Question

Show that $s\propto t^2$ for a freely falling body

Answer 1

Let s is the distance covered by the body and it is moving freely in downward direction under the influence of gravity. u and v are the initial and final velocities of the body.

Distance,

$s=\frac{averagespeed}{time}..........\left(1\right)$

$averagespeed=\frac{u+v}{2}.........\left(2\right)$

Acceleration,

$a=\frac{v-u}{t}$

$v=u+at..........\left(3\right)$

Put the value of v from equation (3) to (1)

$s=\frac{u+u+at}{2t}t$

$s=\frac{2u+at}{2t}t$

$\because u=0$

$s=\frac{at}{2}t$

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

$s\propto t^2$