A point object $O$ and a mirror $M$ move with velocities of $3{\text{cm s}}^{-1}$ and $4{\text{cm s}}^{-1}$, respectively as shown in the figure. $O{O}^{\prime }$ is the normal to the mirror. The speed of the image will be -

1. $-7\hat{i}$
2. $7\hat{i}+2\sqrt{3}\hat{j}$
3. $-3\hat{i}+2\hat{j}$
4. $+7\hat{i}$

The correct option is A $-7\hat{i}$

Given,

${\overrightarrow{v}}_O=3\hat{i}$

${\overrightarrow{v}}_M=-2\hat{i}+2\sqrt{3}\hat{j}$

$\therefore$ The velocity of object with respect to mirror , ${\overrightarrow{v}}_{OM}={\overrightarrow{v}}_O-{\overrightarrow{v}}_M$

${\overrightarrow{v}}_{OM}=3\hat{i}-(-2\hat{i}+2\sqrt{3}\hat{i})=5\hat{i}-2\sqrt{3}\hat{j}$

And the velocity of image with respect to mirror , ${\overrightarrow{v}}_{IM}={\overrightarrow{v}}_I-{\overrightarrow{v}}_M$

$\therefore {\overrightarrow{v}}_I={\overrightarrow{v}}_{IM}+{\overrightarrow{v}}_M$

$=(-5\hat{i}-2\sqrt{3}\hat{j})+(-2\hat{i}+2\sqrt{3}\hat{j})$

${\overrightarrow{v}}_I=-7\hat{i}$

Hence, option (c) is the correct answer.