A gun of mass m1, fires a bullet of mass m2 with a horizontal speed vo. The gun is  fitted with a concave mirror of focal length f facing toward a receding bullet. Find the speed of separation of the bullet and the image just after the gun was fired

Let ν1be the speed of gun (or mirror) just after the firing of bullet. From conservation of linear momentum,m2v0=m1v1v1=m2v0m1                     . . . .(i)Now rate at which distance between mirror and bullet is increasing dudt=v1+v0               . . . . (ii)Here, dudt=v1+v0(as at the time of firing, the bullet is at pole).dvdt=dudt=v1+v0           . . . . (iii)Here, dv/dt is the rate at which distance between image (of bullet) and mirror is increasing. So, if ν2 is the absolute velocity of image (towards right), thenv2−v1=dvdt=v1+v0or    v2=2v1+v0      . . .  (iv)Therefore, speed of separation of the bullet and the image will bevr=v2+vo=2v1+vo+vovr=2v1+vo Substituting value of v1 from Eq. (i), we have vr=21+m2m1v0