Question

A hammer is dropped into a mine. Its velocities at depths d, 2d and 3d are in the ratio

• A. 1:2:3
• B. 1:1:1
• C. 1:4:9
• D. 6:3:2
• E. 1:2:√3

Answer 1

Answer: (E)

The hammer is dropped into a mine, and we are asked to find the ratio of its velocities at different depths \( d \), \( 2d \), and \( 3d \). 

When an object is in free fall, its velocity at any point is given by the equation: \[ v = \sqrt{2gh} \] Where: - \( v \) is the velocity, - \( g \) is the acceleration due to gravity, - \( h \) is the height or depth from the point of release. Now, at depths \( d \), \( 2d \), and \( 3d \), the velocities are: - At depth \( d \): \( v_1 = \sqrt{2gd} \) - At depth \( 2d \): \( v_2 = \sqrt{2g(2d)} = \sqrt{4gd} \) - At depth \( 3d \): \( v_3 = \sqrt{2g(3d)} = \sqrt{6gd} \) The ratio of velocities at these depths is: \[ v_1 : v_2 : v_3 = \sqrt{2gd} : \sqrt{4gd} : \sqrt{6gd} \] Simplifying the ratio: \[ v_1 : v_2 : v_3 = 1 : 2 : \sqrt{3} \] This shows that the velocities at depths \( d \), \( 2d \), and \( 3d \) are in the ratio 1:2:√3

Correct Answer:

Correct Answer: (E) 1:2:√3