Question

Find the velocity of image $\left(\text{in}m/s\right)$, when the motion of the object and the mirror are as shown in the figure below.

• A. $6\hat{i}+10\hat{j}$
• B. $-6\hat{i}+10\hat{j}$
• C. $6\hat{i}-10\hat{j}$
• D. $-6\hat{i}-10\hat{j}$

Answer 1

The correct option is B $-6\hat{i}+10\hat{j}$

We know that, in the direction normal to the mirror i.e along $X-$ direction,

Relative velocity of image with respect to mirror = $-($Relative velocity of object with respect to mirror$)$

$\text{i.e}{\overrightarrow{v}}_{ix}-{\overrightarrow{v}}_m=-({\overrightarrow{v}}_{ox}-{\overrightarrow{v}}_m)$

$\Rightarrow {\overrightarrow{v}}_{ix}=2{\overrightarrow{v}}_m-{\overrightarrow{v}}_{ox}$

From the data given in the question,

${\overrightarrow{v}}_m=2\hat{i}\text{m/s},{\overrightarrow{v}}_{ox}=10\sqrt{2}\cos {45}^{\circ }\hat{i}\text{m/s}=10\hat{i}\text{m/s}$

${\overrightarrow{v}}_{ix}=2\times 2\hat{i}-\left(10\hat{i}\right)=-6\hat{i}\text{m/s}$

In the direction parallel to the surface of the mirror i.e along $Y-$ direction,
${\overrightarrow{v}}_{oy}={\overrightarrow{v}}_{iy}$

$\therefore {\overrightarrow{v}}_{iy}=10\sqrt{2}\sin {45}^{\circ }\hat{j}\text{m/s}$

$\therefore {\overrightarrow{v}}_{iy}=10\hat{j}\text{m/s}$

Hence, velocity of image ${\overrightarrow{v}}_i={\overrightarrow{v}}_{ix}+{\overrightarrow{v}}_{iy}$

$\Rightarrow {\overrightarrow{v}}_i=-6\hat{i}+10\hat{j}$