Question

during a daytime football game, you notice that a players reflective helmet forms an image of the sun 4.8 cm behind the surface of the helmet. what is the radius of curvature of the helmet, assuming it to be roughly spherical?

a. 9.6 cm
b. 7.3 cm
c. 6.1 cm
d. 3.2 cm
e. 1.4 cm

Explanation:

Step1: Identify the image distance

The image is formed 4.8 cm behind the surface of the helmet. For a spherical mirror, the image distance $d_i=- 4.8$ cm (negative because the image is behind the mirror surface). For a spherical mirror, when the object is at infinity (the Sun is very far away), the image is formed at the focal - point. So, the focal length $f$ of the spherical mirror (helmet) is equal to the image distance. That is $f = d_i=-4.8$ cm.

Step2: Use the relationship between radius of curvature and focal length

The relationship between the radius of curvature $R$ and the focal length $f$ for a spherical mirror is $R = 2f$.
Substitute $f=-4.8$ cm into the formula: $R = 2\times(- 4.8)$ cm. Since we are interested in the magnitude of the radius of curvature, $|R|=9.6$ cm.

Answer:

A. 9.6 cm