Question

A gun of mass ${}^{\prime }{m}_1{}^{\prime }$ fires a bullet of mass ${}^{\prime }{m}_2{}^{\prime }$ with a horizontal speed ${}^{\prime }{v}_0{}^{\prime }$. 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

• A. $(1+\frac{m_2}{m_1})v_0$
• B. $2(1-\frac{m_2}{m_1})v_0$
• C. $2(1+\frac{2m_2}{m_1})v_0$
• D. $2(1+\frac{m_2}{m_1})v_0$

Answer 1

The correct option is D $2(1+\frac{m_2}{m_1})v_0$

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

$u=-u$ ; $f=-f$

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

wrt to gun the image velocity is

$\frac{d_v}{d_t}=\frac{d_{\frac{fu}{f-u}}}{d_t}=\frac{f^2}{(f-u{)}^2}\times v_o$

as $u=0$( just after impact)

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

Speed of separation of the bullet and the image just after the gun was fired = velocity of image + velocity of bullet

Speed of mirror = $\frac{m_2{v}_o}{m_1}$

wrt to ground ,

Speed of bullet = speed of bullet wrt mirror + speed of mirror

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

Speed of image=speed of image wrt mirror +speed of mirror

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

Speed of separation of the bullet and the image just after the gun was fired=$2(v_0+\frac{m_2{v}_o}{m_1})$

option $D$ is correct