Question

A light bulb burns in front of the center of a 40cm wide plane mirror that is hung vertically on a wall. A man walks in front of the mirror along a line that is parallel to the mirror and twice as far from it as the bulb. The greatest distance he can walk and still see the image of the bulb.
A. 20cm
B. 40cm
C. 60cm
D. 120cm

Answer 1

Hint: More than physics concepts, we have used basic geometry to solve this problem. Consider the reflection at ends of the mirror and find the points on the man’s path by extending the reflected rays and then the distance between those points is the required length. Now you could simply apply the properties for similar triangles and for perpendicular bisectors to find the answer.

Complete step-by-step answer:
In the question, we are given the case of a 40cm wide plane mirror that is hung vertically on a wall. A bulb kept at a distance L is burning in front of it. Now there is a man who is walking parallel to this whole setup and he is twice as far as the bulb from the mirror. We are asked to find the greatest distance this man can walk such that he still sees the image of the bulb.
For better understanding, the situation can be depicted as,

In the figure, PQ is the mirror and the bulb is kept at the point Z. Also, the man is walking along VY. The light rays reflected from the end points on the mirror touch the man’s path at V and Y respectively. Now let us apply some simple geometry.
As PS acts as the perpendicular bisector in$\Delta PZR$,
$RS=SZ=\dfrac{RZ}{2}=\dfrac{40}{2}=20$ ………………………………. (1)
As per the given conditions,
$PW=2PS$
As $\Delta PSR$ and$\Delta PWV$ are similar triangles,
$VW=2SR$
$\Rightarrow VW=2\times 20=40cm$ ……………………………….. (2)
Exactly in the similar way, we will get,
$XY=40cm$ ……………………………. (3)
Also, WX will be same as the width of the mirror, so,
$PQ=ST=WX=40cm$ …………………………………………….. (4)
So, the man can walk from V to Y seeing the image of the bulb burning. This distance is given by,
$VY=VW+WX+XY$
$\Rightarrow VY=40+40+40$
$\therefore VY=120cm$
Therefore, the greatest distance the man can walk and still see the image of the bulb is 120cm.
So, the correct answer is “Option D”.

Note: While drawing the diagrams related to optics make sure that you follow the rule related to it. Here laws of reflection have to be followed while making the diagram. The law used here is – the incident angle is equal to the reflected angle. The rest of the solution is based on simple geometry.