A particle moves a distance $$x$$ in time $$t$$ according to the equation $$x = {\left( {t + 5} \right)^{ - 1}}.$$   The acceleration of particle is proportional to

Solution :
Given, distance $$x = {\left( {t + 5} \right)^{ - 1}}......\left( {\text{i}} \right)$$
Differentiating Eq. (i) w.r.t. $$t,$$ we get
$$\frac{{dx}}{{dt}} = \left( v \right) = \frac{{ - 1}}{{{{\left( {t + 5} \right)}^2}}}\,......\left( {{\text{ii}}} \right)$$
Again, differentiating Eq. (ii) w.r.t. $$t,$$ we get
$$\frac{{{d^2}x}}{{d{t^2}}} = \left( a \right) = \frac{2}{{{{\left( {t + 5} \right)}^3}}}\,......\left( {{\text{iii}}} \right)$$
Comparing Eqs. (ii) and (iii), we get $$\left( a \right) \propto {\left( v \right)^{\frac{3}{2}}}$$