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) n sec. will be

The problem involves the concept of kinematics, specifically the equations of motion.

Given that the body is moving from rest under constant acceleration, we can use the equation of motion:

S = ut + 1/2at^2

where: S is the displacement, u is the initial velocity, a is the acceleration, and t is the time.

Since the body is moving from rest, the initial velocity u = 0. So, the equation simplifies to:

S = 1/2at^2

Now, let's calculate S1 and S2:

S1 = 1/2a(p-1)^2 S2 = 1/2ap^2

The problem asks for the displacement in (p^2 - p + 1) seconds. Let's denote this displacement as S3.

S3 = 1/2a(p^2 - p + 1)^2

To find S3 in terms of S1 and S2, we need to express a in terms of S1 and S2.

From the equations for S1 and S2, we can express a as:

a = 2S1/(p-1)^2 = 2S2/p^2

Setting these two expressions for a equal to each other gives:

2S1/(p-1)^2 = 2S2/p^2

Solving for S1 in terms of S2 and p gives:

S1 = S2(p-1)^2/p^2

Substituting this expression for S1 into the equation for S3 gives:

S3 = 1/2 * 2S2(p-1)^2/p^2 * (p^2 - p + 1)^2

Simplifying this gives:

S3 = S2 * (p-1)^2 * (p^2 - p + 1)^2 / p^2

So, the displacement in (p^2 - p + 1) seconds is S2 times the square of (p-1) times the square of (p^2 - p + 1), all divided by the square of p.