Question

A particle is dropped from rest from a height. The time taken by it to fall through successive vertical distance 1 meter each will then be :-

• A. all equal, being equal to $\frac{2}{g}$ second
• B. in the ratio of square roots of integers $1,2,3, \ldots$
• C. in the ratio of the difference in the square roots of integers $(\sqrt{1}-\sqrt{0}),(\sqrt{2}-\sqrt{1}),(\sqrt{3}-\sqrt{2}),(\sqrt{4}-\sqrt{3})$
• D. in the ratio $\frac{1}{\sqrt{1}}: \frac{1}{\sqrt{2}}: \frac{1}{\sqrt{3}}: \frac{1}{\sqrt{4}}$

Answer 1

Answer: (C)

For first $1\, m$ of fall,
$1=\frac{1}{2} g t_{1}^{2},$
$\therefore t_{1}=\sqrt{\frac{2}{g}}$
For $2\, m$ of fall,
$2=\frac{1}{2} g t^{2},$
$\therefore t=\sqrt{\frac{4}{g}},$
$\therefore t_{2}=t-t_{1}=\sqrt{\frac{4}{g}}-\sqrt{\frac{2}{g}}=(\sqrt{2}-1) \sqrt{\frac{2}{g}}$
For $3\, m$ of fall,
$3=\frac{1}{2} g t^{2},$
$\therefore t=\sqrt{3} \sqrt{\frac{2}{g}},$
$\therefore t_{3}=t-t_{2}=(\sqrt{3}-\sqrt{2}) \sqrt{\frac{2}{g}}$