How can I find $v(t)$ if I'm given $x(v)$?

Question

The given is the distance but the variable is the velocity $x(v)=k \,v^2$. I want to find the velocity $v(t)$.

Answer 1

The given relation implies that at all times $t$, the function $x(t)$ obeys $$ x(t) = k \left( \frac{dx(t)}{dt} \right)^2 \quad \Rightarrow \quad \frac{dx(t)}{dt} = \pm \sqrt{\frac{x(t)}{k}} $$ This is just a differential equation for $x(t)$ and so can be solved for $x(t)$ using standard ODE techniques. (Note, in particular, that it is a separable equation.) Once this is done, the result can be differentiated to find $v(t)$.

Answer 2

$x = kv^2 \\ \displaystyle \Rightarrow v = \frac {dx}{dt} = 2kv \frac {dv}{dt}$

One trivial solution is that $x=v=0$ for all $t$. However, if $v \ne 0$ then we have

$\displaystyle 2k \frac {dv}{dt} = 1 \\ \displaystyle \Rightarrow \frac {dv}{dt} = \frac 1 {2k}$

You can integrate this to find $v$ as a function of $t$. Integrating introduces a constant of integration, which you can set equal to the initial velocity $v|_{t=0}$.