Question

A convex mirror with a radius of curvature of 3 m is used as the rear-view mirror for a vehicle. If a bus is located at 5 m from this mirror, find the position, nature, and size of the image.

Answer 1

Hint The position of the image will find out by the mirror formula. The nature and size of the image will be denoted by considering the sign and by using the magnification formula.

Complete step by step solution
Given, the bus is located from the convex mirror at the distance, $\mathrm{u}=-5\mathrm{m}$
The radius of curvature of the convex mirror, $\mathrm{r}=+3$m
Therefore, the focal length of the convex mirror, $\mathrm{f}=+{\displaystyle \frac{3}{2}}\mathrm{m}$
According to the mirror formula,
${\displaystyle \frac{1}{\mathrm{f}}}={\displaystyle \frac{1}{\mathrm{v}}}+{\displaystyle \frac{1}{\mathrm{u}}}$
On putting values of each term, we get
${\displaystyle \frac{1}{\mathrm{v}}}+\left({\displaystyle \frac{1}{-5}}\right)={\displaystyle \frac{2}{3}}$
${\displaystyle \frac{1}{\mathrm{v}}}={\displaystyle \frac{2}{3}}+{\displaystyle \frac{1}{5}}$
${\displaystyle \frac{1}{\mathrm{v}}}={\displaystyle \frac{10+3}{15}}={\displaystyle \frac{13}{15}}$
∴ $\mathrm{v}={\displaystyle \frac{15}{13}}\approx 1.1538$ m
Since the position of the image is positive. Therefore, the image will form behind the mirror, i.e. the image will be virtual in nature.
Now, according to the magnification formula of the mirror,
$\mathrm{m}={\displaystyle \frac{{\mathrm{h}}_{\mathrm{I}}}{{\mathrm{h}}_{\mathrm{o}}}}=-{\displaystyle \frac{\mathrm{v}}{\mathrm{u}}}$
On putting the values of v and u, we get
${\mathrm{h}}_{\mathrm{I}}=-\left({\displaystyle \frac{{\displaystyle \frac{15}{13}}}{-1.5}}\right){\mathrm{h}}_{\mathrm{o}}={\displaystyle \frac{10}{13}}{\mathrm{h}}_{\mathrm{o}}$
Since the magnification of the convex mirror is positive and less than 1. Therefore, the nature of the image formed by the convex mirror will be erect and diminished.
Therefore, the image formed by the convex mirror will form at the distance 1.1538 m behind the mirror and the nature of the image will be erect, diminished, and virtual.

Note While calculating the position of the image, the sign convention must be used correctly. The sign convention plays an important role to tell the nature of the image as well.