Question

Derive the equation for displacement-time relation. Graphically.

Answer 1

Derivation of the second equation of motion:

Graph of a particle whose initial velocity is u and final velocity is v. Total time taken is t.

Distance covered by the object in the given time t is given by the area of the trapezium ABDOE

Let in the given time, t the displacement covered by the moving object = s

The area of trapezium, ABDOE

Displacement (s) = Area of $△$ABD + Area of rectangle ADOE

Or,

s=12×AB×AD+AE×OEs=\frac 12 \times AB\times AD +AE\times OEs=21​×AB×AD+AE×OE

Since, $AB=DC=at$, $AD=OE=t$ and $AE=u$ therefore we get,

s=12×at×t+u×ts=\frac 12 \times at \times t + u\times ts=21​×at×t+u×t

Or,

$s=ut+\frac{1}{2}a{t}^2$

Hence the equation is derived.