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 as shown in the figure.

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

Relation between u, v, a, and t

=
Or,
Or, at = v-u
∴ v= u + at ........... (i)
This is the first equation of motion.

Relation between s, u, v and t

Or,

Or,
As both equations are equal
Or,
Or, 2s = (u + v) × t
∴........(ii)
This is the second equation of the motion.

Relation between s, u, a and t

We already have,
V= u + at ........(i)
.......(ii)
Putting value of v from equation (i) in (ii)
Or, s =
Or, s = × t
Or, s = (2u × t + at × t )\(\frac{1}{2}\)
∴ s = ut + \(\frac{1}{2}\)a\(t^2\) ........(iii)
This is the third equation of motion.

Relation between u, v, a and s

We have,
v = u + at..............(i)
s = \(\frac{u +v}{t}\)× t
Putting the value of t from (i) in the equation (ii),
s = \(\frac{v+ u}{2}\) \( \times \frac{v - u}{a}\)

Or, s = \(\frac{v^2- u^2}{2a}\)

\(\therefore\) v² = u² + 2as..................(iv)

This is the fourth equation of motion.