Question

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.

Answer 1

Given,
The focal length of the concave mirror is f and M is the mass of the gun. Horizontal speed of the bullet is V.

Let the recoil speed of the gun be V_g
Using the conservation of linear momentum we can write,
`MV_g = mV`
⇒ `V_g =  m/M `V
Considering the position of gun and bullet at time t = t,
For the mirror, object distance, u = − (Vt + V_gt)
Focal length, f = − f
Image distance, v = ?
 Using Mirror formula, we have:
`1/v + 1/u = 1/f`

⇒ `1/v + 1/u = 1/f `

⇒ `1/v = 1/-f - 1/u` 
⇒  `1/v = -1/f + 1/ ( Vt + V_ g t)`

⇒ `1/v = (-(Vt + V_g t) + f)/( (Vt + V_g t) f`

⇒ `v = (-(Vt - V _g t )  tf) / ( f - (V + Vg) t)`
The separation between image of the bullet and bullet at time t is given by:

`v = u -(( V + V_g )tf )/ ( f- ( V + V g) t + ( V + Vg ) t`

`= (V + Vg ) t [ f/ (f-(V + Vg )t) + 1]`

`= 2( 1 + m/M )Vt`
Differentiating the above equation with respect to 't' we get,
`d ( v -u ) = 2 ( 1 + m/M ) V`
Therefore, the speed of separation of the bullet and image just after the gun was fired is  
`2 ( 1 + m/M ) V`.