Derive the second equation of motion s = ut + (1/2)at2 using the velocity-time graph.
Consider the linear motion of a body with an initial velocity u. Let the body accelerate uniformly and acquire a final velocity v after time t. The velocity–time graph is a straight line AB as shown below.
At t = 0, initial velocity = u = OA
At t = t, final velocity = v = OC
The distance S travelled in time t = area of the trapezium OABD
s = (1/2) x (OA + DB) × OD
s = (1/2) x (u + v) × t
Since v = u + at,
s = (1/2) x (u + u + at) × t
s = ut + (1/2) at2