Third 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.

Third Equation

v² = u² + 2aS

where,

v = final velocity,

S = displacement,

u = initial velocity,

a = acceleration (must be constant),

This equation DOES NOT relate to final velocity.

Analytical Proof

We know that,

a = dv/dt

v = dS/dt

Taking their ratio, we get,

a/v = dv/dS

Hence,

a = vdv/dS

Cross multiplying ‘dS’ and integrating both sides,

S∫0 a . dS = v∫u v . dv ⇒ aS = v² - u²/2

v² = u² + 2aS

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

(Substituting t = (v - u)/a, from first equation of motion)

S = u(v - u/a) + 1/2(v - u)(v - u/a)

Rearranging the terms,

v² = u² + 2aS