The body has an initial velocity u at a point A and then its velocity changes at a uniform rate from A to B in time t. In other words, there is a uniform acceleration a from A to B, and after time t its final velocity becomes v which is equal to BC in the graph. The time t is represented by OC.

Suppose the body travels a distance s in time t. In the figure, the distance travelled by the body is given by the area of the space between the velocity-time graph AB and the time axis OC, which is equal to the area of the figure OABC.

Thus, Distance travelled = Area of figure OABC = Area of rectangle OADC + area of triangle ABD

Now, we will find out the area of rectangle OADC and area of triangle ABD.

(i) Area of rectangle OADC =OA x OC

= u x t = ut

(ii) Area of triangle ABD = \(\frac{1}{2}\) x Area of rectangle AEBD

= \(\frac{1}{2}\) x AD x BD

= \(\frac{1}{2}\) x t x at

= \(\frac{1}{2}at^2\)

Distance travelled, s = Area of rectangle OADC + area of triangle ABD

s = \(ut + \frac{1}{2}at^2\)