A gun of mass M fires a bullet of mass m with a horizontal speed V. The gun is fitted with a concave mirror of focal length f facing towards the receding bullet. Find the speed of separation of the bullet and the image just after the gun was fired.

Explanation

speed of recoil of gun then according to conservation of linear momentum we can write the following :
`MV. = mVimplies V.=mV//M`
Speed of bullet with respect to mirror fitted with gun
`= V. + V=(mV)/(M) + V= V(1+ (m)/(M))`
Mirror formula can be written as :
`(1)/(v)+ (1)/(u) = (1)/(f)`
`implies (1)/(u) = (1)/(f) - (1)/(u) `
` implies v= ( uf)/(u-f)`
On differentiating the above result we get the following :
`implies ( dv)/(dt) = ((u-f) xx f(du)/(dt)- fu xx ((du)/(dt)-0))/((u-f)^(2))`
`implies (dv)/(dt)=(-f^(2) (du)/(dt))/((u-f)^(2))`
Initially u=0 hence
`(du)/(dt ) =(du)/(dt)`
Hence initial speed of image with respect to mirror is the same as speed of object with respect to mirror but in opposite direction . Hence both the speed are to be added to get relative speed . Speed of object with respect to mirror is calculated as `V(1+(m)/(M))` , hence relative speed of separation of bullet and its image will be `2V(1+(m)/(M))`