Question

How to derive the equations for motion

Answer 1

you can derive these equations by using graph

first consider a motion going in a uniform acceleration in the graph as shown below ,draw a line perpendicular to the e in x axis and y axis as shown and mark it as e and c in y axis and x axis respectively 
we know must know that 
 AO =initial velocity 
AE=final velocity 
therfore 
BD=AE-AO
 BD=V-U--------1{when BD is a slope }
we know  that 
BD=EC*OC{velocity bd =acceleration *time}
BD =AT---------2
so like we equate 1 and 2 
v-u=at 
v=u+at ----------EQUATION 1
likewise we now the height is equal to area of AOEC
therefore 
height [s]=area of the rectangle +area of the triangle 
BD =AO*OC{area of the rectangle }
BD =ut ------1
bd=1/2*a*at {area of the triangle }
therefore 
BD =ut+1/2at^2-------EQUATION 2
likewise let we take 
area of the trapezuim
s=v-u*t/a
s=v-u*v+u/2a{from equation 2 t =v+u/a}
therefore 
v^2-u^2 =2as----EQUATION 3

Answer 2

your answer is in attached picture