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\[m\ddot{x}=ma\cdot f'(x)\]
\[\int \frac{d\dot{x}}{dt}dx=a\int f'(x)dx\]
\[\int \frac{d\dot{x}dx}{dt}\cdot\frac{dxd\dot{x}}{dt}\cdot\frac{dt}{dx}dx=a\int f'(x)dx\]
\[\int \dot{x}d\dot{x}=a\int f'(x)dx\]
\[\frac12\dot{x}^2=a(f(x)-f(0))\]
\[|\dot{x}|=\sqrt{2a(f(x)-f(0))}\]