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Ch 03: Vectors and Coordinate Systems
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoKnight Calc 5th EditionCh 03: Vectors and Coordinate SystemsProblema 33
Capitolo 3, Problema 33

A cannonball leaves the barrel with velocity v = (75î ＋ 45ĵ). At what angle is the barrel tilted above horizontal?

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Step 1: Understand the problem. The velocity vector of the cannonball is given as v = (75î + 45ĵ), where î represents the horizontal component and ĵ represents the vertical component. The goal is to find the angle θ that the barrel is tilted above the horizontal.
Step 2: Recall the formula for the angle of a vector relative to the horizontal axis. The angle θ can be calculated using the tangent function: tan(θ) = (vertical component) / (horizontal component). In this case, tan(θ) = 45 / 75.
Step 3: Use the inverse tangent function (arctan or tan⁻¹) to find the angle θ. The formula is θ = tan⁻¹(45 / 75). This will give the angle in radians or degrees, depending on the calculator or method used.
Step 4: Ensure the units are consistent. Since the components of the velocity vector are both in the same units, no conversion is necessary before applying the formula.
Step 5: Interpret the result. The angle θ represents the tilt of the barrel above the horizontal. If needed, convert the angle from radians to degrees using the formula: degrees = radians × (180/π).

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Velocity Vector

Velocity is a vector quantity that describes the rate of change of an object's position. It has both magnitude and direction, represented in this case by the components v = (75î + 45ĵ), where î and ĵ are unit vectors in the horizontal and vertical directions, respectively. Understanding how to interpret and manipulate velocity vectors is crucial for analyzing projectile motion.
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Angle of Projection

The angle of projection is the angle at which an object is launched relative to the horizontal axis. It can be calculated using the components of the velocity vector, specifically through the tangent function: θ = arctan(v_y/v_x), where v_y and v_x are the vertical and horizontal components of the velocity. This angle is essential for predicting the trajectory of the projectile.
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Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, relate the angles of a triangle to the ratios of its sides. In the context of projectile motion, these functions are used to determine relationships between the angle of projection and the components of velocity. Mastery of these functions is necessary for solving problems involving angles and distances in physics.
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