Introduction

Relevance of the Theme

Relative Velocity is a crucial concept in Physics that allows us to understand how the movements of different objects interact with each other. It is the basis for understanding everyday situations, from simple encounters and overtakings in traffic to complex phenomena in astrophysics.

Contextualization

Relative Velocity is at the core of the study of Kinematics, the branch of Physics that focuses on the description of movements, and is a fundamental link between the subthemes 'Uniform Motion' and 'Uniformly Varied Motion'. Without a solid understanding of Relative Velocity, we cannot deepen our understanding of more advanced subjects such as the conservation of linear momentum and the relativity of motion. Thus, this theme serves as a bridge between the basic notions of motion and the more complex theories that will be addressed throughout the course.

Theoretical Development

Components

• Absolute Velocity: The speed at which an object moves in relation to a fixed reference point. It is a linear measure, expressed in units of length divided by time.

• Reference Frame: It is the point of view, the coordinate system, from which the velocity is calculated. It can be changed to any other point without altering the object's velocity.

• Relative Velocity: It is the velocity of an object in relation to another, both in motion or not. This is the speed perceived from the viewpoint of the second object.

• Vectors: Quantities that have magnitude, direction, and sense. Distinguishing relative velocity from absolute velocity requires an understanding of vectors and their properties.

• Vector Algebra: Vector calculus involves a series of operations that allow us to manipulate vectors. This helps us calculate relative velocity accurately.

Key Terms

• Relative Motion: The movement of an object in relation to another. What may seem like a straight-line motion in relation to one object may appear as a curve to another moving object.

• Composition of Velocities: The process of vectorially combining the velocities of different objects to determine their relative motion. It requires an understanding of the parallelogram law.

Examples and Cases

• Case of the Boat and the River: If a boat is moving on a river at a certain speed, its relative velocity in relation to the riverbank is different. This is because the river water, which is the 'reference frame' for velocity, is also moving.

• Case of Two Moving Cars: Two cars may be moving at constant speeds, but an observer in one of the cars will perceive that the other car is moving at a different relative velocity, depending on the observer's position and the speed of their own car.

• Case of a Runner and Their Reflection in a Mirror: A runner running in front of a mirror will see their reflection running in the opposite direction, apparently at the same speed. However, their reflection does not have an actual velocity, it is simply a reflected image of the runner and therefore stationary in relation to the mirror (the 'reference frame').

These examples illustrate the importance of the concept of relative velocity in properly understanding and describing movements. This is a crucial skill in Physics that will be developed and applied in various other topics.

Detailed Summary

Key Points

• Absolute Velocity vs. Relative Velocity: The velocity of an object in relation to a fixed reference point is absolute velocity, while the velocity in relation to another object, which may be in motion or not, is relative velocity. Relative velocity can be equal, smaller, or greater than absolute velocity, depending on the movements of the objects and the reference frames.

• Reference Frame, the Observer: Determining relative velocity requires an understanding of the concept of reference frame. The reference frame is the point of view, the coordinate system, from which the velocity is calculated. Changing the reference frame does not change the object's velocity.

• Vectors and Vector Algebra: Relative velocity is a vector quantity, meaning it includes information of direction and sense, in addition to magnitude (or module). Calculating relative velocity, therefore, requires the use of vector algebra, which is the mathematical manipulation of vectors.

• Composition of Velocities: The composition of velocities is the process of vectorially combining the velocities of different objects to determine relative velocity. This is a central topic in this study and requires an understanding of the parallelogram law.

Conclusions

• Importance of Relative Velocity: Relative velocity is a fundamental concept in kinematics and is crucial for understanding many physical phenomena. Without a solid understanding of this concept, many motion situations would be misinterpreted or not understood.

• Developed Skills: The study of relative velocity develops skills in understanding vectors, manipulating vector quantities, and solving problems in trigonometry and vector algebra. These skills will be useful in many other physics and mathematics topics.

Suggested Exercises

1. Calculation of Relative Velocity: Two bicycles are moving in the same direction. The first one has a speed of 10 km/h and the second one has a speed of 15 km/h. If an observer is on the first bicycle, what would be the relative velocity of the second bicycle? And if the observer is on the second bicycle?

2. Change of Reference Frame: A car is moving at a speed of 20 m/s to the east in relation to the ground. If a driver inside this car throws a ball up and catches it back when it returns, what would be the relative velocity of the ball for the driver during this process? And for an observer at rest on the sidewalk?

3. Composition of Velocities in Different Directions: An airplane is moving at a speed of 500 km/h to the north, while the wind it is flying in has a speed of 100 km/h to the west. What is the relative velocity of the airplane in relation to the ground? And the direction in which the airplane is moving? (Hint: Use the parallelogram law to add the velocity vectors of the airplane and the wind)