Second Equation of Motion

Description:

The Concept

• Equations of Motion are designed to analyze motions with constant acceleration.

• They relate kinematic variables velocity, displacement, acceleration and time in different ways.

• In daily life, there are many instances of the motion with constant acceleration

• Freely falling objects

• Objects slipping down the incline plane

• Car starting from rest, etc.

Second Equation

S = ut + 1/2 at²

Where,

S = displacement

u = initial velocity

a = acceleration (must be constant),

t = time.

This equation DOES NOT relate to final velocity.

Analytical Proof

We know that,

v = u + at

Also,

v = dS/dt

‘S’ is the displacement.

Therefore, equating both,

dS/dt = u + at

S∫0 dS = t∫0 (u + at) . dt

S = ut∫0 dt + a t∫0 t . dt

S = ut + 1/2 at²

Graphical Proof

Following is a v - t graph displaying constant acceleration. (Slope of the curve is constant)

At t = 0 seconds, the particle’s velocity is u m/s.

At t = t seconds, the particle’s velocity is v m/s.

Area under the curve of v − t graph gives displacement.

Now,

S = Area of Rectangle + Area of Triangle

S = (u - 0) × (t - 0) + 1/2(v - u)(t - 0)

S = ut + 1/2(v - u)(t)

(Substituting v − u = at, from first equation of motion and rearranging the terms)

S = ut + 1/2 at²