Ch 03: Vectors and Coordinate Systems

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 03: Vectors and Coordinate Systems

Problem 31

Chapter 3, Problem 31

Ruth sets out to visit her friend Ward, who lives 50 mi north and 100 mi east of her. She starts by driving east, but after 30 mi she comes to a detour that takes her 15 mi south before going east again. She then drives east for 8 mi and runs out of gas, so Ward flies there in his small plane to get her. What is Ward's displacement vector? Give your answer (a) in component form, using a coordinate system in which the y-axis points north, and (b) as a magnitude and direction.

Verified step by step guidance

Step 1: Define the coordinate system. Place Ruth's starting point at the origin (0, 0). The y-axis points north, and the x-axis points east. Ward's location is at (100, 50) in this coordinate system, since he is 100 mi east and 50 mi north of Ruth's starting point.

Step 2: Trace Ruth's path. She first drives 30 mi east, moving her to (30, 0). Then, she takes a detour 15 mi south, moving her to (30, -15). Finally, she drives 8 mi east, ending at (38, -15).

Step 3: Calculate the displacement vector from Ruth's final position to Ward's position. Subtract Ruth's final coordinates from Ward's coordinates: $(x_{Ward} - x_{Ruth}, y_{Ward} - y_{Ruth}) = (100 - 38, 50 - (-15))$.

Step 4: Simplify the displacement vector to component form. The x-component is $100 - 38$, and the y-component is $50 - (-15)$. Write the displacement vector as $(\Delta x, \Delta y)$.

Step 5: To find the magnitude and direction of the displacement vector, use the Pythagorean theorem for magnitude: $\sqrt{(\Delta x)^2 + (\Delta y)^2}$, and use the arctangent function for direction: $\theta = \tan^{-1}(\Delta y / \Delta x)$. Ensure the angle is measured relative to the positive x-axis.

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Key Concepts

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

Displacement Vector

A displacement vector represents the change in position of an object from its initial point to its final point. It is defined by both magnitude and direction, indicating how far and in which direction the object has moved. In this scenario, the displacement vector will be calculated based on the coordinates of Ruth's final position relative to her starting point.

Coordinate System

A coordinate system is a framework used to define the position of points in space. In this problem, a two-dimensional Cartesian coordinate system is employed, where the x-axis represents east-west movement and the y-axis represents north-south movement. Understanding how to translate movements into this system is crucial for accurately determining displacement.

Magnitude and Direction

Magnitude refers to the length or size of a vector, while direction indicates the orientation of that vector in space. For displacement vectors, the magnitude can be calculated using the Pythagorean theorem, and the direction can be expressed as an angle relative to a reference axis. This is essential for providing a complete description of Ward's displacement vector in the problem.

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