## Physics 9th: Motion | Derivation of Equations of Motion by Graphical Method

### Derivation of Equations of Motion by Graphical Method

### TO DERIVE v = u + at BY GRAPHICAL METHOD

This is a graph of uniform acceleration with ‘u’ as initial velocity and ‘v’ as final velocity.

Initial velocity = u = OP

Final velocity = v = RN

= RQ + QN

v = u + QN …..(i)

Acceleration, a = slope of line PN

…..(ii)

Putting the value of QN from equation (ii) into equation (i), we get v = u + at

### TO DERIVE BY GRAPHICAL METHOD

In the above speed-time graph, the distance travelled is given by

Distance travelled = Area of figure OPNR

= Area of DPNQ + Area of rectangle OPQR

(1) Area of triangle PNQ =

=

= [As v = u + at and v – u = at]

Area of DPNQ =

(2) Area of rectangle OPQR = OP ´ PQ

= u ´ t

= ut

Distance travelled = Area of DPNQ + Area of rectangle OPQR

### TO DERIVE BY GRAPHICAL METHOD

In the above speed time graph distance travelled (S) = Area of trapezium OPNR

(sum of parallel sides) ´ height

(OP + RN) ´ OR

(u + v) ´ t

(v + u) t …(i)

But v = u + at

at = v – u

… (ii)

Putting this value of ‘t’ from equation (ii) into equation (i) we get that ___

(v + u)

[As (a + b) (a – b) = ]

2as =