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Laws of Reflection

An object is ...

Question

An object is kept on the principal axis of a convex mirror of focal length $10 cm$ at a distance of $10 cm$ from the pole. The object starts moving at a velocity $20 mm/sec$ towards the mirror at angle $30^{\circ}$ with the principal axis. What will be the speed of its image and direction with the principal axis at that instant.

speed

= \frac{5 \sqrt{7}}{4} mm/sec

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speed

= \frac{5 \sqrt{7}}{2} mm/sec

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{tan}^{- 1} (\frac{2}{\sqrt{3}})

with the principal axis

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{tan}^{- 1} (\frac{1}{\sqrt{3}})

with the principal axis

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Solution

The correct option is C ${tan}^{- 1} (\frac{2}{\sqrt{3}})$ with the principal axis

Let $a$ be the position of object and $b$ be the position of image.

We have , $b = \frac{a f}{a - f} = \frac{(- 10) (10)}{- 10 - 10} = + 5 cm$

$∴ v_{i x} = - (\frac{b^{2}}{a^{2}}) v_{o x}$

$v_{i x} = - {(\frac{5}{- 10})}^{2} \times 20 \times \frac{\sqrt{3}}{2} = - \frac{5 \sqrt{3}}{2} mm/sec$

and

$v_{i y} = - (\frac{b}{a}) v_{o y} = - (\frac{5}{- 10}) \times 20 \times \frac{1}{2} = 5 mm/s$

Thus, $tan θ = ∣ ∣ ∣ \frac{v_{i y}}{u_{i x}} ∣ ∣ ∣ = \frac{5}{5 \sqrt{3} / 2} = \frac{2}{\sqrt{3}}$

and

$v_{i} = \sqrt{{(\frac{5 \sqrt{3}}{2})}^{2} + (5)^{2}} = \frac{5 \sqrt{7}}{2} mm/s$

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