Question

Derive the following equations considering uniform acceleration(by graphical method): s = ut+ 1/2 at 2 .

Answer 1

Let AB represents velocity-time graph of a particle moving with uniform acceleration(a) and the initial velocity (u).BC is drawn perpendicular to X-axis and AD is drawn parallel to X-axis.
From the graph:
OA=CD=u: initial velocity
OC=AD=t:time
OE=BC=v:final velocity

CB=CD+DB
v=u+DB
:. DB=v-u

slope of velocity time graph gives acceleration.
therefore, a= DB/ AD
                  a=DB/t
{SUBSTITUTING EQUATION I : DB= v-u( a= v-u/t)}
            therefore,DB= at

Area enclosed by velocity time graph gives displacement.
Therefore, displacement(s)= Area enclosed by the trapezium OABC
                                           = Area of rectangle OADC+ Area of triangle
                                               ABD
                                           = OA*OC+1/2*AD*BD
                                          =  u *t + 1/2*t*AT
                                          =  ut +1/2 at 2
therefore s=ut+ 1/2 at 2