# Derive the following equation of motion by the graphical method : v2 = u2 + 2as, where the symbols have their usual meanings.

Velocity–Time graph to derive the equations of motion.

We have just seen that the distance travelled s by a body in time t is given by the area of the figure

OABC which is a trapezium. In other words,

Distance travelled, s = Area of trapezium OABC

Now, OA + CB = u + v and OC = t. Putting these values in the above relation, we get:

...... (7)

We now want to eliminate t from the above equation. This can be done by obtaining the value of t from the first equation of motion. Thus, v = u + at (First equation of motion)
And, at = vu or

Now, putting this value of t in equation (7) above, we get:

or 2as = v2 – u2[because (v + u) × (v – u) = v2 – u2]

or v2 = u2 + 2as

This is the third equation of motion.

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