a function of $(x,y)$ , for example $y = mx +c$,
And I also have a function for velocity in time manner, for example $v = 2t$
Basically I want to draw $f(x)$ in some delta time $t_0 - t_1$ with speed $v(t)$. So if $v(t)$ is quadratic function, I'll have an acceleration in drawing the $f(x)$
So, how do I get $x(t)$ and $y(t)$ ??
The distance along the curve $y$ starting from $x=x_0$ is
$$S(x) = \int_{x_0}^x \sqrt{1+y'^2} dx$$
If the particle travels with velocity $v(t)$ along $y$ starting from $x_0$ then
$$S(x) = \int_0^t v(t') dt'$$
defines $x(t)$ implicitly (and $y(t) = f(x(t))$).
For example if $y = x$ and $v = at$ then
$$S(x) = \sqrt{2}(x-x_0) = \frac{at^2}{2}$$
so
$$x(t) = x_0 + \frac{at^2}{2\sqrt{2}}$$