Physics Explanation:

Galileo's law of odd numbers states that the distances traversed during equal time intervals by a body falling from rest stand to one another in the ratio of odd numbers beginning with unity (1:3:5:7...).

The position of a freely falling object is given by:

y = ½gt²

If we take y₀ = ½gτ² as the position after the first time interval τ, then:

• After 1τ: position = 1y₀
• After 2τ: position = 4y₀ (distance from previous interval = 3y₀)
• After 3τ: position = 9y₀ (distance from previous interval = 5y₀)
• After 4τ: position = 16y₀ (distance from previous interval = 7y₀)
• And so on...

This shows the distances in successive intervals follow the ratio 1:3:5:7... demonstrating Galileo's law of odd numbers.

Key observations:

• The distance fallen increases with the square of time
• Each time interval's distance is proportional to consecutive odd numbers
• The total distance fallen after n intervals is n²y₀
• The incremental distance during the nth interval is (2n-1)y₀