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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.


Solution

At the time of firing MV=MV

V=mMV

Consider at time t = t,

u = -(Vt+V^{'}t)

f = -f

Then, v = ?

Using Mirror formula,

1v+1u=1f

1v+1f1u=1f+1(Vt+Vt)

=(Vt+Vt)+f(Vt+Vt)f

v=(V+V)tff(V+V)t

So, separation between image and bullet at time t,

vu=(V+V)tff(V+V)t+(V+V)t

=(V+V)t[ff(V+V)t+1]

=2(1+mM)Vt

Differentiating w.r.t 't'

d(vu)dt=2(1+mM)V

Hence speed of separation of the bullet and image =2(1+mM)V.

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