Lesson Plan | Traditional Methodology | Vectors: Decomposition

Objectives

Duration: (10 - 15 minutes)

The purpose of this stage is to present clear and specific objectives of the lesson to the students, preparing them for the content that will be covered. By understanding the importance of vector decomposition and learning the necessary mathematical tools, students will be better prepared to follow the explanation and apply the knowledge to practical problems.

Main Objectives

1. Understand the importance of vector decomposition into perpendicular coordinates.

2. Learn to use basic trigonometry, such as sine and cosine, to decompose vectors.

3. Recognize practical situations where vector decomposition is applied.

Introduction

Duration: (10 - 15 minutes)

The purpose of this stage is to contextualize the topic of the lesson and spark the students' interest by showing the practical relevance of vector decomposition. Presenting everyday examples and curiosities helps make the content more accessible and engaging, preparing students for the expository part of the lesson.

Context

To start the lesson on vector decomposition, it is important to remember that vectors are physical quantities that have magnitude and direction, such as the speed of a car, the force applied to an object, or the acceleration of an athlete. Understanding how to decompose vectors is essential for solving problems in physics and various engineering fields, as forces and movements often occur in more than one direction simultaneously. Imagine a boat navigating in a river with current: decomposing the vectors helps understand the resulting motion of the boat considering the motor force and the current force.

Curiosities

Did you know that vector decomposition is used in video game graphics to calculate the physics of movements? For example, when jumping off a cliff in a game, the character is affected by gravity and also by the horizontal force of running. Developers use vector decomposition to ensure that the movement occurs realistically.

Development

Duration: (45 - 50 minutes)

The purpose of this stage is to provide a detailed and practical understanding of vector decomposition. By addressing the essential topics, the teacher ensures that students have a solid foundation for understanding how and why vectors are decomposed. The proposed questions allow students to apply the acquired knowledge, reinforcing learning through guided practice.

Covered Topics

1. Definition and Representation of Vectors: Explain that a vector is a quantity that has magnitude and direction. Show how vectors are typically represented graphically as arrows, where the length of the arrow represents the magnitude and the direction of the arrow indicates the direction of the vector. 2. Components of a Vector: Detail that any vector in a plane can be decomposed into two perpendicular components, usually along the x and y axes. These components are called the horizontal and vertical components of the vector. 3. Basic Trigonometry: Introduce the sine and cosine functions, which are used to decompose vectors. Explain that the sine of an angle in a right triangle is the ratio of the length of the side opposite the angle to the hypotenuse, while the cosine is the ratio of the length of the adjacent side to the hypotenuse. 4. Vector Decomposition: Show step-by-step how to decompose a vector into its components using sine and cosine. Use a practical example, such as the force applied to an object on an inclined plane. Draw the vector and the components on the board, and solve to find the magnitudes of the horizontal and vertical components. 5. Practical Application: Discuss how vector decomposition is used in practical situations, such as calculating the resultant force on an object, analyzing motion on inclined planes, or determining paths in video games.

Classroom Questions

1. Given a vector of magnitude 10 units that makes an angle of 30 degrees with the horizontal, find the horizontal and vertical components of this vector. 2. A force vector of 50 N acts at 45 degrees above the horizontal. Determine the horizontal and vertical components of this force. 3. Explain how vector decomposition can be used to analyze the motion of a projectile launched at a certain angle with the horizontal.

Questions Discussion

Duration: (20 - 25 minutes)

The purpose of this stage is to review and consolidate the knowledge acquired by the students during the lesson. By discussing the proposed questions and engaging students with reflective inquiries, the teacher ensures that the concepts have been clearly and practically understood, promoting a deeper and more lasting learning experience.

Discussion

• 🔍 Question 1: Given a vector of magnitude 10 units that makes an angle of 30 degrees with the horizontal, find the horizontal and vertical components of this vector.

Explanation: Horizontal Component (Vx): Vx = V * cos(θ) = 10 * cos(30°) = 10 * (√3/2) ≈ 8.66 units. Vertical Component (Vy): Vy = V * sin(θ) = 10 * sin(30°) = 10 * (1/2) = 5 units.

Therefore, the components of the vector are approximately 8.66 units horizontally and 5 units vertically.

• 🔍 Question 2: A force vector of 50 N acts at 45 degrees above the horizontal. Determine the horizontal and vertical components of this force.

Explanation: Horizontal Component (Fx): Fx = F * cos(θ) = 50 * cos(45°) = 50 * (√2/2) ≈ 35.36 N. Vertical Component (Fy): Fy = F * sin(θ) = 50 * sin(45°) = 50 * (√2/2) ≈ 35.36 N.

Therefore, the components of the force are approximately 35.36 N in both horizontal and vertical directions.

• 🔍 Question 3: Explain how vector decomposition can be used to analyze the motion of a projectile launched at a certain angle with the horizontal.

Explanation: Motion Analysis: When launching a projectile at an angle θ with the horizontal, decomposing the initial velocity vector allows separating the motion into two perpendicular directions: horizontal and vertical. Horizontal Component of Velocity (Vx): Vx = V * cos(θ). This component is responsible for horizontal motion, which is uniform if we neglect air resistance. Vertical Component of Velocity (Vy): Vy = V * sin(θ). This component is responsible for vertical motion, which is uniformly accelerated due to gravity. Application: Decomposition allows calculating the projectile's trajectory, the maximum range, and the maximum height reached. By using the equations of motion, we can predict the position of the projectile at any point in time.

Student Engagement

1. ❓ How would you determine the components of a vector that is at 60 degrees above the horizontal? 2. ❓ Why is it important to decompose vectors when solving physics problems? 3. ❓ What other practical situations could benefit from vector decomposition? 4. ❓ How could air resistance affect vector decomposition in analyzing the motion of a projectile? 5. ❓ Can you think of an example in everyday life or technology where vector decomposition is crucial?

Conclusion

Duration: (5 - 10 minutes)

The purpose of this stage is to review and consolidate the knowledge acquired during the lesson, ensuring that students have a clear and practical understanding of the concepts covered. Summarizing the key points and discussing the relevance of the topic helps solidify the content and demonstrates the importance of vector decomposition in various practical applications.

Summary

• Definition and representation of vectors as quantities that have magnitude and direction.
• Decomposition of vectors into horizontal and vertical components.
• Use of basic trigonometry, such as sine and cosine, to decompose vectors.
• Practical resolution of vector decomposition examples.
• Discussion about practical applications of vector decomposition in various situations.

The lesson connected theory with practice by using detailed examples and solving practical problems on the board. Vector decomposition was demonstrated through everyday situations, such as the force applied on an inclined plane and the motion of a projectile, showing how theory is applied in solving real problems.

Vector decomposition is crucial in everyday life, especially in fields like engineering, physics, and video game development. For example, the physics of movements in video games uses this technique to create realistic movements. Moreover, understanding vector decomposition allows analyzing and predicting the behavior of moving objects, such as the trajectory of a projectile or the forces acting on an inclined plane.