# Derive the expressions for equations of motion.

Let us consider a body whose initial velocity is u, moves for a time t with a constant acceleration a, and so acquires a final velocity v. It moves a distance s in time t. Now, from the definition of acceleration, the change in velocity = v – u
Therefore the rate of change of velocity = acceleration a = or        v – u = at ----(1)
v = u + at ----(2)
Since distance = average × velocity × time
s = ----(3)
Substituting the value of v from equation (2), we get
s = = s = ut + at2 ----(4)
From the equation (1), we have
v – u = at
and from (3) we have
2s = (u + v) × t
u + v = ----(5)
Multiplying (1) by (5), we get
(v – u) (v + u) = at × v2 – u2 = 2as ----(6)
The equations (2), (4) and (6) are the equations of motion.

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