A body starts moving from rest with constant acceleration covers displacement S 1 in first ( p - 1 ) seconds…

For the displacement $S_1$ of the body, it can be written that

$S_1=\frac{1}{2}a(p-1{)}^2 ...\left(1\right)$

And, for the displacement $S_2$ of the body, it follows that

$S_2=\frac{1}{2}a(p{)}^2 ...\left(2\right)$

If $t$ be the required time, it follows that

$S_1+S_2=\frac{1}{2}a{t}^2 ...\left(3\right)$

Equations (1), (2) and (3) imply that

$$\begin{gathered} (p-1{)}^2+p^2=t^2 \\ \Rightarrow t=\sqrt{\mathit{2}{p}^{\mathit{2}}\mathit{-}\mathit{2}p+1} \mathrm{s} \end{gathered}$$