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)

$PQ=QR=h$
$h=\frac{1}{2}g{t}_1^2$ and $2h=\frac{1}{2}g{\left(t_1+t_2\right)}^2\left(\because u=0\right)$
$\frac{1}{2}=\frac{t_1^2}{{\left(t_1+t_2\right)}^2}$ or $\frac{1}{\sqrt{2}}=\frac{t_1}{\left(t_1+t_2\right)}$
$t_1+t_2=\sqrt{2}{t}_1$ or $t_1\left(\sqrt{2}-1\right)=t_2$
$\therefore \frac{t_1}{t_2}=\frac{1}{\left(\sqrt{2}-1\right)}$ .