1. Use the equation of motion for displacement from rest

Since the body starts from rest with constant acceleration $a$,

$s = ut + \frac{1}{2}at^2 = \frac{1}{2}at^2$

because $u=0$.

2. Write $S_1$ and $S_2$

• Displacement in first $(p-1)$ seconds:

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

• Displacement in first $p$ seconds:

$S_2 = \frac{1}{2}ap^2$

3. Find $S_1 + S_2$

$S_1 + S_2 = \frac{1}{2}a\left[(p-1)^2 + p^2\right]$

Expand:

$(p-1)^2 + p^2 = p^2 - 2p + 1 + p^2 = 2p^2 - 2p + 1$

So,

$S_1 + S_2 = \frac{1}{2}a(2p^2 - 2p + 1)$

4. Let the required time be $t$

If the body makes displacement $S_1+S_2$ in time $t$, then

$\frac{1}{2}at^2 = \frac{1}{2}a(2p^2 - 2p + 1)$

Cancel $\frac{1}{2}a$ from both sides:

$t^2 = 2p^2 - 2p + 1$

Hence,

$t = \sqrt{2p^2 - 2p + 1}$

Since time is positive,

$t = \sqrt{\left(2p^2 - 2p + 1\right)}\;\text{s}$

5. Match with options

This corresponds to Option D.