Question

Image of the object is formed as shown in the figure. If the mirror starts moving towards the object with velocity $v$, then velocity of image will be

A. $v$ towards right
B. $v$ towards left
C. $2v$ Towards right
D. $2v$ Towards left

Answer 1

Hint: When an object is placed in front of plane mirror, the distance of image formed behind the mirror is same as that of distance of object from mirror, and hence the speed at which either object or the plane mirror moves towards or away from each other, the speed of image will be same, as speed is a scalar quantity but velocity is a vector quantity so, we will find the direction of speed and its relative speed with respect to the mirror and object approaching speed.

Formula used:
Relative speed if two bodies are moving towards each other with speeds of ${v_1}(and){v_2}$ is given by ${v_{relative}} = {v_1} - {v_2}$ and if the same two bodies are moving against each other than relative speed is given by ${v_{relative}} = {v_1} + {v_2}$.

Complete step by step answer:
As given in the question, mirror starts to move with a velocity of $v$ towards the object so, let us assume that,
Velocity of mirror is ${v_{mirror}} = v$
Velocity of object is ${v_{object}} = 0$
So, relative speed between the mirror and the object is ${v_{object}} - {v_{mirror}}$ which is the velocity of approaching between object and the mirror and it’s calculated as
${v_{object}} - {v_{mirror}} \to (i)$

Now, let us suppose that the velocity of the image be ${v_{image}}$ then the relative speed between the plane mirror and the image is ${v_{mirror}} - {v_{image}}$ which is calculated as
${v_{mirror}} - {v_{image}} \to (ii)$
Now, since the speed of approaching between object and mirror will be equal to relative speed between mirror and the image formed, so equate both equations $(i)\,and\,(ii)$
${v_{object}} - {v_{mirror}} = {v_{mirror}} - {v_{image}}$
$\Rightarrow {v_{object}} + 2{v_{mirror}} = {v_{image}}$
Now, as
${v_{object}} = 0$ And ${v_{mirror}} = v$
$\Rightarrow 0 + 2v = {v_{image}}$
$\therefore {v_{image}} = + 2v$
So, image will start to move in right direction with a speed of ${v_{image}} = + 2v$

Hence, the correct option is C.

Note: Remember, that in order to form the image even when either mirror or ova=object starts to move towards or against each other, the relative speed between the object and plane mirror and the relative speed between the plane mirror and image formed behind the plane mirror. A relative speed is the difference in speeds of two bodies which are either moving towards or against each other.