Introduction

Relevance of the Topic

Kinematics: Instantaneous Velocity is a fundamental concept in Physics that allows us to understand and describe motion. Without understanding this concept, it would be impossible to determine the exact velocity of an object at a specific moment in time. Instantaneous velocity is the basis for understanding other physical concepts such as acceleration, force, and work. It is the gateway that allows us to enter a vast universe of constantly moving physical phenomena.

Contextualization

In terms of Physics, kinematics is the first and one of the most important areas of study. It provides the foundation for understanding how things move in the universe. The concept of instantaneous velocity is the first deep dive into the study of motion. Next, we will explore acceleration, force, and work, all concepts that directly depend on instantaneous velocity. Moreover, the study of instantaneous velocity leads us directly to the calculations and equations that form the basis of Physics, which leads us to the teaching of Mathematics within Physics. Finally, understanding velocity at a specific moment in time allows us to analyze the motion of objects at a more refined level, resulting in a better understanding of the universe around us.

Theoretical Development

Components

• Motion: Represents the change in position of a body relative to a reference over time.

• Average Velocity: It is the ratio between the total displacement and the total elapsed time for that displacement. It is calculated using the formula Vm = d/t, where "Vm" represents the average velocity, "d" the displacement, and "t" the time.

• Instantaneous Velocity: It is the limit that the average velocity approaches, when we consider increasingly smaller time intervals. It can be different from the average velocity over a small time interval. In formula, Vi = lim (Δt → 0) Δd/Δt, where "Vi" is the instantaneous velocity, "Δd" is the displacement, and "Δt" is the time interval.

• Graphical Analysis of Motion: It is the representation of motion in graphs. In the case of velocity, the analysis is done on the velocity versus time graph.

Key Terms

• Rate of Change: Refers to the rate at which one quantity changes in relation to another.

• Limit: It is a value that a function or sequence approaches as the argument or index approaches a certain value. In the context of instantaneous velocity, the limit is used to express the velocity of a body at a specific point in time, making the time interval tend to zero.

• Derivative: In calculus, the derivative is a rate of change. Instantaneous velocity is the derivative of space (displacement) as a function of time.

Examples and Cases

• Moving Car: Suppose a car moving in a straight line, where it is possible to measure the distance traveled and the time elapsed at any moment. The instantaneous velocity of the car at a given moment is its velocity at that exact moment and can be different from the average velocity of a time interval.

• Runner on a Track: If a runner completes a full lap on a track, the distance traveled will be equal to the length of the track. However, the instantaneous velocity of the runner at each moment of this journey will be different, due to the different orientations and inclinations of the runner along the track.

• Throwing a Ball: When throwing a ball, we initially push the ball slowly and then release it abruptly. The instantaneous velocity of the ball when released will be higher than the velocity before being released. This illustrates that instantaneous velocity does not only depend on the recent movement of the object, but also on any forces acting on it.

Detailed Summary

Relevant Points

• Average Velocity versus Instantaneous Velocity: Average velocity measures the rate of displacement in relation to a finite amount of time. Instantaneous velocity, on the other hand, is the exact velocity of an object at a precise point in time, and is calculated by taking the limit of the average velocity as the time interval becomes infinitesimally small. This crucial understanding must be grasped, as it forms the basis of many other physical concepts.

• Velocity Calculations: The formula for average velocity (V = d/t) is straightforward, but the formula for instantaneous velocity (Vi = lim (Δt → 0) Δd/Δt) may seem more complex at first glance. However, by understanding limits and how they apply in velocity calculations, the process becomes clearer.

• Graphical Analysis of Motion: The graphical representation of a movement, especially in the velocity versus time graph, offers a valuable visual perspective. The use of this graph can help to clearly visualize the difference between instantaneous velocity and average velocity.

Conclusions

• The Importance of Instantaneous Velocity: Instantaneous velocity is fundamental in the study of Physics, as it allows us to understand the motion of objects at a deeper level. Understanding that the velocity of an object can change at any point in time is essential for understanding the principles of acceleration, force, and work.

• Instantaneous Velocity is a Derivative Concept: Understanding that instantaneous velocity is a derivative concept helps to establish the transition between Mathematics and Physics. This will become especially relevant as we explore acceleration, which is the rate of change of velocity.

• Applications of the Theory: The theory of instantaneous velocity has practical applications in many fields, from car trips in traffic to rocket launches into space. Proficient use of instantaneous velocity is, therefore, an essential skill that will be used repeatedly throughout the study of Physics.

Exercises

1. Exercise 1: A vehicle initially moves at a speed of 10 m/s. After 5 seconds, it increases its speed to 20 m/s. What is the instantaneous velocity of the vehicle after 6 seconds?

2. Exercise 2: A cyclist covers a distance of 25 km in 2 hours. What is his average velocity? If we examine a 15-minute interval between the first hour and an hour and a half, what would be his instantaneous velocity at that point?

3. Exercise 3: A pendulum performs a simple periodic motion, returning and passing through the equilibrium point every 2 seconds. The instantaneous velocity of the pendulum at the equilibrium point is zero. However, the instantaneous velocity of the pendulum when it is halfway between the equilibrium position and the point of maximum displacement is the highest. Explain why.