Problem 44 - University Physics with Modern Physics

Question

Ariver flows from south to north at 5.0 \(\mathrm{km} / \mathrm{h}\) . On this river, a boat is heading east to west perpendicular to the current at 7.0 \(\mathrm{km} / \mathrm{h}\) . As viewed by an eagle hovering at rest over the shore, how fast and in what direction is this boat traveling?

Step-by-Step Solution

Answer

The boat travels at 8.6 km/h, 35.5 degrees north of west.

Step 1: Understand the Problem

First, identify the velocities given in the problem. The river flows from south to north at a speed of 5.0 km/h. The boat travels east to west (perpendicular to the current) at 7.0 km/h. We want to find the resultant velocity from the perspective of an eagle hovering over the shore.

Step 2: Identify the Components

The problem involves two perpendicular components: the river's velocity vector pointing north (5.0 km/h) and the boat's velocity vector pointing west (7.0 km/h). These form the two sides of a right triangle.

Step 3: Calculate the Magnitude of the Resultant Velocity

Use the Pythagorean theorem to find the magnitude of the resultant velocity. Since these two vectors are perpendicular, the magnitude \( v \) can be calculated by:\[ v = \sqrt{(5.0)^2 + (7.0)^2} \]Substitute the values:\[ v = \sqrt{25 + 49} = \sqrt{74} \approx 8.6 \text{ km/h} \]

Step 4: Determine the Direction of the Resultant Velocity

To find the direction, we use trigonometry. The angle \( \theta \) north of west is calculated as:\[ \theta = \tan^{-1}\left(\frac{5.0}{7.0}\right) \]Calculate the angle:\[ \theta \approx \tan^{-1}(0.714) \approx 35.5^\circ \]

Step 5: State the Resultant Velocity

The magnitude of the boat's velocity as observed by the eagle is approximately 8.6 km/h, and it is directed 35.5 degrees N of W.

Key Concepts

Vector Addition

In the world of physics, understanding vector addition is crucial, particularly when dealing with relative velocities. Vectors are quantities that have both magnitude and direction. In our exercise, we dealt with two vectors: the river's current flowing north at 5.0 km/h and the boat's movement heading west at 7.0 km/h. To determine the resultant velocity as seen by the eagle, these two vectors must be combined using vector addition, a fundamental concept in physics.

When vectors are perpendicular, as in this problem, they form a right triangle, which simplifies calculations. Vector addition in this scenario allows us to calculate the resultant velocity, which represents the actual path and speed of the boat as perceived over the shore. If the vectors were not perpendicular, the approach would involve breaking them down into components along the same axis for easy addition.

• The river's velocity is a vector pointing north.
• The boat's velocity is a vector pointing west.

Combining these gives the resultant vector, showing how the boat moves relative to the shore. Understanding vector addition helps solve complex problems involving movement in different directions.

Pythagorean Theorem

The Pythagorean theorem plays a pivotal role in finding the magnitude of the resultant velocity when dealing with perpendicular vectors. In our exercise, both the river and the boat's velocities create a right triangle, with the river's flow and the boat's speed forming the two shorter sides. The theorem states that the square of the hypotenuse (resultant velocity) is equal to the sum of the squares of the other two sides.

In mathematical terms, the hypotenuse is calculated as follows:

• Identify the two components: the velocity of the river (5.0 km/h) and the velocity of the boat (7.0 km/h).
• Use the formula: \[ v = \sqrt{(5.0)^2 + (7.0)^2} \]
• Plug in the numbers to get: \[ v = \sqrt{25 + 49} = \sqrt{74} \approx 8.6 \text{ km/h} \]

This calculation illustrates how the Pythagorean theorem allows you to find the actual speed of the boat from the perspective of a hovering observer, seamlessly blending geometry with physics to solve real-world problems.

Trigonometry

Trigonometry is vital in determining the direction of the resultant vector, especially when vectors are at right angles. In our problem, after finding the magnitude of the resultant velocity, the next step was to determine its direction. This is done using trigonometric functions, specifically tangent, as it relates the angles to the ratios of two sides of a right triangle.

For this exercise, the angle \( \theta \) (direction of the boat's resultant velocity relative to north of west) is calculated using:

• The formula: \[ \theta = \tan^{-1}\left(\frac{5.0}{7.0}\right) \]
• Calculate the angle using inverse tangent to determine:\[ \theta \approx \tan^{-1}(0.714) \approx 35.5^\circ \]

Understanding the direction is as critical as finding the magnitude for fully comprehending the boat's motion. Trigonometry helps bridge the gap between abstract mathematical calculations and their practical implications in navigation and motion analysis.