Question

Show by using the graphical method that : $s=ut+\frac{1}{2}a{t}^2$ where the symbols have their usual meanings.

Answer 1

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)at²

Distance travelled (s) is,
So, s = Area of rectangle OADC + Area of triangle ABD
$\boxed{s=ut+\frac{1}{2}a{t}^2}$
This is the second equation of motion.
Where
(s) - Displacement
(u) - Initial velocity
(a) - Acceleration
(t) - Time