Question

The captain of a boat wants to travel directly across a river that flows due east with a speed of \(1.00 \mathrm{~m} / \mathrm{s}\). He starts from the south bank of the river and heads toward the north bank. The boat has a speed of \(6.10 \mathrm{~m} / \mathrm{s}\) with respect to the water. What direction (in degrees) should the captain steer the boat? Note that \(90^{\circ}\) is east, \(180^{\circ}\) is south, \(270^{\circ}\) is west, and \(360^{\circ}\) is north.

Short answer

Answer: The captain should steer the boat at approximately \(350.4^{\circ}\).

Step-by-step solution

Identify the given information

We are given the speed of the boat with respect to the water (\(v_{bw} = 6.10 \mathrm{~m} / \mathrm{s}\)) and the eastward speed of the river (\(v_{r} = 1.00 \mathrm{~m} / \mathrm{s}\)). The goal is to find the angle the captain should steer the boat to travel directly north.

Break down the problem using vectors

We'll use vector addition to analyze the problem. The boat's velocity with respect to the water has two components: one in the eastward direction and one in the northward direction. We also have the eastward velocity of the river. We need to find the angle at which the boat's northward component cancels out the eastward component due to the river's flow.

Setup the right triangle

Let's consider a right triangle formed by the boat's velocity vector with respect to the water (\(v_{bw}\)), its eastward component (\(v_{bx}\)), and its northward component (\(v_{by}\)). The angle we want to find is the angle between \(v_{bw}\) and \(v_{by}\). Let's call this angle \(\theta\). From trigonometry, we have: \(v_{bx} = v_{bw} \sin(\theta)\) and \(v_{by} = v_{bw} \cos(\theta)\)

Use vector addition to solve for the angle

We know that the eastward component of boat's velocity, \(v_{bx}\), should cancel out the eastward velocity of the river, \(v_{r}\). That means \(v_{bx} = v_{r}\), or: \(v_{bw} \sin(\theta) = v_{r}\) Now we can solve for the angle \(\theta\): \(\theta = \arcsin(\frac{v_{r}}{v_{bw}})\)

Calculate the angle

Plug in the given values for \(v_{r}\) and \(v_{bw}\): \(\theta = \arcsin(\frac{1.00 \mathrm{~m} / \mathrm{s}}{6.10 \mathrm{~m} / \mathrm{s}})\) Calculating this value, we get: \(\theta \approx 9.6^{\circ}\)

Convert the angle to match the given notation

Since the angle is measured from the northward direction, we have to convert it to match the notation given in the problem statement, where \(90^{\circ}\) is east, \(180^{\circ}\) is south, \(270^{\circ}\) is west, and \(360^{\circ}\) is north. To do this, we simply subtract the calculated angle from \(360^{\circ}\): \(360^{\circ} - 9.6^{\circ} = 350.4^{\circ}\) The captain should steer the boat at an angle of approximately \(350.4^{\circ}\) to travel directly across the river.

Key concepts

Velocity Components

Understanding velocity components is crucial when analyzing motion in physics. Velocity, a vector quantity, has both magnitude and direction and can be broken down into simpler parts along the axes of a coordinate system. For example, a velocity vector can be decomposed into horizontal (east-west) and vertical (north-south) components.

Let's consider a riverboat scenario. A boat crosses a river and maintains a constant speed with respect to the water, but due to the river's current, the actual path of the boat is altered. By breaking down the boat's velocity into components—parallels to the river's flow and perpendicular to it—we create a toolset to predict and calculate the boat's true course.

Importance of Understanding Components

Velocity components help us manage complex movements seen in problems like the riverboat situation, allowing students to analyze each part of the motion separately. This dissection into horizontal and vertical motions simplifies calculations and helps to apply the correct trigonometric functions to find the desired angle or speed.

Riverboat Problem

The riverboat problem is a classic illustration of vector addition and relative motion in physics. It examines a boat's ability to reach its destination across a river while being affected by the current. The objective is often to determine the angle at which to steer the boat or the resultant path it takes.

With the goal to reach a point directly across from its starting position, the boat must be steered at an angle upstream to counteract the river's flow. The problem might appear straightforward at first glance, but it's a practical application of vector addition that requires students to consider both the speed of the boat and the speed of the river flow.

Real-World Applications

The principles used to solve the riverboat problem are not confined to the classroom; they have real-world implications in navigation and transportation. It teaches students to anticipate how moving objects interact with environmental forces, an essential skill for anyone pursuing a career in fields like engineering, navigation, or physics itself.

Trigonometry in Physics

Trigonometry provides powerful tools for solving physics problems involving right triangles, which are prevalent when dealing with vector components. The functions of sine, cosine, and tangent relate the angles of a triangle to the ratios of its sides, which is invaluable when we need to find an unknown side or angle based on other given measurements.

Using trigonometry in physics, especially in the riverboat problem, helps describe the relationship between the boat's velocity, the river's current, and the intended direction of travel. Calculating the correct angle at which to steer the boat involves applying the sine function, which in this case relates the angle to the ratio of the river's speed to the boat's speed.

Simplifying Complex Problems

Comprehending these relationships is crucial for students. Trigonometry not only simplifies complex scenarios but also equips learners with a methodology for dissecting various physics problems into more manageable parts. Sin, cos, and tan become foundational tools to tackle a wide range of physics challenges, affirming their practicality beyond theoretical mathematics.