Concave Mirror and Object Motion Speed Calculation

To find the magnitude of the mirror's velocity $V_m$, we set up a one-dimensional coordinate system along the principal axis of the mirror, taking the direction of incident light (to the right) as positive.

1. Find the position of the image relative to the mirror:

Given:

• Focal length of the concave mirror, $f = -10\text{ cm}$
• Distance of object relative to mirror, $u = -30\text{ cm}$

Using the mirror formula: $\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$

Substitute the values: $\frac{1}{v} + \frac{1}{-30} = \frac{1}{-10}$

$\frac{1}{v} = -\frac{1}{10} + \frac{1}{30} = -\frac{2}{30} = -\frac{1}{15}$

$v = -15\text{ cm}$

Thus, the real image is formed $15\text{ cm}$ in front of the mirror pole.

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2. Relate the velocities of the object, mirror, and image:

Differentiating the mirror formula with respect to time $t$: $\frac{d}{dt}\left(\frac{1}{v} + \frac{1}{u}\right) = \frac{d}{dt}\left(\frac{1}{f}\right)$

$-\frac{1}{v^2}\frac{dv}{dt} - \frac{1}{u^2}\frac{du}{dt} = 0$

$\frac{dv}{dt} = -\left(\frac{v}{u}\right)^2 \frac{du}{dt}$

Here:

• $\frac{du}{dt} = v_{o/m} = v_o - v_m$ (velocity of object relative to mirror)
• $\frac{dv}{dt} = v_{i/m} = v_i - v_m$ (velocity of image relative to mirror)

Therefore, the velocity relation in the laboratory frame is: $v_i - v_m = -\left(\frac{v}{u}\right)^2 (v_o - v_m)$

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3. Calculate the magnitude of $V_m$:

Given:

• Velocity of object, $v_o = +15\text{ cm s}^{-1}$ (towards the mirror, so in the $+x$ direction)
• Velocity of image, $v_i = 0\text{ cm s}^{-1}$ (instantaneously at rest)

Substitute these into the equation: $0 - v_m = -\left(\frac{-15}{-30}\right)^2 (15 - v_m)$

$-v_m = -\frac{1}{4}(15 - v_m)$

$-4v_m = -15 + v_m$

$-5v_m = -15 \implies v_m = 3\text{ cm s}^{-1}$

The magnitude of $V_m$ is $3\text{ cm s}^{-1}$.