Question

A body starts moving from rest with constant acceleration and covers displacement $S_1$ in the first $(p - 1)$ seconds and $S_2$ in the first $p$ seconds. The displacement $S_1 + S_2$ will be made in time:

• A. $\sqrt{2p^2 - 2p + 1} \, s$
• B. $(2p + 1) \, s$
• C. $(2p - 1) \, s$
• D. $(2p^2 - 2p + 1) \, s$

Answer 1

Answer: (A)

$\sqrt{2p^2 - 2p + 1} \, s$

Answer 2

Step 1: Calculate $S_1$ in the First $(p - 1)$ Seconds

Since the body starts from rest, using the formula $S = \frac{1}{2}at^2$,

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

Step 2: Calculate $S_2$ in the First $p$ Seconds

Using the same formula for $S_2$,

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

Step 3: Total Displacement $S_1 + S_2$

If $S_1 + S_2$ represents the displacement in time $t$, then:

$S_1 + S_2 = \frac{1}{2}at^2$

Substitute $S_1$ and $S_2$ values:

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

Simplify by canceling $\frac{1}{2}a$:

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

Step 4: Solve for $t$

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

So, the correct answer is: $\sqrt{2p^2 - 2p + 1} \, s$