# A particle moves towards a concave mirror of focal length 30 cm along its axis and with a constant speed of 4 cm/ sec. What is the speed of its image when the particle is at 90 cm from the mirror? Option 1) 2 cm/ sec Option 2) 8 cm/sec Option 3) 1 cm/sec Option 4) 4 cm/sec

Relation between object and image velocity if object is moving along principal axis -

$\underset{V_{im}}{\rightarrow}$ $=$ $\frac{-v^{2}}{u^{2}}$ $\underset{V_{om}}{\rightarrow}$

$\underset{V_{im}}{\rightarrow}$ = $\underset{V_{i}}{\rightarrow}$ - $\underset{V_{m}}{\rightarrow}$

$\underset{V_{om}}{\rightarrow}$ = $\underset{V_{o}}{\rightarrow}$ - $\underset{V_{m}}{\rightarrow}$

- wherein

$\underset{V_{o}}{\rightarrow}$ = velocity of light

$\underset{V_{i}}{\rightarrow}$ = velocity of image

$\underset{V_{m}}{\rightarrow}$ = velocity of mirror

$\underset{V_{om}}{\rightarrow}$ = velocity of object w.r.t. mirror

$\underset{V_{im}}{\rightarrow}$ = velocity of image w.r.t. mirror

$\frac{dv}{dt}$  =   $\frac{-v^{2}}{u^{2}}. \frac{du}{dt} ;$

$u= -90 f= 30 cm$

$\frac{1}{v} = \frac{1}{f} -\frac{1}{u}=\frac{-1}{30}+\frac{1}{90} =\frac{-2}{90}$

$v = -45 cm$

$\Rightarrow \frac{dv}{dt} = -\left ( \frac{-45}{-90} \right )^{2}\times4 cm/s$

$= -1 cm/sec$

Option 1)

2 cm/ sec

Incorrect

Option 2)

8 cm/sec

Incorrect

Option 3)

1 cm/sec

Correct

Option 4)

4 cm/sec

Incorrect

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