Question

A ball is thrown up vertically with a certain velocity so that, it reaches a maximum height h. Find the ratio of the times in which it is at height $\frac h3$ while going up and coming down respectively.

• A. $\frac {\sqrt 2-1}{\sqrt 2+1}$
• B. $\frac {\sqrt 3-\sqrt 2}{\sqrt 3+\sqrt 2}$
• C. $\frac{\sqrt 3-1}{\sqrt3+1}$
• D. $\frac 13$

Answer 1

Answer: (B)

$\frac {\sqrt 3-\sqrt 2}{\sqrt 3+\sqrt 2}$

Answer 2

Velocity, $v=\sqrt {2gh}$

$\frac h3=\sqrt {2gh}t−\frac 12gt^2$

$\frac 12gt^2+\frac h3-\sqrt {2gh}t=0$

$\frac {t1}{t2}=\frac {\sqrt {2gh}+\sqrt {gh−2gh/3} }{\sqrt {2gh}−\sqrt {2gh−2gh/3}}$

$=\frac {\sqrt 2+\frac {2}{\sqrt 3} }{\sqrt 2−\frac {2}{\sqrt 3}}$

$=\frac {\sqrt 3-\sqrt 2}{\sqrt 3+\sqrt 2}$

So, the correct option is (B): $\frac {\sqrt 3-\sqrt 2}{\sqrt 3+\sqrt 2}$.