Question

Two bodies of masses \( m_1 \) and \( m_2 \) are dropped from two different heights \( h_1 \) and \( h_2 \) respectively. The ratio of the times taken by the two masses to touch the ground is (neglect air resistance)

• A. \( \frac{h_1}{h_2} \)
• B. \( \frac{m_1 h_1}{m_2 h_2} \)
• C. \( \frac{m_1 h_2}{m_2 h_1} \)
• D. \( \sqrt{\frac{h_1}{h_2}} \)

Hint:

The time to fall from a height under gravity depends only on the height, not on the mass of the object.

Answer 1

The Correct Option is D

The time taken by a body to fall freely under gravity from a height \( h \) is given by: \[ t = \sqrt{\frac{2h}{g}} \] Where \( g \) is the acceleration due to gravity. For two bodies dropped from heights \( h_1 \) and \( h_2 \), the times taken for the bodies to reach the ground are: \[ t_1 = \sqrt{\frac{2h_1}{g}}, \quad t_2 = \sqrt{\frac{2h_2}{g}} \] The ratio of the times is: \[ \frac{t_1}{t_2} = \frac{\sqrt{\frac{2h_1}{g}}}{\sqrt{\frac{2h_2}{g}}} = \sqrt{\frac{h_1}{h_2}} \] Thus, the ratio of the times is \( \sqrt{\frac{h_1}{h_2}} \). \[ \boxed{\sqrt{\frac{h_1}{h_2}}} \]