# Third Equation of Motion by Graphical Method

## Third Equation of Motion by Graphical Method

Consider an object is moving with a uniform acceleration “a” along a straight line. The initial and final velocities of the object at time t = 0 and t = t are u and v respectively. During time t, let s be the total distance travelled by the object. In uniformly acceleration motion the velocity – time graph of an object is a straight line, inclined to the time axis.

OD = u, OC = v and OE = DA = t

Let, the Initial velocity of the object = u

Let, the object is moving with uniform acceleration, a

Let, the 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.

Third Equation of Motion: The Distance covered by the object moving with uniform acceleration is given by the area of trapezium ABDOE.

∴ Area of trapezium ABDOE = ½ x (Sum of Parallel Slides + Distance between Parallel Slides)

Distance (s) = ½ (DO + BE) x OE = ½ (u + v) x t … (3)

Now from equation (2): $$a=\frac{v-u}{t}$$,

∴ $$t=\frac{v-u}{a}$$ … 4

Now, substitute equation (4) in equation (3) we get:

$$s=\frac{1}{2}\left( u+v \right)\times \left( \frac{v-u}{a} \right)$$,

s = ½a (v + u) (v – u)

2as = (v + u) (v – u)

2as = v² – u²

v² = u² + 2as

This Expression gives the relation between position and velocity.