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. `S_1 + S_2`
B. `S_1 S_2`
C. `S_1 - S_2`
D. `S_1//S_2`

Explanation

To solve the problem, we need to find the displacement of a body moving from rest under constant acceleration for a specific time interval. Let's break down the steps:

### Step 1: Understand the given information
- The body starts from rest, so the initial velocity \( u = 0 \).
- The acceleration is constant, denoted as \( a \).
- We need to find the displacement in the time interval of \( (p^2 - p + 1) \) seconds.

### Step 2: Calculate the displacement in the first \( (p - 1) \) seconds
Using the formula for displacement under constant acceleration:
\[ S_1 = ut + \frac{1}{2} a t^2 \]
For \( t = (p - 1) \):
\[ S_1 = 0 \cdot (p - 1) + \frac{1}{2} a (p - 1)^2 = \frac{1}{2} a (p - 1)^2 \]

### Step 3: Calculate the displacement in the first \( p \) seconds
Using the same formula:
\[ S_2 = u t + \frac{1}{2} a t^2 \]
For \( t = p \):
\[ S_2 = 0 \cdot p + \frac{1}{2} a p^2 = \frac{1}{2} a p^2 \]

### Step 4: Calculate the total displacement in \( (p^2 - p + 1) \) seconds
Using the displacement formula again:
\[ S = ut + \frac{1}{2} a t^2 \]
For \( t = (p^2 - p + 1) \):
\[ S = 0 \cdot (p^2 - p + 1) + \frac{1}{2} a (p^2 - p + 1)^2 \]

### Step 5: Expand \( (p^2 - p + 1)^2 \)
\[ (p^2 - p + 1)^2 = (p^2 - p + 1)(p^2 - p + 1) = p^4 - 2p^3 + 3p^2 - 2p + 1 \]
Thus,
\[ S = \frac{1}{2} a (p^4 - 2p^3 + 3p^2 - 2p + 1) \]

### Step 6: Relate \( S \) to \( S_1 \) and \( S_2 \)
We know that:
\[ S = S_1 + S_2 \]
Substituting the values of \( S_1 \) and \( S_2 \):
\[ S = \frac{1}{2} a (p - 1)^2 + \frac{1}{2} a p^2 \]
Now, simplifying \( S_1 + S_2 \):
\[ S_1 + S_2 = \frac{1}{2} a ((p - 1)^2 + p^2) \]
Calculating \( (p - 1)^2 + p^2 \):
\[ (p - 1)^2 + p^2 = (p^2 - 2p + 1) + p^2 = 2p^2 - 2p + 1 \]
Thus,
\[ S_1 + S_2 = \frac{1}{2} a (2p^2 - 2p + 1) \]

### Final Result
From the above calculations, we can conclude that:
\[ S = S_1 + S_2 \]
Thus, the displacement in \( (p^2 - p + 1) \) seconds is:
\[ S = S_1 + S_2 \]

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

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