Question

A room (cubical) is made of mirrors. An insect is moving along the diagonal on the floor such that the velocity of image of insect on two adjacent wall mirrors is 10 $cms^{-1}$. The velocity of image of insect in celling mirror is

Answer 1

$5\sqrt{2}\;{\rm cm/s}$

Answer 2

Solution Overview

Let the insect’s motion along the floor (xy–plane) have coordinates $(x,y,0)$. Two adjacent wall mirrors are, say, the planes $x=0$ and $y=0$. Their plane‐mirror images (via reflection) are located at:

$$\text{Image in } x=0:\;(-x,y,0) \quad ; \quad \text{Image in } y=0:\;(x,-y,0)$$

For a fixed (stationary) mirror, the component of the image’s velocity normal to the mirror is twice that of the object. The distance between the insect and its image in the $x=0$ mirror is $2x$; so

$$\frac{d(2x)}{dt}=2\frac{dx}{dt}=10 \quad\Longrightarrow\quad \frac{dx}{dt}=5 \,{\rm cm/s}.$$

Similarly, for the other wall $y=0$ we get:

$$\frac{dy}{dt}=5 \,{\rm cm/s}.$$

Since the insect is moving along the diagonal of the floor (so $x=y$), the speed of the insect is

$$v=\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}=\sqrt{5^2+5^2}=5\sqrt{2}\;{\rm cm/s}.$$

Now consider the ceiling mirror. The ceiling is the horizontal plane $z=L$ (with $L$ the side of the cubical room). An insect on the floor at $(x,y,0)$ has its image in the ceiling at

$$(x,\, y,\, 2L-0)=(x,y,2L).$$

Since the insect’s motion is confined to the floor the vertical coordinate of the image is fixed (i.e. $2L$ is constant). Thus the image’s motion is purely horizontal and replicates the insect’s horizontal motion. Therefore the speed of the image in the ceiling mirror is

$$5\sqrt{2}\;{\rm cm/s}.$$

Summary

• Explanation (minimal):
  For a plane mirror the image’s velocity normal to the mirror is twice the insect’s corresponding velocity. Given the wall images change at 10 cm/s, we deduce that along each horizontal direction the insect’s velocity component is 5 cm/s. Thus its speed is $5\sqrt{2}$ cm/s. In the ceiling mirror (a horizontal plane), the insect has no perpendicular (vertical) component, so its image moves horizontally with the same speed $5\sqrt{2}$ cm/s.