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

The Correct Option is A

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: \(\sqrt{2p^2 - 2p + 1} \, s\)