A small block of mass m and a concave mirror of radius R fitted with a stand lie on a smooth horizontal table with a separation d between them. The mirror together with its stand has a mass m. The block is pushed at t = 0 towards the mirror so that it starts moving towards the mirror at a constant speed V and collides with it. The collision is perfectly elastic. Find the velocity of the image (a) at a time `t lt d/V,` (b) at a time `t gt d/V`.

Explanation

a. At time t=t,
`uy=-(d-Vt)`
Here `dgtVt, t=-R/2`
By mirror fomula `1/v+1/u=1/f`

`rarr 1/v=1/f-1/u=-1/R+1/(d-Vt)`
`=(-2(d-Vt)+R)/(R(d-Vt)`
`v=(-R(d-Vt))/(R-2(d-Vt))`
differentiating w.r.t 't'
`(dv)/(dt)=(-RV[R-2(d-Vt)]-2V[R(d-Vt)])/([R-2d(d-Vt)]^2)`
`=-(R^2V)/([R-2(d-Vt)]^2)`
This is the required speed of mirror.
b. similar as above using `u=(d-Vt)`.