An object and a concave mirror of focal length f = 10 cm both move along the principal axis of the mirror with constant speeds. The object moves with speed V 0 = 15 cm s - 1 towards the mirror with re

Question

An object and a concave mirror of focal length $f = 10 cm$ both move along the principal axis of the mirror with constant speeds. The object moves with speed $V_{0} = 15 cm s^{- 1}$ towards the mirror with respect to a laboratory frame. The distance between the object and the mirror at a given moment is denoted by $u$ . When $u = 30 cm$ , the speed of the mirror $V_{m}$ is such that the image is instantaneously at rest with respect to the laboratory frame, and the object forms a real image. The magnitude of $V_{m}$ is _______ $cm s^{- 1}$ .

TestHub · Accepted Answer

Magnification expression for concave mirror is given by,
$m = \frac{f}{f - u}$
$= \frac{(- 10)}{(- 10) - (- 30)} = \frac{- 1}{2}$
For concave mirror, we can write,
$v_{i, m} = - m^{2} v_{o . m}$
$\Rightarrow (v_{o} - v_{m}) \times m^{2} + (v_{i} - v_{m}) = 0$
As given in the question, $v_{i} = 0$ . Therefore,
$\Rightarrow (v_{o} - v_{m}) \times \frac{1}{4} = v_{m}$
$\Rightarrow v_{o} = 5 v_{m}$
$\Rightarrow v_{m} = \frac{v_{o}}{5}$
$= \frac{15}{5}$
$= 3 cm s^{- 1}$

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