Textbook Question
For the vectors A and B in Fig. E1.24, use a scale drawing to find the magnitude and direction of the vector difference A − B.
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Ch 01: Units, Physical Quantities & Vectors
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610
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Table of contents
- Ch 01: Units, Physical Quantities & Vectors41
- Ch 02: Motion Along a Straight Line67
- Ch 03: Motion in Two or Three Dimensions47
- Ch 04: Newton's Laws of Motion23
- Ch 05: Applying Newton's Laws59
- Ch 06: Work & Kinetic Energy14
- Ch 07: Potential Energy & Conservation8
- Ch 08: Momentum, Impulse, and Collisions18
- Ch 09: Rotation of Rigid Bodies42
- Ch 10: Dynamics of Rotational Motion48
- Ch 11: Equilibrium & Elasticity27
- Ch 12: Fluid Mechanics31
- Ch 13: Gravitation35
- Ch 14: Periodic Motion32
- Ch 15: Mechanical Waves25
- Ch 16: Sound & Hearing32
- Ch 17: Temperature and Heat36
- Ch 18: Thermal Properties of Matter38
- Ch 19: The First Law of Thermodynamics33
- Ch 20: The Second Law of Thermodynamics22
- Ch 21: Electric Charge and Electric Field36
- Ch 22: Gauss' Law24
- Ch 23: Electric Potential31
- Ch 24: Capacitance and Dielectrics18
- Ch 25: Current, Resistance, and EMF26
- Ch 26: Direct-Current Circuits30
- Ch 27: Magnetic Field and Magnetic Forces18
- Ch 28: Sources of Magnetic Field10
- Ch 29: Electromagnetic Induction28
- Ch 30: Inductance17
- Ch 31: Alternating Current21
- Ch 32: Electromagnetic Waves1
- Ch 33: The Nature and Propagation of Light20
- Ch 34: Geometric Optics34
- Ch 35: Interference19
- Ch 36: Diffraction25
- Ch 37: Special Relativity20
- Ch 38: Photons: Light Waves Behaving as Particles15
- Ch 39: Particles Behaving as Waves26
- Ch 40: Quantum Mechanics I: Wave Functions21
- Ch 41: Quantum Mechanics II: Atomic Structure25
- Ch 42: Molecules and Condensed Matter18
- Ch 43: Nuclear Physics16
- Ch 44: Particle Physics and Cosmology23
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook
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Young & Freedman Calc 14th EditionCh 01: Units, Physical Quantities & VectorsProblem 26
Young & Freedman Calc 14th EditionCh 01: Units, Physical Quantities & VectorsProblem 26Chapter 1, Problem 26
A spelunker is surveying a cave. She follows a passage 180 m straight west, then 210 m in a direction 45° east of south, and then 280 m at 30° east of north. After a fourth displacement, she finds herself back where she started. Use a scale drawing to determine the magnitude and direction of the fourth displacement.
Verified step by step guidance1
Begin by understanding the problem: The spelunker makes three displacements and returns to her starting point after a fourth displacement. We need to find the magnitude and direction of this fourth displacement.
Break down each displacement into its vector components. For the first displacement, 180 m straight west, the vector is (-180, 0) in Cartesian coordinates, assuming east is positive x and north is positive y.
For the second displacement, 210 m at 45° east of south, calculate the components using trigonometry. The x-component is 210 * sin(45°) and the y-component is -210 * cos(45°), as it is southward.
For the third displacement, 280 m at 30° east of north, calculate the components similarly. The x-component is 280 * sin(30°) and the y-component is 280 * cos(30°), as it is northward.
Add the components of the three displacements to find the resultant vector. The fourth displacement will be the negative of this resultant vector, as it brings the spelunker back to the starting point. Calculate the magnitude and direction using the Pythagorean theorem and inverse tangent function.
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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 vectors to determine a resultant vector. In this problem, the spelunker's movements can be represented as vectors, each with a magnitude and direction. By adding these vectors graphically or mathematically, we can find the resultant vector that returns her to the starting point.
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Vector Addition By Components
Trigonometry in Physics
Trigonometry is essential for resolving vectors into their components, especially when dealing with angles. The spelunker's movements involve directions given in degrees, requiring the use of sine and cosine functions to break down each vector into horizontal and vertical components, facilitating accurate vector addition.
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Scale Drawing
A scale drawing is a graphical representation that maintains proportionality between the actual distances and the drawn distances. In this problem, a scale drawing helps visualize the spelunker's path and determine the magnitude and direction of the fourth displacement by accurately plotting each vector and measuring the resultant vector.
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Related Practice
Textbook Question
Vector A has y-component Ay = +9.60 m. A makes an angle of 32.0° counterclockwise from the +y-axis. (a) What is the x-component of A? (b) What is the magnitude of A?
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Textbook Question
Compute the x- and y-components of the vectors A, B, C, and D in Fig. E1.24.
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Textbook Question
Let θ be the angle that the vector A makes with the +x-axis, measured counterclockwise from that axis. Find angle θ for a vector that has these components: Ax = 2.00m, Ay = −1.00 m
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Textbook Question
You are using water to dilute small amounts of chemicals in the laboratory, drop by drop. How many drops of water are in a 1.0-L bottle?
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Textbook Question
For the vectors A and B in Fig. E1.24, use a scale drawing to find the magnitude and direction of the vector sum A + B
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