Question

Let $A , B , C , D$ be points on a vertical line such that $AB =$ $B C=C D$. If a body is released from position $A$, the times of descent through $AB , BC$ and $CD$ are in the ratio.

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

Answer 1

Answer: (B)

$S=A B=\frac{1}{2} g t_1^2 $
$\Rightarrow 2 S=A C=\frac{1}{2} g\left(t_1+t_2\right)^2$
and $3 S=A D=\frac{1}{2} g\left(t_1+t_2+t_3\right)^2$
$t_1=\sqrt{\frac{2 S}{g}}$
$t_1+t_2=\sqrt{\frac{4 S}{g}}, t_2=\sqrt{\frac{4 S}{g}}-\sqrt{\frac{2 S}{g}}$
$t_1+t_2+t_3=\sqrt{\frac{6 S}{g}}$
$t_3=\sqrt{\frac{6 S}{g}}-\sqrt{\frac{4 S}{g}}$
$t _1: t _2: t _3:: 1:(\sqrt{2}-\sqrt{1}):(\sqrt{3}-\sqrt{2})$