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 determine the elapsed time between the two observations, we need to understand the relationship between the angle of incidence and the angle of reflection.
1. Let's start by drawing a diagram to visualize the situation.
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------------------|----\----------- (Floor)
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| | Mirror
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Here, the wall is represented by the vertical line, and the mirror is shown as a diagonal line. The rays of sunlight incident on the mirror are represented by the lines coming from the left side (east) and are reflected off the mirror onto the floor.
2. Let's denote the angle of incidence as θ and the angle of reflection as φ.
We can see that the angle of incidence and the angle of reflection are equal, as per the law of reflection.
Therefore, θ = φ.
3. Now, let's consider the morning observation, where the reflected ray strikes the floor at a distance of 3.13 m from the base of the wall.
We have a right-angled triangle formed by the mirror, the wall, and the incident ray of sunlight. The distance from the base of the wall to the point of incidence is 1.81 m, and the distance from the point of incidence to the reflection point is 3.13 m. Therefore, the total distance from the base of the wall to the reflection point is 1.81 + 3.13 = 4.94 m.
In this triangle, we can determine the tangent of θ:
tan(θ) = opposite/adjacent = 4.94 m / 1.81 m.
Using a scientific calculator or any online tangent calculator, we find that θ ≈ 71.65 degrees.
4. Now, let's consider the second observation, where the reflected ray strikes the floor at a distance of 1.19 m from the wall.
Again, we have a right-angled triangle formed by the mirror, the wall, and the incident ray of sunlight. The distance from the base of the wall to the point of incidence remains the same (1.81 m).
However, the distance from the point of incidence to the reflection point has changed to 1.19 m. Therefore, the total distance from the base of the wall to the reflection point is 1.81 + 1.19 = 3 m.
In this triangle, we can determine the tangent of θ:
tan(θ) = opposite/adjacent = 3 m / 1.81 m.
Using a scientific calculator or any online tangent calculator, we find that θ ≈ 58.05 degrees.
5. Now, we need to find the difference between the two angles of incidence (θ).
Δθ = θ2 - θ1 = 58.05 degrees - 71.65 degrees.
= -13.6 degrees.
Note: We subtracted θ1 from θ2 because the rotation of the Earth is from east to west, so the angles decrease over time.
6. We know that the Earth rotates at a rate of 15 degrees per hour.
Therefore, the elapsed time (t) can be calculated by dividing the difference in angles by the rate of rotation:
t = Δθ / Rate of rotation
≈ -13.6 degrees / 15 degrees per hour
≈ -0.907 hours.
However, we have a negative time, which doesn't make sense in this context. Therefore, we need to consider the absolute value (magnitude) of the time elapsed:
Elapsed time = |t| = |-0.907| hours
≈ 0.907 hours.
Thus, approximately 0.907 hours have elapsed between the two observations.
To solve this problem, we need to understand the concept of the angle of incidence and the angle of reflection.
Let's denote the angle between the incident ray and the reflected ray as θ. Since the mirror is facing due east, the angle of incidence is equal to the angle of reflection.
Given that the reflected ray strikes the floor at a distance of 3.13 m from the base of the wall early in the morning, and at a distance of 1.19 m later in the morning, we can use trigonometry to find the value of θ.
Early in the morning:
tan(θ) = 3.13 m / 1.81 m
θ = tan^(-1)(3.13 / 1.81)
θ ≈ 59.27 degrees
Later in the morning:
tan(θ) = 1.19 m / 1.81 m
θ = tan^(-1)(1.19 / 1.81)
θ ≈ 33.2 degrees
Now, we need to find the difference in the angles of incidence between these two observations.
Angle difference = 59.27 degrees - 33.2 degrees
Angle difference ≈ 26.07 degrees
We also know that the earth rotates at a rate of 15.0 degrees per hour.
Time = Angle difference / Rate of rotation
Time = 26.07 degrees / 15.0 degrees per hour
Time ≈ 1.74 hours
Therefore, approximately 1.74 hours or 1 hour and 44 minutes have elapsed between the two observations.