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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Ch 03: Vectors and Coordinate Systems
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796
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Table of contents
- Ch 01: Concepts of Motion35
- Ch 02: Kinematics in One Dimension75
- Ch 03: Vectors and Coordinate Systems58
- Ch 04: Kinematics in Two Dimensions60
- Ch 05: Force and Motion23
- Ch 06: Dynamics I: Motion Along a Line53
- Ch 07: Newton's Third Law27
- Ch 08: Dynamics II: Motion in a Plane52
- Ch 09: Work and Kinetic Energy56
- Ch 10: Interactions and Potential Energy50
- Ch 11: Impulse and Momentum56
- Ch 12: Rotation of a Rigid Body71
- Ch 13: Newton's Theory of Gravity52
- Ch 14: Fluids and Elasticity44
- Ch 15: Oscillations55
- Ch 16: Traveling Waves37
- Ch 17: Superposition39
- Ch 18: A Macroscopic Description of Matter53
- Ch 19: Work, Heat, and the First Law of Thermodynamics60
- Ch 20: The Micro/Macro Connection53
- Ch 21: Heat Engines and Refrigerators34
- Ch 22: Electric Charges and Forces40
- Ch 23: The Electric Field39
- Ch 24: Gauss' Law32
- Ch 25: The Electric Potential63
- Ch 26: Potential and Field57
- Ch 27: Current and Resistance42
- Ch 28: Fundamentals of Circuits39
- Ch 29: The Magnetic Field47
- Ch 30: Electromagnetic Induction60
- Ch 31: Electromagnetic Fields and Waves32
- Ch 32: AC Circuits63
- Ch 33: Wave Optics32
- Ch 34: Ray Optics57
- Ch 35: Optical Instruments30
- Ch 36: Special Relativity38
- Ch 37: The Foundations of Modern Physics25
- Ch 38: Quantization49
- Ch 39: Wave Functions and Uncertainty31
- Ch 40: One-Dimensional Quantum Mechanics35
- Ch 41: Atomic Physics18
- Ch 42: Nuclear Physics27
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook
Chapter 3, Problem 35a
Jack and Jill ran up the hill at 3.0 m/s. The horizontal component of Jill's velocity vector was 2.5 m/s. What was the angle of the hill?
Verified step by step guidance1
Identify the components of Jill's velocity vector. The horizontal component is given as 2.5 m/s, and the total velocity is 3.0 m/s. The angle of the hill can be determined using trigonometric relationships.
Recall the trigonometric definition of cosine: \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \). Here, the adjacent side corresponds to the horizontal component of Jill's velocity (2.5 m/s), and the hypotenuse corresponds to the total velocity (3.0 m/s).
Rearrange the formula to solve for the angle \( \theta \): \( \theta = \cos^{-1}\left(\frac{\text{horizontal component}}{\text{total velocity}}\right) \).
Substitute the given values into the equation: \( \theta = \cos^{-1}\left(\frac{2.5}{3.0}\right) \).
Use a calculator or mathematical software to compute the inverse cosine of the ratio. This will give you the angle of the hill in degrees or radians, depending on the context.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Velocity Components
Velocity can be broken down into horizontal and vertical components. In this scenario, Jill's velocity vector has a horizontal component of 2.5 m/s, which is part of her overall velocity. Understanding how to resolve a vector into its components is crucial for analyzing motion on inclined surfaces.
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05:53
Calculating Velocity Components
Trigonometric Functions
Trigonometric functions, particularly sine and cosine, are essential for relating the angles of a triangle to the ratios of its sides. In this case, the angle of the hill can be determined using the tangent function, which relates the vertical rise to the horizontal run. This relationship is fundamental in physics for analyzing inclined planes.
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08:30Intro to Wave Functions
Inclined Plane Dynamics
Inclined planes are surfaces that are tilted at an angle to the horizontal. The dynamics of objects on inclined planes involve gravitational forces, normal forces, and friction. Understanding how these forces interact helps in calculating the angle of the incline based on the components of motion, such as velocity.
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06:59Intro to Inclined Planes
Related Practice
Textbook Question
Jack and Jill ran up the hill at 3.0 m/s. The horizontal component of Jill's velocity vector was 2.5 m/s. What was the vertical component of Jill's velocity?
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Textbook Question
A cannon tilted upward at 30° fires a cannonball with a speed of 100 m/s. What is the component of the cannonball's velocity parallel to the ground?
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Textbook Question
Kami is walking through the airport with her two-wheeled suitcase. The suitcase handle is tilted 40° from vertical, and Kami pulls parallel to the handle with a force of 120 N. (Force is measured in newtons, abbreviated N.) What are the horizontal and vertical components of her applied force?
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Textbook Question
A cannonball leaves the barrel with velocity v = (75î ＋ 45ĵ). At what angle is the barrel tilted above horizontal?
Textbook Question
You are fixing the roof of your house when a hammer breaks loose and slides down. The roof makes an angle of 35° with the horizontal, and the hammer is moving at 4.5 m/s when it reaches the edge. What are the horizontal and vertical components of the hammer's velocity just as it leaves the roof?
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