Question

A reflecting, spherical Christmas tree ornament has a diameter of 9.0 cm. A child looks at the ornament from a distance of 24 cm. Describe the image she sees.
Where is the image located?

Answer 1

A reflecting, spherical Christmas tree ornament has a diameter of $9.0\mathrm{cm}$ . A child looks at the ornament from a distance of $24\mathrm{cm}$ . Describe the image she sees. Where is the image located?

Solution.

Spherical Christmas tree ornament is the convex mirror.

The symbols in the diagram:

$d_{o}$ - a distance from the object to the mirror;

$d_{i}$ - a distance from the image to the mirror;

$f$ - a focal length of the mirror;

$AB$ - object;

$A_{1}B_{1}$ - image.

The mirror equation:

$$\frac {1}{d _ {o}} + \frac {1}{d _ {i}} = \frac {1}{f}.$$

$f$ is negative, because the mirror is convex.

$d_{i}$ is negative, because the image is formed behind the mirror.

$$\frac {1}{d _ {o}} - \frac {1}{d _ {i}} = - \frac {1}{f}.$$

A focal length of the mirror:

$$f = \frac{r}{2};$$

$r$ - the radius of the spherical mirror.

$$r = \frac{d}{2};$$ $$f = \frac{d}{4}.$$ $$\frac{1}{d_o} - \frac{1}{d_i} = -\frac{4}{d}.$$ $$\frac{1}{d_i} = \frac{1}{d_o} + \frac{4}{d};$$ $$\frac{1}{d_i} = \frac{d + 4 d_o}{d \cdot d_o};$$ $$d_i = \frac{d \cdot d_o}{d + 4 d_o}.$$ $$d_i = \frac{0.09 \cdot 0.24}{0.09 + 4 \cdot 0.24} = 0.02(m) = 2(cm).$$

Answer: A child sees the virtual upright diminished image of himself. The image is located behind the mirror. The distance from the image to the mirror is: $d_i = 2\mathrm{cm}$.