A body is moving from rest under constance accelration 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)^(th)` sec will be

Options

A. `S_1+S_2`
B. `S_1S_2`
C. `S_1-S_2`
D. `(S_1)/(S_2)`

Explanation

To solve the problem step by step, we will use the equations of motion under constant acceleration.

### Step 1: Understand the Problem
We need to find the displacement of a body moving from rest under constant acceleration during the \((p^2 - p + 1)^{th}\) second. We denote the displacement in the first \((p-1)\) seconds as \(S_1\) and in the first \(p\) seconds as \(S_2\).

### Step 2: Write the Formula for Displacement
The displacement \(S\) under constant acceleration can be calculated using the formula:
\[ S = ut + \frac{1}{2} a t^2 \]
where \(u\) is the initial velocity, \(a\) is the acceleration, and \(t\) is the time. Since the body starts from rest, \(u = 0\).

### Step 3: Calculate \(S_1\)
For the first \((p-1)\) seconds:
\[ S_1 = 0 \cdot (p-1) + \frac{1}{2} a (p-1)^2 = \frac{1}{2} a (p-1)^2 \]

### Step 4: Calculate \(S_2\)
For the first \(p\) seconds:
\[ S_2 = 0 \cdot p + \frac{1}{2} a p^2 = \frac{1}{2} a p^2 \]

### Step 5: Calculate the Displacement in the \((p^2 - p + 1)^{th}\) Second
The displacement in the \(n^{th}\) second is given by:
\[ S_n = u + \frac{1}{2} a (2n - 1) \]
Substituting \(n = p^2 - p + 1\):
\[ S_{p^2 - p + 1} = 0 + \frac{1}{2} a (2(p^2 - p + 1) - 1) \]
\[ = \frac{1}{2} a (2p^2 - 2p + 2 - 1) = \frac{1}{2} a (2p^2 - 2p + 1) \]

### Step 6: Relate \(S_{p^2 - p + 1}\) to \(S_1\) and \(S_2\)
Now, we need to express \(S_{p^2 - p + 1}\) in terms of \(S_1\) and \(S_2\):
\[ S_1 + S_2 = \frac{1}{2} a (p-1)^2 + \frac{1}{2} a p^2 \]
Combining these:
\[ = \frac{1}{2} a \left((p-1)^2 + p^2\right) \]
Calculating \((p-1)^2 + p^2\):
\[ (p-1)^2 + p^2 = (p^2 - 2p + 1) + p^2 = 2p^2 - 2p + 1 \]
Thus:
\[ S_1 + S_2 = \frac{1}{2} a (2p^2 - 2p + 1) \]

### Conclusion
We find that:
\[ S_{p^2 - p + 1} = S_1 + S_2 \]
Thus, the displacement in the \((p^2 - p + 1)^{th}\) second is equal to \(S_1 + S_2\).

### Final Answer
The displacement in the \((p^2 - p + 1)^{th}\) second is:
\[ S_{p^2 - p + 1} = S_1 + S_2 \]