Ch 03: Motion in Two or Three Dimensions

Young & Freedman Calc - University Physics 15th Edition

Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook

Young & Freedman Calc 15th Edition

Ch 03: Motion in Two or Three Dimensions

Problem 37

Chapter 3, Problem 37

A canoe has a velocity of 0.40 m/s southeast relative to the earth. The canoe is on a river that is flowing 0.50 m/s east relative to the earth. Find the velocity (magnitude and direction) of the canoe relative to the river.

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Identify the given velocities: The velocity of the canoe relative to the earth is 0.40 m/s southeast, and the velocity of the river relative to the earth is 0.50 m/s east.

Break down the southeast velocity into its components. Since southeast is at a 45-degree angle between south and east, use trigonometry to find the east and south components. The east component is $0.40 \times \cos(45^\circ)$ and the south component is $0.40 \times \sin(45^\circ)$.

Express the river's velocity as a vector. Since it is flowing east, its vector is $0.50 \text{ m/s} $ in the east direction and $0 \text{ m/s} $ in the south direction.

Subtract the river's velocity vector from the canoe's velocity vector to find the canoe's velocity relative to the river. This involves subtracting the east component of the river's velocity from the east component of the canoe's velocity, and the south component remains unchanged.

Calculate the magnitude and direction of the resulting velocity vector. Use the Pythagorean theorem to find the magnitude: $ \sqrt{(\text{east component})^2 + (\text{south component})^2} $. Determine the direction using the arctangent function: $ \tan^{-1}(\frac{\text{south component}}{\text{east component}}) $.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Relative Velocity

Relative velocity is the velocity of an object as observed from a particular frame of reference. It is calculated by vectorially subtracting the velocity of the reference frame from the velocity of the object. In this problem, the canoe's velocity relative to the river is found by subtracting the river's velocity from the canoe's velocity relative to the earth.

Vector Addition and Subtraction

Vector addition and subtraction involve combining or resolving vectors based on their magnitude and direction. This is crucial for determining the resultant velocity of the canoe relative to the river. The southeast velocity of the canoe and the eastward velocity of the river must be treated as vectors, using trigonometric methods to find the resultant vector.

Trigonometry in Physics

Trigonometry is used in physics to resolve vectors into components and to find angles and magnitudes. In this scenario, trigonometric functions help determine the direction and magnitude of the canoe's velocity relative to the river. By breaking down the southeast vector into its east and south components, we can accurately calculate the resultant vector using trigonometric identities.

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