a
(a) \(\frac{{dt}}{{dx}} = 2\alpha x + \beta \Rightarrow v = \frac{1}{{2\alpha x + \beta }}\)

\(a = \frac{{dv}}{{dt}} = \frac{{dv}}{{dx}}.\frac{{dx}}{{dt}}\)

\(a = v\frac{{dv}}{{dx}} = \frac{{ - v.2\alpha }}{{{{(2\alpha x + \beta )}^2}}} = - 2\alpha .v.{v^2} = - 2\alpha {v^3}\)

\(\therefore \) Retardation \( = 2\alpha {v^3}\)