First 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.

First Equation of Motion

v = u + at

where,

u = intial velocity,

v = final velocity

a = acceleration (must be constant),

t = time.

This equation DOES NOT relate to displacement.

Analytical Proof

We know that,

a = dv/dt

Cross multiplying ‘dt’, and integrating both sides,

dv = a.dt

v∫u dv = t∫0 a . dt ⇒ v∫u dv = a . t∫0 dt

v - u = at

v = u + at

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.
• Slope of this graph gives acceleration.

a = v - u/t - 0 (Acceleration is slope of the v - t graph)

Rearranging the terms,

v = u + at