Question

An object is kept on the principal axis of a convex mirror of focal length $10\text{cm}$ at a distance of $10\text{cm}$ from the pole. The object starts moving at a velocity $20\text{mm/sec}$ towards the mirror at angle ${30}^{\circ }$ with the principal axis. What will be the speed of its image and direction with the principal axis at that instant.

• A. speed $=\frac{5\sqrt{7}}{4}\text{mm/sec}$
• B. speed $=\frac{5\sqrt{7}}{2}\text{mm/sec}$
• C. ${\tan }^{-1}\left(\frac{2}{\sqrt{3}}\right)$ with the principal axis
• D. ${\tan }^{-1}\left(\frac{1}{\sqrt{3}}\right)$ with the principal axis

Answer 1

Answer: (B), (C)

${\tan }^{-1}\left(\frac{2}{\sqrt{3}}\right)$ with the principal axis


Let $a$ be the position of object and $b$ be the position of image.

We have , $b=\frac{af}{a-f}=\frac{(-10)\left(10\right)}{-10-10}=+5\text{cm}$

$\therefore v_{ix}=-\left(\frac{b^2}{a^2}\right)v_{ox}$

$v_{ix}=-{\left(\frac{5}{-10}\right)}^2\times 20\times \frac{\sqrt{3}}{2}=-\frac{5\sqrt{3}}{2}\text{mm/sec}$

and

$v_{iy}=-\left(\frac{b}{a}\right)v_{oy}=-\left(\frac{5}{-10}\right)\times 20\times \frac{1}{2}=5\text{mm/s}$

Thus, $\tan \theta =\left|\frac{v_{iy}}{u_{ix}}\right|=\frac{5}{5\sqrt{3}/2}=\frac{2}{\sqrt{3}}$

and

$v_i=\sqrt{{\left(\frac{5\sqrt{3}}{2}\right)}^2+(5{)}^2}=\frac{5\sqrt{7}}{2}\text{mm/s}$