A small mirror is attached to a vertical wall, and it hangs a distance of 1.81 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 3.13 m from the base of the wall. Later on in the morning, the ray is observed to strike the floor at a distance of 1.19 m from the wall. The earth rotates at a rate of 15.0˚ per hour. How much time (in hours) has elapsed between the two observations?
To find the time elapsed between the two observations, we need to determine the change in the angle of incidence of the sunlight on the mirror. Here's how we can do it step-by-step:
1. Draw a diagram: Sketch a diagram to visualize the situation. Label the relevant distances and angles.
2. Determine the angle of incidence: In this case, we have a right-angled triangle formed by the mirror, the floor, and the ray of sunlight incident on the mirror. The angle of incidence can be found using the tangent function:
tan(θ) = opposite/adjacent
From the given information, we know that the distance from the base of the wall to the point where the ray strikes the floor is 3.13 m in the early morning. We also know that the distance from the floor to the mirror is 1.81 m.
So, tan(θ) = 1.81/3.13
Solving for θ, we get θ = tan^(-1)(1.81/3.13) ≈ 30.13°
Therefore, the angle of incidence in the early morning is approximately 30.13°.
3. Determine the angle of reflection: The angle of reflection is equal to the angle of incidence. So, in this case, the angle of reflection in the early morning is also approximately 30.13°.
4. Determine the change in the angle of incidence: Now, we need to find the change in the angle of incidence between the early morning and later in the morning. To do this, we can use the distance the ray strikes the floor from the wall.
The given information states that the distance is 1.19 m. Considering the right-angled triangle formed by the mirror, the floor, and the ray of light, the tangent function can be used again:
tan(θ) = opposite/adjacent
tan(θ) = 1.81/1.19
Solving for θ, we get θ = tan^(-1)(1.81/1.19) ≈ 56.97°
Therefore, the angle of incidence later in the morning is approximately 56.97°.
5. Determine the change in angle: The change in angle between the two observations is the difference between the angles of incidence.
Change in angle = 56.97° - 30.13° ≈ 26.84°
6. Determine the time elapsed: Recall that the Earth rotates at a rate of 15.0° per hour. We can now find the time elapsed between the two observations by dividing the change in angle by the rotation rate:
Time elapsed = Change in angle/Rotation rate
Time elapsed = 26.84°/15.0° per hour ≈ 1.79 hours
Therefore, approximately 1.79 hours (or 1 hour and 47 minutes) have elapsed between the two observations.
Note: The calculation of angles and time elapsed was done using approximate values.