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

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, 2 tau, 3 tau`.... which are given in second column of Table 3.2. 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 distance traversed in successive `tau s`. we find that the distances are in the simple ratio `1 : 3 : 5 : 7 : 9 : 11`.... as shown in the last column. This law was established by Galileo Galilei (1564 - 1642) who was the first to make quantitative studies of free fall.