Question

A linear object $AB$ is placed along the axis of a concave mirror. The object is moving towards the mirror with speed $U$. The speed of the image of the point $A$ is $4U$ and the speed of the image of $B$ is also $4U$ but in opposite direction. If the center of the line $AB$ is at a distance $L$ from the mirror then find out the length of the object.

• A. $\frac{3L}{2}$
• B. $\frac{5L}{2}$
• C. $L$
• D. None of these

Answer 1

Answer: (C)

$L$

Using Mirror Formula:

$\frac{1}{v}=\frac{1}{f}-\frac{1}{u}v=\frac{fu}{u-f}dv=\frac{fdu(u-f)-\left(du\right)fu}{{(u-f)}^2}\frac{dv}{du}=\frac{{-f}^2}{{(u-f)}^2}$

Let the length of object be 2a (a is positive)

$\therefore u_A=-L-a\&u_B=-L+a$

$\therefore -4=\frac{{-f}^2}{{(-L-a-f)}^2}=\frac{{-f}^2}{{(-L+a-f)}^2}If;(-L-a-f)=(-L+a-f)\Rightarrow a=0;notpossible\therefore (-L-a-f)=-(-L+a-f)\Rightarrow f=-L-4=\frac{{-f}^2}{{(-L-a-f)}^2}\Rightarrow -4=\frac{{-(-L)}^2}{{(-L-a-(-L\left)\right)}^2}\Rightarrow 4=\frac{L^2}{a^2}\Rightarrow a=L/2\Rightarrow 2a=L$