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

The Correct Option is C

$ 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)} $ .