Question

A point object is placed on the principal axis at 60 cm in front of a concave mirror of length 40 cm on the principal axis. If the object is moved with a velocity of 10 cm/s along the principal axis, then the velocity (only magnitude ) of image is

• A. 20 cm/s
• B. 30 cm/s
• C. 40 cm/s
• D. 60 cm/s

Answer 1

The correct option is C 40 cm/s

$\begin{array}{c}4=-60cm\\ f=-40cm\\ \frac{du}{dt}=10cm{s}^{-1}\\ \text{Now, by mirror formula,}\\ \Rightarrow \frac{1}{v}+\frac{1}{u}=\frac{1}{p}\\ \Rightarrow \frac{1}{v}-\frac{1}{60}=\frac{-1}{40}\\ \therefore \frac{1}{v}=\frac{1}{60}-\frac{1}{40}\\ \therefore \frac{1}{v}=\frac{2-3}{120}\\ \therefore v=-120cm\end{array}$

Now, again use mirror formula,

$\therefore \frac{1}{v}+\frac{1}{4}=\frac{1}{f}$

differentiate with respect to time,

$\begin{array}{c}\Rightarrow \frac{-1}{v^2}\frac{dv}{dt}-\frac{1}{u^2}\frac{du}{dt}=0\\ \therefore \frac{1}{(-120{)}^2}\frac{dv}{dt}-\frac{1}{(1-6{)}^2}\times 10=0\end{array}$

$\begin{array}{cc}\Rightarrow & \frac{dv}{dt}=\frac{-10\times {120}^{}\times {120}^{}}{60\times 60}\\ \therefore & \frac{dv}{dt}=-40{cms}^{-1}\end{array}$

$\begin{array}{c}\text{Magnitude}=40{cms}^{-1}\\ \text{Hence, (c) option is correct.}\end{array}$