Derive the following equations for a uniformly accelerated motion – (i) v = u + at (ii) S = ut + 1/2 at² (iii) v2 = u2 + 2aS where the symbols have their usual meanings.

(i) Acceleration = change\ in\ velocity\over time\ taken

= final\ velocity - initial\ velocity\over time\ taken

or, a = v - u\over t

or, at = v – u

or, v = u + at

(ii) Distance travelled = Avg velocity × time

= v + u\over 2 × t

= (u + at) + u\over 2 × t

= 2u + at\over 2 × t

= 2ut + at²\over 2

= ut + 1/2 at²

(iii) Distance travelled = Avg velocity × time

S = v + u\over 2 × t

But, from v = u + at, we have t = v - u\over a

∴ S = v + u\over 2 × v - u\over a

= v^2 + u^2\over 2a