A small mirror is attached to a vertical wall, and it hangs...

A small mirror is attached to a vertical wall, and it hangs a distance of $1.80 \mathrm{~m}$ above the floor. The mirror is facing due east, and a ray of sunlight strikes the mirror early in the morning and then again later in the morning. The incident and reflected rays lie in a plane that is perpendicular to both the wall and the floor. Early in the morning, the reflected ray strikes the floor at a distance of $3.86 \mathrm{~m}$ from the base of the wall. Later on in the moming, the ray is observed to strike the floor at a distance of $1.26 \mathrm{~m}$ from the wall. The earth rotates at a rate of $15.0^{\circ}$ per hour. How much time (in hours) has elapsed between the two observations?

Key Concepts

Law of Reflection

This concept states that when a light ray is incident on a plane mirror, the angle of incidence is equal to the angle of reflection. This principle is fundamental in analyzing how light rays interact with reflective surfaces, allowing the determination of the paths of reflected rays based on the incoming ray's direction.

Geometric Ray Diagram Analysis

The problem involves constructing geometric models of the ray paths by using the mirror’s position, the mirror’s height, and the points where the rays hit the floor. By drawing right triangles and relating distances and angles, one can analyze the reflection geometry to determine the angles at which the rays approach and leave the mirror.

Trigonometric Relationships in Reflection

Trigonometry is used to relate the physical distances (such as the height of the mirror and the distance along the floor where the ray strikes) to the angles of incidence and reflection. Functions like the tangent can be applied to these right triangles to connect spatial measurements with angular values, which is key to solving the problem.

Earth’s Rotation and Angular Velocity

The problem leverages the fact that the Earth rotates at a rate of 15° per hour. This angular velocity provides a link between the change in the observed angles of the sunlight reflection and the time elapsed. Understanding how angular displacement corresponds to time is essential for converting the geometric changes in ray orientation into a temporal difference.

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