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 solve this problem, we need to consider the geometry of the situation and make use of the information given.
Let's start by visualizing the scenario described. We have a vertical wall with a mirror attached to it. The mirror hangs a distance of 1.81 m above the floor. The incident ray of sunlight strikes the mirror, gets reflected, and then strikes the floor. We are given two instances when this happens: early in the morning and later in the morning.
From the information provided, we know that the incident and reflected rays lie in a plane perpendicular to both the wall and the floor. This means that the angle of incidence is equal to the angle of reflection.
In the initial observation, the reflected ray strikes the floor at a distance of 3.13 m from the base of the wall. In the later observation, the ray strikes the floor at a distance of 1.19 m from the wall. We can denote these distances as d1 and d2, respectively.
Let's consider the triangle formed by the wall, the mirror, and the floor in the initial observation. We can use trigonometry to find the angle of incidence and the angle of reflection.
Using the tangent function, we have:
tan(angle of incidence) = (1.81) / (3.13)
Solving this equation, we find that the angle of incidence is approximately 30.49°.
Since the angle of incidence is equal to the angle of reflection, we can apply the same formula to the later observation:
tan(angle of reflection) = (1.81) / (1.19)
Solving this equation, we find that the angle of reflection is approximately 58.71°.
Now, we need to determine the time elapsed between the two observations. We can use the fact that the Earth rotates at a rate of 15.0° per hour.
Between the two observations, the angle of reflection has changed by:
58.71° - 30.49° = 28.22°
To find the time elapsed, we divide this angle change by the rate of rotation:
Time elapsed = 28.22° / 15.0° per hour
Calculating this, we find that the time elapsed between the two observations is approximately 1.88 hours.
Therefore, approximately 1.88 hours (or 1 hour and 53 minutes) have elapsed between the early morning and later morning observations.