Question

A stone is allowed to fall from the top of a tower and covers half the height of the tower in the last second of its journey. The time taken by the stone to reach the foot of the tower is

• A. $(2-\sqrt{2})s$
• B. $4s$
• C. $(2+\sqrt{2})s$
• D. $(2\pm \sqrt{2})s$

Answer 1

Answer: (D)

$(2\pm \sqrt{2})s$

Let total distance traveled be $h$.
Now, Velocity at midpoint, $v=\sqrt{2gs}=\sqrt{gh}$

Velocity at lowest point, $V=v=\sqrt{2gs}=\sqrt{2gh}$

Now, $V=v+gt$ $(t=1s)$

Now, $\sqrt{2gh}=\sqrt{gh}+g$

$\Rightarrow \sqrt{h}=\frac{\sqrt{g}}{\sqrt{2}-1}$

Also, $h=\frac{1}{2}g{t}^2$

$\Rightarrow t^2-\frac{2h}{g}=0$

$t=\pm \sqrt{\frac{2h}{g}}=\pm \frac{\sqrt{2}}{\sqrt{2}-1}t=2\pm \sqrt{2}$