Equations of Motion by Graphical Method - Derivation of Displacement - Velocity Relation by Graphical Method

The velocity of an accelerated object changes with time. The change in velocity can be due to a change in direction, magnitude, or both.

Velocity-time graph

From the graph in the figure, the distance covered by the object in time t can also be calculated using the area of trapezium DOEB.

s = area of trapezium DOEB

s = `"1"/"2"`× sum of lengths of parallel sides × distance between the parallel sides 

Substituting values from the graph:

• Parallel sides are OD=u (initial velocity) and BE=v (final velocity).
• The distance between them is OE=t (time).
• $$\times (u+v)\times t$$

From the acceleration equation a = `"(v-u) "/"t"`

t = `"(v-u) "/"a"`

$$t$$

s = `"1"/"2"`× (u + v)×`"(v-u) "/"a"`

s = `"(v+u) ( v-u)"/"2a"`

Expand (u+v)(v−u):

s = `(v^2-u^2)/(2a)`

Multiply both sides by 2a:

2as = v²−u²

v² = u² + 2as 

This is Newton’s third equation of motion.