(Integration) Physics problem to find the time before a collar comes to rest on a ring with friction.

Question

If unclear, scroll to the end integral which needs to be evaluated.

I need to find $\tau$ from this integration/differential equation but I'm stuck. $$\int\frac{-{\rm d}v}{\sqrt{v^4+R^2g^2}}=\int_0^{\tau}\frac{\mu{\rm d}t}{R}$$

Answer 1

Since the question is asking for distance travelled, the trick is to bring distance along the circular track, say $x$, in as the variable instead of time. So if we take $F = m \dfrac{dv}{dt}$, then note that $F\dfrac{dx}{dt} = mv \dfrac{dv}{dt}$. Now we can remove the explicit dependence on $t$ and obtain $$\int_0^{X} F\,dx = \int_{v_0}^0 mv\,dv.$$