Question

Suppose a buoy supports a light ten feet above the surface of still water. At what point on the water's surface will an observer see the reflection of the light if the observer is fifteen feet from the buoy and five feet above the water's surface?

Short answer

The observer sees the reflection 30 feet from the buoy along the water's surface.

Step-by-step solution

Understand the Problem

We need to determine the point on the water's surface where the reflection of a light is seen by an observer. The observer is 15 feet away from the buoy and 5 feet above the water surface. The light is 10 feet above the water.

Use the Law of Reflection

The angle of incidence equals the angle of reflection. The light's reflection point will be directly beneath it on the water's surface due to the perpendicular nature of reflection from a flat surface. Thus, the reflection point is horizontally under the light and vertically at the water's surface.

Calculate Reflection Point's Location

The light is 10 feet above the water, so its reflection is directly below it at a point (0, 0) on the coordinate plane, where the vertical distance is ignored because it becomes a point on the surface.

Determine the Observer's Position

The observer is 15 feet away horizontally from the buoy/light and 5 feet above the water surface. Place the observer at point (15, 5) in a coordinate system where the buoy is at (0, 0) for reference.

Analyze Observer's Line of Sight

Now, the observer's line of sight to the reflection forms a right triangle with the horizontal (base) as the line from the observer to the reflection on the surface, and the vertical rise as their height above the water. Using similar triangles, we find the surface reflection point on the water.

Calculate Distance to Reflection

Using the similarity of triangles, the horizontal distance to the reflection equals the total distance from observer and the proportional height. Therefore, set up \[\frac{\text{height of light}}{\text{height of observer}} = \frac{\text{horizontal distance of light}}{\text{horizontal distance of observer}}\] and solve for the distance. Solve\[\frac{10}{5} = \frac{d}{15}\] which gives \[d = 2 \times 15 = 30\] feet.

Conclude the Exact Location

Since the total distance calculated is from the observer to the reflection point, and the observer is initially 15 feet horizontally from the buoy, the reflection surface point is 15 feet further along, making it 30 feet horizontally from the buoy directly under the light source.

Key concepts

Law of Reflection

The Law of Reflection is a fundamental principle in physics that governs how light behaves when it encounters a reflective surface, such as water. According to this law, the angle of incidence (the angle at which the incoming light ray strikes the surface) is equal to the angle of reflection (the angle at which the light ray bounces off the surface). Both of these angles are measured relative to an imagined perpendicular line called the normal, which is drawn at the point of contact on the surface.
This principle applies regardless of the properties of the light or the flatness of the reflective surface, assuming the surface is smooth. The law's simplicity makes it a powerful tool for solving reflection problems, such as finding where an observed light will reflect on a surface.

• This explains why when you shine a flashlight on a mirror or calm water, the light reflects in predictable directions.
• Understanding this helps us predict and calculate where reflections will occur in many real-world situations.

Coordinate Geometry

In problems involving precise locations and movements, such as reflections or the positions of objects in space, coordinate geometry is an invaluable tool. Coordinate geometry allows us to assign positions to objects using coordinates, typically in a two-dimensional plane. For instance, in this problem, the buoy and the observer are placed in a coordinate system with the buoy at the origin (0,0) and the observer at (15, 5).
Assigning these coordinates simplifies the interaction between objects, as it allows us to apply mathematical techniques to calculate distances and angles, find midpoints, and perform transformations. This systematic approach aids in visualizing the problem in a more structured way.

• Coordinates help define and communicate the specific location of objects relative to each other.
• It simplifies calculations needed to resolve the problem, such as determining distances or angles using algebraic formulas.

Trigonometry

Trigonometry, the area of mathematics dealing with the relationships between the sides and angles of triangles, plays a critical role in analyzing and solving reflection problems like this one. In this scenario, we use similar triangles to understand how far the reflection is from the buoy and the observer.
By setting up a proportion (or using relationships between sides of triangles), we determine the exact location of the reflected light. In our problem, the similarity of triangles means that ratios of corresponding sides are equal, which helps calculate distances accurately.

• This allows us to establish the relationship between the heights and distances of the light and the observer.
• Trigonometry helps translate real-world problems into solvable mathematical equations.

Problem Solving

Solving physics problems effectively requires a structured approach that breaks down the problem into manageable parts. Begin by thoroughly understanding the problem and identifying all given data and what needs to be found. In our reflection exercise:

• First, identify the positions and heights, such as the light above the water and the observer's location.
• Next, apply the Law of Reflection to determine how light reflects on the surface.
• Then use coordinate geometry to place each element in a way that makes solving the problem simpler and more logical.
• Employ trigonometry to calculate necessary distances using established geometric relationships, such as similar triangles.

By systematically following these steps, problems that initially seem complex become clear and approachable. This structured problem-solving method is valuable not only in physics but in any situation requiring logical and mathematical solutions.