The relation between the velocity of the object and mirror in-plane mirror:

In case of plane mirror, distance of the object from the mirror is equal to distance of image from the mirror.

i.e Distance of Image formed in the mirror is same as the distance of the object formed the surface of the mirror.

Hence, from the mirror property:

$$x_{\mathrm{im}}=-x_{\mathrm{on}},y_{\mathrm{im}}=y_{\mathrm{om}}\text{ and }{z}_{\mathrm{im}}=z_{\mathrm{om}}$$

Here $x_{im}$ means " x coordinate of image with respect to mirror.
Differentiating w.r.t time, we get,

$$v_{(\mathrm{im})x}=-v_{(\mathrm{om})x};v_{(\mathrm{im})y}=v_{(\mathrm{om})y};v_{(\mathrm{im})\mathrm{z}}=v_{(\mathrm{orn})z}$$

Here,
$v_i=$ velocity of the image with respect to the ground.
$v_0=$ velocity of the object with respect to the ground.
$v_{om}=$ velocity of the object with respect to the mirror.
$v_{im}=$ velocity of the object with respect to the mirror.
i.e ${\overrightarrow{v}}_{\text{om }}={\overrightarrow{v}}_{\mathrm{o}}-{\overrightarrow{v}}_{\mathrm{m}}$ and ${\overrightarrow{v}}_{\mathrm{im}}={\overrightarrow{v}}_{\mathrm{i}}-{\overrightarrow{v}}_{\mathrm{m}}$

For x -axis-

$$\begin{array}{rl}& v_{(im)x}=-v_{(\mathrm{om})x}\\ & \Rightarrow v_i-v_{\mathrm{m}}=-(v_{\mathrm{o}}-v_{\mathrm{m}})(\text{ for }x\text{-axis })\end{array}$$

- I.e When the object moves with speed $v$ towards (or away)

• from the plane mirror then image

  also moves toward (or away) with speed $v$. But the relative speed of image w.r.t. the object is $2v$.

  For y -axis and z -axis

  $$v_{(\mathrm{im})y}=v_{(\mathrm{om})y};v_{(\mathrm{im})\mathrm{z}}=v_{(\mathrm{om})z}$$

  | Relative velocity of image w.r.t. mirror | = | Relative velocity of object w.r.t. mirror |

  But $v_1-v_{\mathrm{m}}=(v_{\mathrm{o}}-v_{\mathrm{m}})$
  or $v_{\mathrm{i}}=v_{\mathrm{o}}$
  for $y$-and $z$-axis.

  Here, $v_i=$ velocity of the image with respect to the ground.
  $v_0=$ velocity of the object with respect to the ground.

i.e Velocity of the object is equal to the velocity of the image when the object is moving to parallel to the mirror surface.