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Spherical/Curved mirrors

Question

A linear object AB is placed along the axis of a concave mirror. The object is moving towards the mirror with speed u. The speed of the image of the point A is 4u and the speed of the image of B is also 4u but in opposite direction. If the centre of the line AB is at a distance L from the mirror then find out the length of the object.

Moderate

A

3L/2
B

5L/3
C

L
D

None of these

Solution

For concave mirror, if x and y are object distance and image distance, respectively, we have
$- \frac{1}{x} - \frac{1}{y} = - \frac{1}{|f|} \Rightarrow \frac{1}{x} + \frac{1}{y} = - \frac{1}{|f|} \Rightarrow - \frac{1}{x^{2}} \frac{d x}{d t} - \frac{1}{y^{2}} \frac{d y}{d t} = 0 \Rightarrow |\frac{V_{x}}{V_{y}}| = \frac{x^{2}}{y^{2}}$
For, $|\frac{V_{x}}{V_{y}}| = \frac{1}{4}, \frac{x}{y} = \pm 2$
For $x / y = 2$ , we get $x = 3 |f| / 2$ [for point A] and for $x / y = - 2$ , we get $x = |f| / 2$ [for point B].
As the middle point happens to be focus of the mirror, we get $|f| = L$ .

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