Question

An object at a distance of $20m$ from a convex lens of focal length $0.3m$. The lens forms an image of the object. If the object moves away from the lens at a speed of $5\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right.$, the speed and direction of the image will be

• A. $3.22\times {10}^{-3}\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right.$ towards the lens
• B. $2.26\times {10}^{-3}\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right.$ away from the lens
• C. $1.16\times {10}^{-3}\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right.$ towards the lens
• D. $0.92\times {10}^{-3}\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right.$ away from the lens

Answer 1

The correct option is C

$1.16\times {10}^{-3}\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right.$ towards the lens

Step 1: given data.

Object distance, $u=-20m$

Focal length, $f=0.3m$

The velocity of the object, $V_o=5\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right.$

Step 2: Find the velocity and direction of the image.

Formula used:

$\frac{1}{f}=\frac{1}{v}-\frac{1}{u}$ ….$\left(i\right)$

Where $f$ is focal length,$v$ is image distance, $u$ is object distance.

$m=\frac{v}{u}$ …..$\left(ii\right)$

Where $m$ is the magnification of the lens.

Now,

$\frac{1}{f}=\frac{1}{v}-\frac{1}{u}$ $\left[\because Fromequation\left(i\right)\right]$

$\Rightarrow$$\frac{1}{0.3}=\frac{1}{v}-\frac{1}{\left(-20\right)}$

$\Rightarrow$$\frac{1}{v}=\frac{1}{0.3}+\frac{1}{20}$

$\Rightarrow$$\frac{1}{v}=\frac{197}{60}$

$\Rightarrow$$v=\frac{60}{197}$

Also,

Also, we know that the velocity of an image with respect to the lens.

$V_i=m^2{V}_o$

Where, $V_i$ is the velocity of the image, $V_o$is the velocity of an object, $m$ is magnification.

$\Rightarrow$$V_i={\left(\frac{v}{u}\right)}^2{V}_o$ $\left[\because Fromequation\left(ii\right)\right]$

$\Rightarrow$$V_i={\left(\frac{60}{197\times 20}\right)}^2\times 5$

$\Rightarrow$ $V_i=1.6\times {10}^{-3}\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right.$

Since, the final answer is positive, so the direction is toward the lens

Hence, option C is correct.