A particle of mass $$m$$ moves along $$x$$ axis with its potential energy as $$U(x) = \frac{\alpha}{x^2} - \frac{\beta}{x}$$ where $$\alpha$$ and $$\beta$$ are positive constants. The particle is released from rest at $$x_0 = \frac{\alpha}{\beta}$$. Then

$$U(x)$$ can be expressed as $$U(x) = \frac{\alpha}{x_0^2}\left[\left(\frac{x_0}{x}\right)^2 - \frac{x_0}{x}\right]$$

velocity of the particle $$v(x)$$ as a function of $$x$$ can be expressed as $$v(x) = \left[\frac{2\alpha}{mx_0^2}\left\{\frac{x_0}{x} - \left(\frac{x_0}{x}\right)^2\right\}\right]^{1/2}$$

the maximum speed of the particle is $$v_{\max} = \sqrt{\frac{\alpha}{2mx_0^2}}$$

the total energy of the particle $$KE(x) + U(x)$$ is zero