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Ch 04: Kinematics in Two Dimensions
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook
All textbooksKnight Calc 5th EditionCh 04: Kinematics in Two DimensionsProblem 60b
Chapter 4, Problem 60b

Ships A and B leave port together. For the next two hours, ship A travels at 20 mph in a direction 30° west of north while ship B travels 20° east of north at 25 mph. What is the speed of ship A as seen by ship B?

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Step 1: Break down the velocity vectors of both ships into their respective components. For ship A, calculate the northward and westward components of its velocity using trigonometric functions. The northward component is given by \( v_{A, \text{north}} = v_A \cdot \cos(30°) \), and the westward component is \( v_{A, \text{west}} = v_A \cdot \sin(30°) \). Similarly, for ship B, calculate the northward and eastward components using \( v_{B, \text{north}} = v_B \cdot \cos(20°) \) and \( v_{B, \text{east}} = v_B \cdot \sin(20°) \).
Step 2: Determine the relative velocity components of ship A as seen by ship B. Subtract the velocity components of ship B from those of ship A. For the northward direction, \( v_{\text{relative, north}} = v_{A, \text{north}} - v_{B, \text{north}} \). For the east-west direction, \( v_{\text{relative, east-west}} = v_{A, \text{west}} - (-v_{B, \text{east}}) \), since ship B's eastward velocity is opposite to ship A's westward velocity.
Step 3: Combine the relative velocity components to find the magnitude of the relative velocity. Use the Pythagorean theorem: \( v_{\text{relative}} = \sqrt{(v_{\text{relative, north}})^2 + (v_{\text{relative, east-west}})^2} \).
Step 4: If needed, calculate the direction of the relative velocity vector. Use the inverse tangent function: \( \theta_{\text{relative}} = \tan^{-1}\left(\frac{v_{\text{relative, east-west}}}{v_{\text{relative, north}}}\right) \). This gives the angle of the relative velocity vector with respect to the northward direction.
Step 5: Interpret the result. The magnitude calculated in Step 3 represents the speed of ship A as seen by ship B, and the direction calculated in Step 4 (if required) provides the orientation of the relative velocity vector.

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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 one object as observed from another moving object. It is calculated by subtracting the velocity vector of the observer from the velocity vector of the object being observed. This concept is crucial for understanding how the speed and direction of one ship appear from the perspective of another ship.
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Vector Addition

Vector addition involves combining two or more vectors to determine a resultant vector. In this context, the velocities of ships A and B must be represented as vectors, taking into account their magnitudes and directions. This process allows us to find the relative velocity of ship A as seen from ship B.
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Trigonometry in Physics

Trigonometry is essential in physics for resolving vectors into their components. By using sine and cosine functions, we can break down the velocities of ships A and B into their northward and eastward components. This breakdown is necessary to accurately calculate the relative speed and direction of ship A from the perspective of ship B.
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Related Practice
Textbook Question

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Textbook Question

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Textbook Question

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Textbook Question

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Textbook Question

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Textbook Question

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