Question

An object is present on the principal axis of a concave mirror of focal length 15 cm. Object is at a distance of 20 cm from mirror. If velocity of object is 5cms–1 towards mirror and velocity of mirror is 5cms–1 towards object then velocity of image at the same insant is

Answer 1

The velocity of the image at that instant is 90 cm/s.

Answer 2

Solution

1. Mirror Formula:

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

   For $f = 15\,\text{cm}$ and $u = 20\,\text{cm}$:

   $$\frac{1}{15} = \frac{1}{20} + \frac{1}{v} \quad\Longrightarrow\quad \frac{1}{v} = \frac{1}{15} - \frac{1}{20} = \frac{4-3}{60} = \frac{1}{60}$$

   Thus, $v = 60\,\text{cm}$.

2. Determining the Rate of Change of $u$:

   Let the positions of the mirror and object be $x_m$ and $x_o$ respectively so that $u = x_o - x_m$.

   Given:

   • Object speed $= 5\,\text{cm/s}$ towards the mirror (i.e. $v_o = -5\,\text{cm/s}$ if we choose the positive direction away from the mirror).
   • Mirror speed $= 5\,\text{cm/s}$ towards the object (i.e. $v_m = +5\,\text{cm/s}$ because the mirror moves in the positive direction relative to our coordinate system).

   Therefore,

   $$\frac{du}{dt} = \frac{dx_o}{dt} - \frac{dx_m}{dt} = (-5) - (+5) = -10\,\text{cm/s}$$

   (The negative value indicates that the object–mirror distance is decreasing.)

3. Differentiating the Mirror Equation:

   Differentiate

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

   with respect to time (noting that $f$ is constant):

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

   Thus,

   $$\frac{dv}{dt} = \frac{v^2}{u^2} \frac{du}{dt} \quad \text{with an extra negative sign}$$

   Correcting the sign,

   $$-\frac{dv}{dt}\cdot \frac{1}{v^2} = -\frac{du}{dt}\cdot \frac{1}{u^2} \quad \Longrightarrow \quad \frac{dv}{dt} = -\frac{v^2}{u^2}\frac{du}{dt}$$

4. Substitute the Values:

   $$\frac{dv}{dt} = -\frac{60^2}{20^2} \times (-10) = -\frac{3600}{400} \times (-10) = -9 \times (-10) = 90\,\text{cm/s}$$

Explanation (Minimal):

• Use mirror formula: $\frac{1}{15} = \frac{1}{20} + \frac{1}{v}$ ⟹ $v = 60\,\text{cm}$.
• Differentiate: $\frac{dv}{dt} = -\frac{v^2}{u^2}\frac{du}{dt}$.
• Relative speed: $\frac{du}{dt} = -5 - 5 = -10\,\text{cm/s}$.
• Compute: $\frac{dv}{dt} = -\frac{60^2}{20^2}(-10)=90\,\text{cm/s}$.