Velocity - Time Graph And Physical Equation

Consider about a body which moves from A to B in time t with uniform velocity. The velocity of the body at A and B is u and v respectively.

The slope of the velocity time graph = Acceleration = a
= Slope of AB
= BD/AD
From figure it is clear that
OA = u
BC = v
OC = t
OA = CD
BD = BC - CD
BC = v - u
Therefore,
a = (v - u)/t
v = u + at
t = (v - u)/a
u = v - at

The area described by body to move from A to B
= Distance covered by body = S
= Area of Trapezium
= (OA + BC)xOC/2
= (u + v)x t/2                ------------------- (i)
Now, putting the value of v.
(u + u + at)xt/2
S  = ut + 1/2 at^2

Now, from (i)
S = (u + v)(v - u)/2a
2aS = v^2 - u^2
Therefore,
v^2  = u^2 + 2aS

The required physical equation of velocity time graph are

a = (v -  u)/t
S = ut + 1/2at^2
v^2 = u^2 + 2aS