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 41

Chapter 3, Problem 41

The treasure map in FIGURE P3.41 gives the following directions to the buried treasure: 'Start at the old oak tree, walk due north for 500 paces, then due east for 100 paces. Dig.' But when you arrive, you find an angry dragon just north of the tree. To avoid the dragon, you set off along the yellow brick road at an angle 60° east of north. After walking 300 paces you see an opening through the woods. In which direction should you walk, as an angle west of north, and how far, to reach the treasure?

Verified step by step guidance

Step 1: Represent the problem using a coordinate system. Place the old oak tree at the origin (0, 0). The treasure is located at (0, 500) after walking 500 paces north, and then at (100, 500) after walking 100 paces east.

Step 2: Represent your new path along the yellow brick road. You walk 300 paces at an angle θ east of north. Break this displacement into components: the northward component is 300 * cos(θ), and the eastward component is 300 * sin(θ).

Step 3: Determine your current position after walking along the yellow brick road. Add the components of your displacement to the origin: your position is (300 * sin(θ), 300 * cos(θ)).

Step 4: Calculate the vector displacement from your current position to the treasure's location. The displacement vector is given by (100 - 300 * sin(θ), 500 - 300 * cos(θ)).

Step 5: Use the Pythagorean theorem to find the distance to the treasure: distance = sqrt((100 - 300 * sin(θ))^2 + (500 - 300 * cos(θ))^2). To find the direction, calculate the angle west of north using the arctangent function: angle = arctan((100 - 300 * sin(θ)) / (500 - 300 * cos(θ))).

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Key Concepts

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

Vector Addition

Vector addition is the process of combining two or more vectors to determine a resultant vector. In this scenario, the directions given in the treasure map can be represented as vectors, where the north and east movements are components of the overall path. Understanding how to add these vectors graphically or mathematically is essential for determining the final position relative to the starting point.

Trigonometry

Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles. In this problem, trigonometric functions can be used to resolve the angle and distance walked along the yellow brick road into its northward and eastward components. This understanding is crucial for calculating the final direction and distance needed to reach the treasure.

Coordinate System

A coordinate system provides a framework for defining the position of points in space using numerical values. In this context, establishing a coordinate system with a defined origin (the old oak tree) allows for precise calculations of movements in different directions. By converting the movements into coordinates, one can easily determine the final position and the necessary adjustments to reach the treasure.

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Related Practice

Textbook Question

A crate, seen from above, is pulled with three ropes that have the tensions shown in FIGURE P3.44. Tension is a vector directed along the rope, measured in newtons (abbreviated N). Suppose the three ropes are replaced with a single rope that has exactly the same effect on the crate. What is the tension in this rope? Write your answer in component form using unit vectors.

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Textbook Question

Tom is climbing a 3.0-m-long ladder that leans against a vertical wall, contacting the wall 2.5 m above the ground. His weight of 680 N is a vector pointing vertically downward. (Weight is measured in newtons, abbreviated N.) What are the components of Tom's weight parallel and perpendicular to the ladder?

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Textbook Question

A jet plane taking off from an aircraft carrier has acceleration a = ( 15 m/s², 22° above horizontal). What are the horizontal and vertical components of the jet's acceleration?

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Textbook Question

FIGURE P3.43 shows three ropes tied together in a knot. One of your friends pulls on a rope with 3.0 units of force and another pulls on a second rope with 5.0 units of force. How hard and in what direction must you pull on the third rope to keep the knot from moving? Give the direction as an angle below the negative x-axis.

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Textbook Question

The bacterium E. coli is a single-cell organism that lives in the gut of healthy animals, including humans. When grown in a uniform medium in the laboratory, these bacteria swim along zig-zag paths at a constant speed of 20 μm/s. FIGURE P3.42 shows the trajectory of an E. coli as it moves from point A to point E. What are the magnitude and direction of the bacterium's average velocity for the entire trip?

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Textbook Question

Your neighbor Paul has rented a truck with a loading ramp. The ramp is tilted upward at 25°, and Paul is pulling a large crate up the ramp with a rope that angles 10° above the ramp. If Paul pulls with a force of 550 N, what are the horizontal and vertical components of his force? (Force is measured in newtons, abbreviated N.)

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