Question

An observer moves towards a stationary plane mirror at a speed of $4\,m{s^{-1}}$ the speed of image with respect to the mirror is ________________.

Answer 1

The mirror and the object move towards each other. Hence there is a relative velocity between the object with respect to the mirror. Similarly, the image will also behave in similar fashion as that of the object with respect to the mirror. Further accordingly from the relative velocity of the image with respect to the mirror we can determine its velocity with respect to ground.

Complete step by step answer:
An observer moves towards a stationary plane mirror at a speed of $4\,m{s^{-1}}$. We need to find out the speed of the image with respect to the mirror.

Let the velocity of the observer with respect to mirror be, ${v_{om}} = - 4\,m{s^{-1}}$
$\Rightarrow - \dfrac{{dx}}{{dt}}$ (As the observer move towards the mirror that is towards negative x axis)
Let the velocity of the image with respect to mirror be,
${v_{im}} =4m{s^{-1}}$
$\Rightarrow x' = x$
$\Rightarrow {v_{im}} = \dfrac{{dx'}}{{dt}} = \dfrac{{dx}}{{dt}} \\\ \Rightarrow {v_{im}} = 4\,m{s^{-1}}$ $\left( {{\text{since }}x' = x} \right)$
Now,
${v_{io}} = {v_{im}} + {v_{mo}}$
$\Rightarrow {v_{io}} = {v_{im}} - {v_{om}}$
$\Rightarrow {v_{io}} = 4 - \left( { - 4} \right)$
$\therefore {v_{io}} = 8{\text{ m/s}}$.

Additional information:
Relative Velocity and Relative acceleration: Relative generally means the comparison between any two objects regarding their different characteristics. Relative motion can be defined as the comparison between the motions of a single object to the motion of another object moving with the same velocity. Relative motion can be easily found out with the help of the concept of relative velocity, relative acceleration or relative speed.

An object is in a static state when the relative motion of the object with regard to another object has the relative velocity with the earth, but the relative motion of the object remains constant. Basically, relative motion includes the comparison of relative acceleration of the rigid bodies and the relative velocities of the two objects. We can simply say that the comparisons between the motions of any two objects are called relative motion.

Note: Students please not that if the observer is moving towards mirror which is in positive x direction of axis, then the method to solve the problem is the same way and the answer will be also same the only change to do is change the sign of the velocity of observer from negative to positive and the velocity of image with respect to mirror from positive to negative. And rearrange the equation of velocity of image with respect to the observer. You will get an answer with a negative sign and while taking the magnitude of that you will get the speed.