Question

A ball dropped from a point A falls down vertically to C, through the midpoint B. The descending time from A to B and that from A to C are in the ratio

• A. 1 : 1
• B. 1 : 2
• C. 1 : 3
• D. 1 : $\sqrt{2}$
• E. 1 : $\sqrt{3}$

Hint:

In free fall from rest, time varies as square root of distance.

Answer 1

The Correct Option is D

Concept: For free fall from rest: \[ s = \frac{1}{2}gt^2 \Rightarrow t = \sqrt{\frac{2s}{g}} \] Thus, time is proportional to square root of displacement.

Step 1: Let total height = H. Then midpoint B is at height \(H/2\).

Step 2: Time from A to C. \[ t_2 = \sqrt{\frac{2H}{g}} \]

Step 3: Time from A to B. \[ t_1 = \sqrt{\frac{2(H/2)}{g}} = \sqrt{\frac{H}{g}} \]

Step 4: Take ratio. \[ \frac{t_1}{t_2} = \frac{\sqrt{H/g}}{\sqrt{2H/g}} = \frac{1}{\sqrt{2}} \]

Step 5: Final ratio. \[ 1 : \sqrt{2} \]