Question 34: Ray Optics

Question

An object and a concave mirror of focal length $f=10 \mathrm{~cm}$ both move along the principal axis of the mirror with constant speeds. The object moves with speed $V_{0}=15 \mathrm{~cm} \mathrm{~s}^{-1}$ towards the mirror with respect to a laboratory frame. The distance between the object and the mirror at a given moment is denoted by $u$. When $u=30 \mathrm{~cm}$, the speed of the mirror $V_{m}$ is such that the image is instantaneously at rest with respect to the laboratory frame, and the object forms a real image. The magnitude of $V_{m}$ is _________ $\mathrm{cm} \,\mathrm{s}^{-1}$.

Accepted Answer

Magnification,
$$
\begin{aligned}
& m  =\frac{f}{u-f} \\
& =\frac{10}{30-10}=\frac{1}{2}
\end{aligned}
$$
Also $\quad m^2=\frac{v_m}{v_0-v_m}$
[where, $v_m=$ speed of mirror]
$$
\begin{aligned}
& m^2\left(v_{\mathrm{o}}-v_{\mathrm{m}}ight)  =v_m \\
& \Rightarrow \left(v_0-v_might) 	imes \frac{1}{4}  =v_{\mathrm{m}} \\
& \Rightarrow v_0  =5 v_m \\
& \Rightarrow v_m  =\frac{v_0}{5}=\frac{15}{5} \\
& \Rightarrow v_m  =3 \mathrm{~cm} / \mathrm{s}
\end{aligned}
$$

Ray Optics

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