Displacement in N^th Second

Description:

We have the kinematic equations for a body having uniform acceleration −

v = u + at

S = ut + 1/2at²

2aS = v² - u²

To derive an equation for displacement in n^th second, let us consider a body in motion with uniform acceleration. The displacement in every second will be different as the velocity of the body is increasing uniformly.

Assume S₂ is the displacement in n seconds

From kinematic equation −

S = ut + 1/2 at²

We get −

S₂ = un + 1/2 an²

Assume S₁ is the displacement in n − 1 seconds

From kinematics equation −

S = ut + 1/2 at²

We get −

S₁ = u(n - 1) + 1/2 a(n - 1)²

Hence, displacement in n^th second is given by −

S_n = S₂ - S₁

S_n = un + 1/2 an² - [u(n - 1) + 1/2 a(n - 1)²]

S_n = un + 1/2 an² - [un - u + 1/2 a(n² - 2n + 1)]

S_n = un + 1/2 an² - [un - u + 1/2 an² - an + 1/2 a]

S_n = un + 1/2 an² - un + u - 1/2 an² + an - 1/2 a

S_n = u + an - 1/2 a

S_n = u + 1/2 a(2n - 1)