Question

A particle is dropped from rest from a large height. Assume g to be constant throughout, the motion. The time taken by it to fall through successive distances of 1 m each will be

• A. All equal, being equal to $\sqrt{2/g}$ second
• B. In the ratio of the square roots of the integers 1, 2, 3,
• C. In the ratio of the difference in the square roots of the integers, i.e., $\sqrt{1}$,$(\sqrt{2}-\sqrt{1})$, $(\sqrt{3}-\sqrt{2})$,$(\sqrt{4}-\sqrt{3})$,...
• D. In the ratio of the reciprocals of the square roots of the integers, i.e.,$\frac{1}{\sqrt{1}}$,$\frac{1}{\sqrt{2}}$,$\frac{1}{\sqrt{3}}$,...

Answer 1

The correct option is C In the ratio of the difference in the square roots of the integers, i.e., $\sqrt{1}$,$(\sqrt{2}-\sqrt{1})$, $(\sqrt{3}-\sqrt{2})$,$(\sqrt{4}-\sqrt{3})$,...

For first $1m$

$1=\frac{1}{2}\times g\times t^2$

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

For first $1m$

$2=\frac{1}{2}\times g\times t^2$

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

for second $1m$

$t_2=t-t_1$

$=\sqrt{\frac{2}{g}}(\sqrt{2}-\sqrt{1})$

similarly,

$t_3=\sqrt{\frac{2}{g}}(\sqrt{3}-\sqrt{2})$

Hence option $c$ is correct.