The distance traversed, during equal intervals of time, of a body falling from rest, stand to one another in the same ratio as the odd numbers beginning with unity[ namely,` 1:3:5:7….].`Prove it.

Explanation

Let us divide the time interval of motion of an object under free fall into many equal intervals `tau` and find out the distance traversed during successive intervals of time. Since initial velocity
is zero, we have `y =- 1/2 g t ^(2)`
Using this equation we can calculate the position of the object after different time intervals `0,tau , 2tau 3 tau....`Which are given in second column of table. If we take `(-1//2) g tau ^(2) as y _(0) `the position coordinate after first time interval `tau,` then third column gives the positions in the unit of `y_(0).` The fourth column gives the distances traversed in successive `taus.` We find that the distances are in the simple ratio `1:3:5:7:9:11...` as show in the last column. This law was established by Galileo Galei (1564-1642) who was the first to make quantitative studies of free fall.