Question

A gun of mass $m_1$ fires a bullet of mass $m_2$ with a horizontal speed $v_0$. The gun is fitted with a concave mirror of focal length $f$ facing toward a receding bullet. Find the speed of separations of the bullet and the image just after the gun was fired.

Answer 1

$\frac{1}{v}+\frac{1}{u}=\frac{1}{f}$

$u=-u;f=-f$

$v=\frac{fu}{f-u}$

w.r.t to gun the image velocity is

$\frac{d_v}{d_t}=\frac{d\frac{fu}{f-u}}{d_t}=\frac{f^2}{{\left(f-u\right)}^2}\times v_0$

as $u=0$ $($just after impact$)$

$\frac{d_v}{d_t}=v_0$

sped of separation of the bullet and the image just after the gun was fired$=$velocity

of image+velocity of bullet

speed of mirror$=\frac{m_2{v}_0}{m_1}$

w.r.t to ground$,$

speed of bullet$=$speed of bullet wrt mirror $+$ sped of mirror

$=v_0+\frac{m_2{v}_0}{m_1}$

speed of image$=$speed of image we.t mirror$+$speed of mirror

$=v_0+\frac{m_2{v}_0}{m_1}$

speed of separation of the bullet and the image just after the gun was fired$==2\left(v_0+\frac{m_2{v}_0}{m_1}\right).$