# Mechanics differential equation

This question is related to a mechanics problem for which I have already done the dynamics side of it and have now reduced it down to this but I'm having trouble evaluating this differential equation.

$$\frac{ds}{dt}=\sqrt{(2g\sin\alpha) s}$$

I can see that I need to separate variables but I'm not sure how to proceed from here.

Any help would be appreciated.

• Is $\alpha$ function of $t$ or of $s$ ? If not the ODE is separable : use the straightforwardl integration method. – JJacquelin Mar 10 '16 at 16:48
• $\alpha$ was just the angle that the plane was inclined at, so no it isn't a function of $t$ or $s$. – MHW Mar 10 '16 at 16:53

$$\frac{ds}{dt}=\sqrt{(2g\sin\alpha) s}$$
$$\frac{ds}{\sqrt{s}}=\sqrt{2g\sin\alpha}\:dt$$
$$\int\frac{ds}{\sqrt{s}}=\sqrt{2g\sin\alpha}\int dt$$ $$2\sqrt{s}=\sqrt{2g\sin\alpha}\:t+\text{constant}$$