Derive the third equation of motion v^{2} - u^{2} = 2as using velocity-time graph.

Consider the velocity-time graph for a body having some non zero initial velocity at time t = 0.

u: velocity at time t_{1}

v: velocity at time t_{2}

a: uniform acceleration of the body along the straight line

Displacement covered during the time interval t_{2} - t_{1} = Area ABt_{1} t_{2}

S = Area of triangle ABA’ + Area of rectangle AA’t_{2} t_{1} = Area of trapezium

S = ½ x Sum of parallel sides x Perpendicular distance

… (Equation 1)

From the first equation of motion, v = u + at;

Substituting in equation 1, we get

or, v^{2} – u^{2} = 2as

which is the required equation of motion.