Question

Two men $P$, and $Q$ start with velocities $v$ at the same time from the junction of two roads inclined at ${45}^{\circ }$ to each other. If they travel by different roads, the rate at which they are being separated, is

• A. $v\sqrt{2-\sqrt{2}}$
• B. $v\sqrt{2+\sqrt{2}}$
• C. $\sqrt{2}v$
• D. $v$

Answer 1

The correct option is A

$v\sqrt{2-\sqrt{2}}$

Let $R$, and $S$ be the position of men $P$, and $Q$ respectively at any time $t.$

Since velocities are same $\Rightarrow OR=OS=x$

Given $\frac{dx}{dt}=v$

Let $SR=y$

Now, in $△ORS$ by cosine rule

$y^2=x^2+x^2-2x^2\cos {45}^{\circ }$

$=2x^2-x^2\sqrt{2}$

$\Rightarrow y=x\sqrt{2-\sqrt{2}}$

$\frac{dy}{dt}=\frac{dx}{dt}\sqrt{2-\sqrt{2}}\Rightarrow v\sqrt{2-\sqrt{2}}$