Equation of Motion of Uniform Acceleration

Equations involving displacement, initial velocity, final velocity, acceleration and time of motion of a moving body are equations of motion.

Consider a body moving in a straight line with uniform acceleration, 

Let,

Displacement = s
Initial velocity = u
Final velocity = v
Acceleration = a
Time taken = t

Relation between u, v, a, and t

 We know, acceleration = (final velocity - iniitial velocity) / time

or, a = (v-u) / t

or, at = v-u

∴ v= u + at ........... (i)

This is the first equation of motion.

Relation between s, u, a and t

v = s / t

v = (u + v )/ 2

so,

    s / t = (u + v) / 2 ....(a)

or,  s / t = (u + u + at) /2

or, s / t = (2u + at) /2

or, s / t = u + 1/2 at

or, s = ut + 1/2 at² 

This is the second equation of motion.

Relation between u, v, a and s

We have,
v = u + at..............(i)

from (i), we get, t = (v-u) / a .. (b)

from (a),

    s =  (u + v) / 2 * t

we can use the (b) in this eqn, we get,

     s = (u + v) / 2 * (v - u) / a

or, s = (v² - u² ) / 2a

or, v² - u² = 2as

This is the third equation of motion.

Relation between s, u , v and t

s = 1/2 (u + v) * t

This is the fourth quation of motion.