Show by using the graphical method that: `s=ut+1/2at^2` where the symbols have their usual meanings.

#### Solution

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 the trapezium OABC

But, Area of the figure OABC = Area of rectangle OADC + Area of triangle ABD

= Area of rectangle OADC + area of triangle ABD

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

(i) Area of rectangle OADC

= (OA) (OC)

= (u) (t)

(ii) Area of triangle ABD,

= (1/2)(AD)(BD)

= (1/2)(t)(at)

= (1/2)at^{2}

Distance travelled (s) is,

So, s = Area of rectangle OADC + Area of triangle ABD

`s = ut + 1/2at^2`

This is the second equation of motion.

Where

(s) - Displacement

(u) - Initial velocity

(a) - Acceleration

(t) - Time