Question

Derive the equation S = ut+1/2at2 using graphical method where the symbols have their usual meaning.

Answer 1

in this graph consider,
OABC as a trapzeium
wkt,
area of trap= 1/2(sum of paraell sides)×h
so, area of trap is the distance covered
so area of trap = s
now paraell sides are AO and BC and h is OC
SO,
S=1/2(AO +BC)×OC
OR, (according to garph)
s=1/2(v+u)×t
(by first equation of motion)
(v=u+at)
s=1/2{u+at+u}t
s=1/2{2u/t+at2}

S=Ut+1/2At2

Answer 2

Explanation:

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