velocity as a function of position

Question

Let's say a particle of mass $m$ moves with a speed $ v(x) = \frac{ \beta }{x}$.

How would one go about calculating quantities such as $\ddot{x}(t), \dot{x}(t)$ and $x(t) $ ?

Answer 1

$v=\dot{x}$ so that

$$x \dot{x} = \beta \implies \frac{d}{dt} x^2 = 2 \beta \implies x(t) = \sqrt{2 \beta t+C}$$

where $C$ is a constant of integration. You may find $\dot{x}$ and $\ddot{x}$ by straightforward differentiation.