Question

Let $A,B,C,D$ be points on a vertical line such that $AB=$ $BC=CD$. 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)

$S=AB=\frac{1}{2}g{t}_1^2$
$\Rightarrow 2S=AC=\frac{1}{2}g{\left(t_1+t_2\right)}^2$
and $3S=AD=\frac{1}{2}g{\left(t_1+t_2+t_3\right)}^2$
$t_1=\sqrt{\frac{2S}{g}}$
$t_1+t_2=\sqrt{\frac{4S}{g}},t_2=\sqrt{\frac{4S}{g}}-\sqrt{\frac{2S}{g}}$
$t_1+t_2+t_3=\sqrt{\frac{6S}{g}}$
$t_3=\sqrt{\frac{6S}{g}}-\sqrt{\frac{4S}{g}}$
$t_1:t_2:t_3::1:(\sqrt{2}-\sqrt{1}):(\sqrt{3}-\sqrt{2})$