Galileo’s law of odd numbers : “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: 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 distances 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, 3tau`... which are given in second column of Table 3.2. If we take `(-1//2)g t^(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 `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.