Question

Points $P$ , $Q$ and $R$ are in a vertical line such that $PQ = QR$ . A ball at $P$ is allowed to fall freely. The ratio of times of descent through $PQ$ and $QR$ is

• A. $1 : 1$
• B. $1 : \sqrt{2}$
• C. $1 : \left(\sqrt{2} - 1\right)$
• D. $1 : \left(\sqrt{2} + 1\right)$

Answer 1

Answer: (C)

$1 : \left(\sqrt{2} - 1\right)$

Answer 2

$PQ = QR = h$ $h=\frac{1}{2}gt^{2}_{1}$ and $2h=\frac{1}{2}g\left(t_{1}+t_{2}\right)^{2}\quad\left(\because u=0\right)$ $\frac{1}{2} =\frac{t^{2}_{1}}{\left(t_{1}+t_2\right)^{2}}$ or $\frac{1}{\sqrt{2}}=\frac{t_{1}}{\left(t_{1}+t_{2}\right)}$ $t_{1}+t_{2}=\sqrt{2}t_{1}$ or $t_{1}\left(\sqrt{2}-1\right)=t_{2}$ $\therefore \frac{t_{1}}{t_{2}}=\frac{1}{\left(\sqrt{2}-1\right)}$ .