Question

A body starts falling from height $h$ and travels distance $\frac{h}{2}$ during last second of motion, then total time of travel (in second) is

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

Answer 1

The correct option is B $2+\sqrt{2}$

$\because h=\frac{1}{2}g{t}^2$ Let total time of fall $=T$
$\therefore h=\frac{1}{2}g{T}^{2}and\frac{h}{2}=\frac{1}{2}g(T-1{)}^2\left[upto\right(T-1\left)s\right]$
$\therefore 2=\frac{T^2}{(T-1{)}^2}\Rightarrow \frac{T}{T-1}=\sqrt{2}$

$T=\sqrt{2}T-\sqrt{2}\Rightarrow (\sqrt{2}-1)T=\sqrt{2}$
$T=\frac{\sqrt{2}}{(\sqrt{2}-1)}=\frac{\sqrt{2}}{(\sqrt{2}-1)}\times \frac{\sqrt{2}+1}{\sqrt{2}+1}$
$=\frac{\sqrt{2}(\sqrt{2}+1)}{(2-1)}=2+\sqrt{2}$