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 38

Chapter 3, Problem 38

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?

Verified step by step guidance

Start by understanding that the acceleration vector is given in polar form with a magnitude of 15 m/s² and a direction of 22° above the horizontal. To find the horizontal and vertical components, we will use trigonometric functions.

The horizontal component of acceleration (aₓ) can be found using the cosine function: aₓ = a * cos(θ), where 'a' is the magnitude of the acceleration (15 m/s²) and 'θ' is the angle (22°).

The vertical component of acceleration (aᵧ) can be found using the sine function: aᵧ = a * sin(θ), where 'a' is the magnitude of the acceleration (15 m/s²) and 'θ' is the angle (22°).

Substitute the given values into the formulas: aₓ = 15 * cos(22°) and aᵧ = 15 * sin(22°).

Evaluate the trigonometric functions (cos(22°) and sin(22°)) to compute the horizontal and vertical components of the acceleration. These will give you the final values for aₓ and aᵧ.

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Key Concepts

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

Vector Components

Vectors can be broken down into their horizontal and vertical components using trigonometric functions. For a vector at an angle, the horizontal component is found using the cosine function, while the vertical component is determined using the sine function. This decomposition is essential for analyzing motion in two dimensions.

Trigonometric Functions

Trigonometric functions, specifically sine and cosine, relate the angles of a triangle to the ratios of its sides. In the context of vector components, cosine gives the adjacent side (horizontal component) and sine gives the opposite side (vertical component) when applied to the angle of the vector. Understanding these functions is crucial for resolving vectors.

Acceleration

Acceleration is the rate of change of velocity of an object and is a vector quantity, meaning it has both magnitude and direction. In this scenario, the acceleration of the jet plane is given in terms of its magnitude and direction, which must be resolved into components to analyze its motion in the horizontal and vertical directions effectively.

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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

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

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?

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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

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?

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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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