A stone falls freely under gravity. It covers distances h₁, h₂ and h₃ in the first 5 seconds, the next 5 secon

Question

A stone falls freely under gravity. It covers distances h₁, h₂ and h₃ in the first 5 seconds, the next 5 seconds and the next 5 seconds respectively. The relation between h₁, h₂ and h₃ is:

Accepted Answer

This problem illustrates Galileo's law of odd numbers. For a body falling freely from rest, the distances traversed during equal intervals of time stand to one another in the same ratio as the odd numbers beginning with unity (1: 3: 5: 7...).

1.  **Identify Conditions:** The stone falls under gravity (uniform acceleration, $a=g$) starting from rest ($u=0$). The time intervals are equal ($t = 5$ s).
2.  **Apply Galileo's Law:** 
    *   Distance covered in the 1st interval ($h_1$) is proportional to 1.
    *   Distance covered in the 2nd interval ($h_2$) is proportional to 3.
    *   Distance covered in the 3rd interval ($h_3$) is proportional to 5.
    *   Therefore, the ratio is $h_1 : h_2 : h_3 = 1 : 3 : 5$ .
3.  **Formulate Relation:**
    From the ratio, we can express $h_1$ in terms of others:
    $$ h_1 = \frac{h_2}{3} $$
    $$ h_1 = \frac{h_3}{5} $$
    Combining these gives: $h_1 = \frac{h_2}{3} = \frac{h_3}{5}$

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