Question

A luminous point object is moving along the principal axis of a concave mirror with a focal length of $12cm$towards it. When its distance from the mirror is $20cm$ its velocity is $4cm/s.$The velocity of the image in $cm/s$ at that instant is:

• A. $6,$towards the mirror
• B. $6,$away from the mirror
• C. $9,$away from the mirror
• D. $9,$ towards the mirror

Answer 1

The correct option is C

$9,$away from the mirror

Step 1; Calculating the value of $v$

Using the mirror formula

$\frac{1}{f}=\frac{1}{v}+\frac{1}{u}$

$$\begin{gathered} f=-12cm \\ u=-20cm \end{gathered}$$

$\frac{1}{v}=\left(-\frac{1}{12}\right)-\left(-\frac{1}{20}\right)$

$v=-30cm$

Step 2: Calculating the velocity at the time $t$

Differentiating the mirror formula

$$\begin{gathered} 0=\left(-\frac{1}{u^2}\right)\frac{du}{dt}–\left(-\frac{1}{v^2}\right)\frac{dv}{dt} \\ \Rightarrow \left(\frac{dv}{dt}\right)=\left(\frac{1}{u^2}\right)\left(\frac{du}{dt}\right)\left(-v^2\right) \\ =-\left[\frac{{\left(30\right)}^2}{{\left(20\right)}^2}\right]\times 4 \\ =–9cm/s \end{gathered}$$

Thus, the required velocity is $9cm/s$away from the mirror.

Hence, option C is correct.