Question

Derive x=ut + 1/2 at^2 by graphical method. ​

Answer 1

Answer:

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=

2

1

×Area of rectangle AEBD

=

2

1

×AD×BD

=

2

1

×t×at

=

2

1

at

2

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

s=ut+

2

1

at

2