Question

The ratio of distances traversed in successive intervals of time when a body falls freely under gravity from certain height is

• A. 1 : 2 : 3
• B. 1 : 5 : 9
• C. 1 : 3 : 5
• D. \(\sqrt{1}\) : \(\sqrt{2}\) : \(\sqrt{3}\)
• E. 1 : 4 : 9

Hint:

Do not confuse the total distance ratio (which is \( 1 : 4 : 9 \), like the squares of time) with the successive interval ratio (which is \( 1 : 3 : 5 \), the odd numbers).

Answer 1

The Correct Option is C

Concept: Galileo's Law of Odd Numbers states that for an object starting from rest and moving with constant acceleration, the distances traveled in equal successive time intervals follow the ratio of odd integers.
• Total Distance after \( t \) intervals: \( S_t = \frac{1}{2} g(t \Delta t)^2 \).
• Distance in a specific interval: \( \Delta S_n = S_n - S_{n-1} \).

Step 1: Calculate distances for successive intervals.
Let the time interval be \( \Delta t = 1 \).
• For the 1st interval: \( S_1 = \frac{1}{2} g(1)^2 = \frac{1}{2}g \).
• For the 2nd interval: \( S_2 - S_1 = \frac{1}{2} g(2)^2 - \frac{1}{2} g(1)^2 = \frac{4}{2}g - \frac{1}{2}g = \frac{3}{2}g \).
• For the 3rd interval: \( S_3 - S_2 = \frac{1}{2} g(3)^2 - \frac{1}{2} g(2)^2 = \frac{9}{2}g - \frac{4}{2}g = \frac{5}{2}g \).

Step 2: Find the ratio.
The ratio of distances in these successive intervals is: \[ \frac{1}{2}g : \frac{3}{2}g : \frac{5}{2}g \quad \dots \] Simplifying by dividing throughout by \( \frac{1}{2}g \): \[ 1 : 3 : 5 \]