Question

A point object O is at the center of curvature of a concave mirror. The mirror starts to move at a speed $u$, in a direction perpendicular to the principal axis. Then, the initial velocity of the image is
A. $2u$, in the direction opposite to that of mirror’s velocity
B. $2u$, in the direction same as that of mirror’s velocity
C. Zero
D. $u$, in the direction same as that of mirror’s velocity

Answer 1

Hint: Use the mirror formula to find the relation between v and f. Then use the formula for the magnification of the mirror. Find the speed of the image using the magnification. Use suitable sign conventions.

Complete step by step answer:
It is given that the height at the center of curvature of a concave mirror. Therefore, $u = 2f$
As per sign conventions, focal length of a concave mirror is taken as negative. Therefore,
$f = - f$ and $u = - 2f$
From mirror formula, we have
$u = - 2f$,
where f is the focal length of the mirror, u is the object distance and v is the image distance.
Putting the values of u and f in the above equation, we get
$\dfrac{1}{v} + \dfrac{1}{{ - 2f}} = \dfrac{1}{{ - f}}\\
\Rightarrow \dfrac{1}{v} = \dfrac{1}{{2f}} - \dfrac{1}{f} \\
\Rightarrow \dfrac{1}{v} = \dfrac{{1 - 2}}{{2f}} \\
\Rightarrow v = - 2f$

Also, magnification of a mirror is defined as the ratio of image distance to that of object distance. Therefore,
\[m = - \dfrac{v}{u}\]
\[ \Rightarrow m = - \left( {\dfrac{{ - 2f}}{{ - 2f}}} \right)\\
\therefore m = - 1\]
The object will move with the speed with respect to the mirror.
Therefore, speed of the image = Magnification × u $ = - 1 \times u = - u$
The negative sign shows here that the image is moving opposite to that of the mirror. As the mirror is also moving with a speed u, therefore the total speed of the image will be $u + u = 2u$.

Hence, the correct option is B.

Note:As the mirror starts moving at a speed $u$, so the object will move with the speed with respect to the mirror. The distance on the left hand side of the principal axis is taken as positive and on the right hand side of the principal axis is taken as negative. The height of the object is always taken as positive and the height of the inverted image is taken as negative. Focal length of a converging mirror is taken negative.