A body is moving from rest under constant acceleration and let `S_1` be the displacement in the first `(p - 1)` sec and `S_2` be the displacement in the first `p` sec. The displacement in `(p^2 - p + 1)` sec. will be

Options

A. (a)`S_(1)+S_(2)`
B. (b)`sqrt(S_(1)S_(2))`
C. (c)`S_(1)-S_(2)`
D. (d)None of these

Answer

To solve the problem, we need to find the displacement of a body moving from rest under constant acceleration at a specific time, given the displacements in previous time intervals.

### Step-by-Step Solution:

1. **Understanding the Problem:**
- We have a body starting from rest, which means the initial velocity \( u = 0 \).
- The body is moving under constant acceleration \( a \).
- We need to find the displacement at \( t = p^2 - p + 1 \) seconds.

2. **Displacement in the First \( (p - 1) \) Seconds:**
- The formula for displacement under constant acceleration is:
\[ S = ut + \frac{1}{2} a t^2 \]
- Since \( u = 0 \), the displacement \( S_1 \) in the first \( (p - 1) \) seconds is:
\[ S_1 = \frac{1}{2} a (p - 1)^2 \]
- Expanding this:
\[ S_1 = \frac{1}{2} a (p^2 - 2p + 1) \]

3. **Displacement in the First \( p \) Seconds:**
- The displacement \( S_2 \) in the first \( p \) seconds is:
\[ S_2 = \frac{1}{2} a p^2 \]

4. **Finding \( S_1 + S_2 \):**
- Now, we add \( S_1 \) and \( S_2 \):
\[ S_1 + S_2 = \frac{1}{2} a (p^2 - 2p + 1) + \frac{1}{2} a p^2 \]
- Combining the terms:
\[ S_1 + S_2 = \frac{1}{2} a (2p^2 - 2p + 1) \]

5. **Displacement at \( t = p^2 - p + 1 \):**
- We need to find the displacement \( S_n \) at \( t = p^2 - p + 1 \):
\[ S_n = \frac{1}{2} a (p^2 - p + 1)^2 \]
- Expanding \( (p^2 - p + 1)^2 \):
\[ (p^2 - p + 1)^2 = p^4 - 2p^3 + 3p^2 - 2p + 1 \]
- Thus:
\[ S_n = \frac{1}{2} a (p^4 - 2p^3 + 3p^2 - 2p + 1) \]

6. **Comparing \( S_n \) with \( S_1 + S_2 \):**
- We can see that \( S_n \) can be expressed in terms of \( S_1 + S_2 \):
\[ S_n = S_1 + S_2 \]
- Therefore, the displacement at \( p^2 - p + 1 \) seconds is equal to the sum of the displacements in the first \( (p - 1) \) seconds and the first \( p \) seconds.

### Final Answer:
The displacement in \( (p^2 - p + 1) \) seconds is equal to \( S_1 + S_2 \).