Question

Consider an observer at a point O, at a distance of $100cm$ from a convex lens of focal length $50cm$. A plane mirror is placed behind the lens at a distance of $150cm$ from the lens. The mirror now starts moving towards the right with a velocity of $10cm{s}^{-1}$. What will be the magnitude of velocity (in $m{s}^{-1}$) of his own image as seen by observer, at the moment when the mirror just starts moving?

(Answer upto two digits after the decimal point.)

Answer 1

Lens formula gives, $\frac{1}{v}-\left(\frac{1}{-100}\right)=\frac{1}{50}\Rightarrow v=100cm$
The mirror forms an image $50cm$ behind itself and the image is moving to right with a velocity of $20m{s}^{-1}$. The image formed by the mirror will act as moving object for the lens.
Again, applying the lens formula gives :
$\frac{1}{v}-\left(\frac{1}{-200}\right)=\frac{1}{50}\Rightarrow v=\frac{200}{3}cm$
For lens, $\frac{dv}{dt}={\left(\frac{v}{u}\right)}^2\frac{du}{dt}=\frac{{\left(200/3\right)}^2}{{\left(200\right)}^2}20=\frac{20}{9}cm{s}^{-1}$