Spherical mirrors: A convex spherical mirror with a focal length of magnitude 25 cm has a 4.0-cm tall flower placed 100 cm in front of it. What is the height of the image of the flower?

Question

Quizplus · Accepted Answer

The height of the image can be found using the mirror equation and the magnification equation. The mirror equation is $ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} $, where $ f $ is the focal length, $ v $ is the image distance, and $ u $ is the object distance. For a convex mirror, the focal length is positive, and the object distance is negative (since it's in front of the mirror). The magnification $ m $ is given by $ m = \frac{v}{u} = \frac{h_i}{h_o} $, where $ h_i $ is the image height and $ h_o $ is the object height.Given: $ f = 25 $ cm (positive for convex mirror), $ u = -100 $ cm (negative since the object is in front of the mirror), and $ h_o = 4.0 $ cm.First, find the image distance $ v $ using the mirror equation. Rearranging the equation gives $ \frac{1}{v} = \frac{1}{f} - \frac{1}{u} $. Plugging in the values gives $ \frac{1}{v} = \frac{1}{25} - \frac{1}{-100} = \frac{4 + 1}{100} = \frac{5}{100} $, so $ v = 20 $ cm.Then, use the magnification equation to find $ h_i $: $ m = \frac{v}{u} = \frac{20}{-100} = -\frac{1}{5} $, and $ m = \frac{h_i}{h_o} $, so $ -\frac{1}{5} = \frac{h_i}{4.0} $. Solving for $ h_i $ gives $ h_i = 4.0 \times -\frac{1}{5} = -0.8 $ cm. The negative sign indicates the image is inverted relative to the object, but since the question asks for the height, we take the absolute value, which is 0.80 cm.

Spherical Mirrors: a Convex Spherical Mirror with a Focal Length