Question

A $2.0\mathrm{cm}$tall object is placed in front of a mirror. A $1.0\mathrm{cm}$tall upright image is formed behind the mirror, $150\mathrm{cm}$from the object. What is the focal length of the mirror?

Short answer

The focal length of the mirror is$1\mathrm{m}$.

Step-by-step solution

Given information

We have given,

Height of the object = $2\mathrm{cm}$

Height of the image = $1\mathrm{cm}$

distance between he object and image = $150\mathrm{cm}$

We have to find the focal length of the mirror.

Simplify

Magnification of the mirror

$$\begin{gathered} \mathrm{m}=\frac{\mathrm{Height}\mathrm{of}\mathrm{the}\mathrm{image}}{\mathrm{Height}\mathrm{of}\mathrm{the}\mathrm{object}}=\frac{\mathrm{h}\text{'}}{\mathrm{h}}=\frac{1}{2} \\ \mathrm{m}=-\frac{\mathrm{image}\mathrm{distance}}{\mathrm{object}\mathrm{distance}}=-\frac{\mathrm{v}}{\mathrm{u}} \\ \mathrm{u}=-2\mathrm{v} \end{gathered}$$

we know that

$\mathrm{v}-\mathrm{u}=150\mathrm{cm}$

then,

$$\begin{gathered} \mathrm{v}+2\mathrm{v}=150 \\ \mathrm{v}=50\mathrm{cm} \\ \mathrm{u}=-100\mathrm{cm} \end{gathered}$$

Using mirror equation

localid="1649081898164" $$\begin{gathered} \frac{1}{\mathrm{f}}=\frac{1}{v}+\frac{1}{u} \\ \frac{1}{f}=\frac{1}{50}+\frac{1}{-100} \\ f=100\mathrm{cm}=1\mathrm{m} \end{gathered}$$