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 33

Chapter 3, Problem 33

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

Verified step by step guidance

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).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

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

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.

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.

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.

Recommended video:

Guided course

08:30

Intro to Wave Functions

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?

views

Textbook Question

Ruth sets out to visit her friend Ward, who lives 50 mi north and 100 mi east of her. She starts by driving east, but after 30 mi she comes to a detour that takes her 15 mi south before going east again. She then drives east for 8 mi and runs out of gas, so Ward flies there in his small plane to get her. What is Ward's displacement vector? Give your answer (a) in component form, using a coordinate system in which the y-axis points north, and (b) as a magnitude and direction.

views

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?

views

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?

views

Textbook Question

Trevon drives with velocity v₁ = (55î - 10ĵ) mph for 1.0 h, then v₂ = (20î ＋ 50ĵ) mph for 2.0 h. What is Trevon's displacement? Write your answer in component form using unit vectors.

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?

views