Contextualization

Vectors are fundamentally important for understanding many concepts in Physics. They are used to represent quantities that have both magnitude and direction, such as velocity or force. The idea of a vector comes with a new operation, called vector subtraction, which is essential for solving many problems in Physics. For example, in Mechanics, we can use vector subtraction to find the difference between two forces applied to an object, allowing us to then calculate the resultant force.

Therefore, it is essential to understand how to differentiate vectors to solve problems. Despite this, vector subtraction is not a trivial operation. It requires knowledge and skills that go beyond simple knowledge of how to add and subtract numbers. In fact, it is an operation that involves the geometry of vectors, and this is where many students struggle.

Vector subtraction also has a wide range of real-world applications. For example, in Engineering, vector subtraction is used to calculate the difference between two forces applied to a structure. In Geography, it is used to determine the direction and magnitude of the difference between two points. Even in sports, the ability to subtract vectors can be useful. For example, in soccer, vector subtraction can help determine the direction and force needed to make a pass or shot.

To assist you in studying vector differentiation, we recommend the following sources:

1. Halliday, Resnick, and Walker. Fundamentals of Physics. Vol 1. 9th ed. LTC, 2012.
2. Tipler, Paul; Mosca, Gene. Physics for Scientists and Engineers. Vol 1. 6th ed. LTC, 2008.
3. Khan Academy. Vectors and vector spaces. Available at: https://pt.khanacademy.org/math/vectors-and-spaces

Like all subjects in Physics, study is the key to mastery. We hope the suggested sources above will be useful for you!

Practical Activity

Activity Title: "Vectors in Action: The Physics of Soccer"

Project Objective

The objective of the project is to apply and understand vector subtraction in the physical context of soccer, having students act as players and analysts, where they must calculate the resultant of vectors to make passes and shots, using vector subtraction.

Detailed Project Description

The project consists of modeling the dynamics of a soccer game using vectors. Each group of students will be divided into two teams (analysts and players). The analysts will be responsible for calculating the resultant of vectors so that the players can make passes and shots. The players will, in practice, execute the passes and shots of the game according to the analysts' instructions.

Required Materials

• One ball
• Two sets of cones to mark the goalposts
• Graph paper
• Ruler
• Compass
• Calculator

Detailed Step-by-Step for Carrying Out the Activity

1. Divide the class into groups of 3-5 students. Each group should subdivide into analysts and players.

2. Mark a soccer field in the available space with the cones. The field scale may vary according to the available space.

3. Each team must then play a soccer match. During the game, the analysts should observe the players' movements and the direction of the passes and shots made.

4. The analysts should then use the graph paper to create a vector diagram representing the movements, passes, and shots.

5. Next, the analysts should calculate the resultant of the vectors using vector subtraction.

6. Based on these calculations, the analysts should guide the players on how they should move and kick the ball to achieve the desired resultant.

7. Repeat steps 3 to 6 several times during the month to allow for a better understanding of the concept of vector subtraction.

Project Deliverables and Written Document

The groups must submit a written report containing the following elements:

1. Introduction: students must explain the relevance of vector subtraction in a practical context like soccer. They should indicate the project's objective and its real-world applications.

2. Development: students need to explain the theory of vectors and the subtraction operation, detail the practical activity carried out, indicate the methodology used, and finally, present and discuss the results obtained.

3. Conclusion: students must conclude the work by summarizing its main points, stating the learnings obtained, and drawing conclusions about the project.

4. Bibliography: students must indicate the sources they relied on to work on the project such as books, web pages, videos, etc.

It is important that the written report is directly connected to the practical activity carried out. Each calculation, each result obtained, each difficulty overcome should be highlighted in the report.

Remember, this project is not only about understanding vector subtraction but also about collaboration and teamwork.