Problem 36 - College Physics

Question

\(\bullet$$\bullet\) A 1.55 -m-tall fisherman stands at the edge of a lake, being watched by a suspicious trout who is 3.50 \(\mathrm{m}\) from the fisherman in the horizontal direction and 45.0 \(\mathrm{cm}\) below the surface of the water. At what angle from the vertical does the fish see the top of the fisherman's head?

Step-by-Step Solution

Answer

The angle from the vertical is approximately 29.74°.

Step 1: Understanding the problem

The problem involves a fisherman and a fish in a lake. The fish is observing the fisherman who is on land. We need to determine at what angle from the vertical the fish sees the top of the fisherman's head. We have the height of the fisherman, the horizontal distance from the fisherman to the fish, and the depth of the fish below the water surface.

Step 2: Visualize the situation

Imagine a right triangle where the top of the triangle is the head of the fisherman and the other endpoints are at the fish's location. The vertical leg of this triangle is the sum of the fisherman's height plus the depth of the fish (converted to meters). The horizontal leg of the triangle is the distance between the fisherman and the fish.

Step 3: Calculate vertical distance

Calculate the vertical distance from the top of the fisherman's head to the fish's position. This is equal to the fish's depth plus the fisherman's height:\[\text{Vertical distance} = \text{Height of fisherman} + \text{Depth of fish}\]\[= 1.55 \text{ m} + 0.45 \text{ m} = 2.00 \text{ m}\]

Step 4: Calculate the angle using trigonometry

Use the tangent function to find the angle. The tangent of the angle is the ratio of the horizontal distance to the vertical distance:\[\tan(\theta) = \frac{\text{Horizontal Distance}}{\text{Vertical Distance}}\]\[\theta = \tan^{-1}\left(\frac{3.50 \text{ m}}{2.00 \text{ m}}\right)\]Use a calculator to find \( \theta \approx 60.26^{\circ} \).

Step 5: Convert to "angle from the vertical"

Since the angle \(\theta\) is measured from the horizontal, the angle from the vertical is:\[90^{\circ} - 60.26^{\circ} = 29.74^{\circ}\]

Key Concepts

Angle Calculation

When solving physics problems involving angles, it's essential to identify what reference point the angle is being measured from, such as vertical or horizontal lines. In problems like the one with the fisherman and the trout, you need to use the geometry of a right triangle to find the angle.
First, you calculate one angle using trigonometry, such as the tangent function, which gives you an angle from the horizontal. This can be confusing if the angle asked for is from another reference, like the vertical. To find the angle from the vertical, subtract the horizontal angle from 90 degrees, because the vertical and horizontal axes are perpendicular.
This calculation turns into:

• Determine the angle with the horizontal using trigonometry.
• Subtract from 90° to get the angle with the vertical.

The process highlights how crucial it is to understand angle orientation, not just their size.

Right Triangle

Understanding the setup as a right triangle is pivotal in physics problems involving visual angles. A right triangle is a triangle with one angle at exactly 90 degrees. In our problem, the hypotenuse connects the top of the fisherman's head to the fish.
The vertical side of the triangle combines the fisherman's full height and the fish's depth. The horizontal side represents the distance between them across the lake. Establishing these relations helps in visualizing and simplifying complex scenarios into manageable calculations.
Utilizing right triangles is common in solving such problems because they model how light travels or how something is visually perceived in the simplest manner. They form the backbone of not just physics problems but numerous geometrical and engineering calculations.

Tangent Function

In trigonometry, the tangent function helps us relate angles to the lengths of the opposite side and adjacent side of a right triangle. In this particular exercise, we employ the tangent function to find out at what angle the trout sees the fisherman.
The tangent \( \tan(\theta) \) is defined as the ratio of the length of the opposite side to that of the adjacent side: \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \). Here, the 'opposite' is the horizontal distance of 3.50 meters, and the 'adjacent' is the vertical distance of 2.00 meters.
Calculating \( \theta = \tan^{-1}(\frac{3.50}{2.00}) \) we find \( \theta \approx 60.26^{\circ} \) from the horizontal. Mastering the tangent function is essential as it applies widely in physics, from simple calculations to complex wave dynamics.

Visualization of Physics Problems

To solve a physics problem effectively, one of the first steps must always be visualization. Having a clear mental or even a physical diagram helps break down a complex problem into simpler, more approachable parts. In this case, the visualization involves imagining a right triangle with specific known measures.
By visualizing where the fish is and where the fisherman is in relation to each other, you can better grasp how the angle calculation should proceed. Whether using sketches, diagrams, or real-world analogies, visualizing allows us to ground abstract numbers into a concrete image.

• It helps identify known and unknown variables.
• It clarifies the relationships between elements of the problem.
• It guides the method of solution, step by step.

Never underestimate the insight a well-made diagram can provide in tackling physics challenges.