Ratio of Time Taken to Drop Bodies from Different Heights

Question:

Two bodies of different masses m₁ and m₂ are dropped from two different heights a and b.What is the ratio of time taken by the two bodies to drop through these distances.

Solution: Ratio of time taken

The two objects are striking the ground with velocities v₁ and v₂ respectively.

$${v_1=\sqrt{2gh_1}}$$

$${v_2=\sqrt{2gh_2}}$$

Time taken by the two objects:

$${h=\frac{1}{2}gt^2}$$

$${t=\sqrt{\frac{2h}{g}}}$$

Here t₁ and t₂ are the time taken by the two objects of masses m₁ and m₂ with heights a and b respectively.

$${t_1=\sqrt{\frac{2a}{g}}}$$

$${t_2=\sqrt{\frac{2b}{g}}}$$

Ratio of time taken by the two objects

$${\frac{t_1}{t_2}=\frac{\sqrt{\frac{2a}{g}}} {\sqrt{\frac{2b}{g}}}}$$

$${\frac{t_1}{t_2}=\sqrt{\frac{a}{b}}}$$

$${t_1 : t_2 = \sqrt{a} : \sqrt{b}}$$