Question

An object is moving with velocity $\frac{1}{2}(\hat{i}+\sqrt{3}\hat{j})\text{m/s}$ such that its image in the plane mirror is moving with velocity $\frac{1}{2}(\hat{i}-\sqrt{3}\hat{j})\text{m/s}$ where $\hat{i}$ and $\hat{j}$ are unit vectors along $+x$ and $+y$ - axis respectively. For the situation described above, identify the incorrect statement.

• A. Mirror is moving along $y$ - axis.
• B. Mirror might be stationary along $x$ - axis.
• C. Normal of mirror is along $y$ - axis
• D. Object is moving in $xy$ - plane.

Answer 1

The correct option is A Mirror is moving along $y$ - axis.

Velocity of object,
${\overrightarrow{v}}_o=\frac{1}{2}(\hat{i}+\sqrt{3}\hat{j})\text{m/s}$
As object have velocities along $x$ and $y$ - axis, thus, it is moving in $xy$ - plane.
Velocity of image,
${\overrightarrow{v}}_i=\frac{1}{2}(\hat{i}-\sqrt{3}\hat{j})\text{m/s}$
It is clear that the velocity of the image is the same as that of object velocity in $x$ - direction.
$\Rightarrow (v_o{)}_x=(v_i{)}_x=\frac{1}{2}\text{m/s}$
Velocity of image is negative of object velocity in $y-$ direction.
$\Rightarrow (v_o{)}_y=-(v_i{)}_y$
$\Rightarrow$ $y$ direction is the direction of the normal to the plane mirror.
$\therefore$ The mirror is lying along $x$ - axis.
Also, the components of image velocity is not changing in magnitude, thus, the mirror is stationary.
Hence, option $\left(a\right)$ is the correct answer.