If $v=dx/dt$ what is $v^2$ equal to? Just working on mechanics problems right now and I realized that I never actually dealt with a problem like this before. 

Usually when I'm solving a differential equation, I would often be able to convert variable $v$ to $dx/dt$. However, this question requires me to convert $v$ into a form related to $dx/dt$.
Well... like that? $$(v = \frac{\mathrm{d}S }{\mathrm{d} t}) \xrightarrow{f: x \mapsto x^2} v^2 =  \frac{\mathrm{d}S }{\mathrm{d} t} \frac{\mathrm{d}S }{\mathrm{d} t} \Leftrightarrow v^2 =  \frac{\mathrm{d}S }{\mathrm{d} t^2} \mathrm{d}S $$

which give us $$v^2 = a * \mathrm{d}S$$

I'm probably wrong, but I though I could give it a try.
Well... like that? $$(v = \frac{\mathrm{d}S }{\mathrm{d} t}) \xrightarrow{f: x \mapsto x^2} v^2 =  \frac{\mathrm{d}S }{\mathrm{d} t} \frac{\mathrm{d}S }{\mathrm{d} t} \Leftrightarrow v^2 =  \frac{\mathrm{d}S }{\mathrm{d} t^2} \mathrm{d}S $$

which give us $$v^2 = a * \mathrm{d}S$$

I'm probably wrong, but I though I could give it a try.