Question

Using v-t graph derive the equation that relates position and velocity.​

Answer 1

Let the initial velocity of the object = u

Let the object is moving with uniform acceleration, a.

Let object reaches at point B after time, t and its final velocity becomes, v

Draw a line parallel to x-axis DA from point, D from where object starts moving.

Draw another line BA from point B parallel to y-axis which meets at E at y-axis.

The distance covered by the object moving with uniform acceleration is given by the area of trapezium ABDO

Therefore,

Area of trapezium ABDOE =  1/2× (sum of parallel sides + distance between parallel sides)

Distance(S)=1/2(DO+BE)×OE

S =1/2(u+v)×t...............(i)

​we know that,

a=v-u/t

from above equation we can say,  

t=v-u/t............(ii)

After substituting the value of t from equation(ii) in equation (i)

S= 1/2a (u+v)(v−u)

​2aS=(u+v)(v−u)

2aS=v ² −u²

Hence Proved.