For a body moving with a uniform acceleration, prove that its average velocity is the arithmetic mean of its initial and final velocities.

Explanation

Let u be the initial velocity of a body and a be its uniform acceleration. After time t the body acquires a velocity b (say) and s its displacement. Hence average velocity of the body ` = (s)/(t)`. Again the arithmetic mean of the initial and the final velocities of the body
`= (u+v)/(2)=(u+u+at)/(2)= u+(1)/(2) at`
`= ((u+(1)/(2)at)t)/(t)=(ut+(1)/(2)at^(2))/(t)=(s)/(t)` (Proved).