Question

Derive graphically S=ut+1/2at2.

Answer 1

Consider the velocity-time graph of a body shown in the figure. 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 traveled 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 traveled = 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×OC
=u×t
=ut

(ii) Area of triangle ABD= 1/2×Area of rectangle AEBD
= 1/2×AD×BD

=1/2×t×at

= 1/2 at^2


Distance travelled, s= Area of rectangle OADC + area of triangle ABD
s=ut+1/2at^2