A body starts moving from rest with constant acceleration covers...

For the displacement ${S}_{1}$ of the body, it can be written that

${S}_{1} = \dfrac{1}{2}a ( p - 1{)}^{2} . . . \left(1\right)$

And, for the displacement ${S}_{2}$ of the body, it follows that

${S}_{2} = \dfrac{1}{2}a ( p{)}^{2} . . . \left(2\right)$

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

${S}_{1} + {S}_{2} = \dfrac{1}{2}a{t}^{2} . . . \left(3\right)$

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

$( p - 1{)}^{2} + {p}^{2} = {t}^{2} ⇒ t = \sqrt{2{p}^{2} - 2p + 1}s$