A particle of unit mass undergoes one-dimensional motion such that its velocity varies according to $$v\left( x \right) = \beta {x^{ - 2n}}$$   where, $$\beta $$ and $$n$$ are constants and $$x$$ is the position of the particle. The acceleration of the particle as a function of $$x,$$ is given by

Solution :
Given, $$v = \beta {x^{ - 2n}}$$
$$\eqalign{ & a = \frac{{dv}}{{dt}} = \frac{{dx}}{{dt}} \cdot \frac{{dv}}{{dx}} \cr & \Rightarrow a = v\frac{{dv}}{{dx}} = \left( {\beta {x^{ - 2n}}} \right)\left( { - 2n\beta {x^{ - 2n - 1}}} \right) \cr & \Rightarrow a = - 2n{\beta ^2}{x^{ - 4n - 1}} \cr} $$