Question

A point object 'O' is kept in front of a concave mirror of focal length 20 cm at a distance of 30 cm from the pole. The object is moving on the principal axis toward the pole with a velocity of 2 m/s.

• A. The instantaneous velocity of image is 8 ms$^{-1}$
• B. The instantaneous velocity of image is 4 ms$^{-1}$
• C. The instantaneous velocity of image with respect to object is 6 ms$^{-1}$
• D. The instantaneous velocity of image with respect to object is 10 ms$^{-1}$

Answer 1

Answer: (A), (D)

A, D

Answer 2

Using the mirror formula $\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$, with $f = -20$ cm and $u = -30$ cm, we find $v = -60$ cm. Differentiating with respect to time, we get $-\frac{1}{v^2}\frac{dv}{dt} - \frac{1}{u^2}\frac{du}{dt} = 0$. Let $v_i = \frac{dv}{dt}$ and $v_o = \frac{du}{dt}$. Since the object moves towards the pole, $v_o = +2$ m/s. The magnification $m = -\frac{v}{u} = -\frac{-60}{-30} = -2$. Then $v_i = -v_o \left(\frac{v}{u}\right)^2 = -v_o m^2 = -(2 \text{ m/s})(-2)^2 = -8$ m/s. The magnitude of the image velocity is 8 m/s. The velocity of the image with respect to the object is $v_i - v_o = -8 \text{ m/s} - 2 \text{ m/s} = -10$ m/s. The magnitude is 10 m/s.