Introduction to Kinematics: Average Speed of Uniformly Varied Motion

Relevance of the Topic

Speed is a fundamental concept in physics and the study of motion. It is the measure of how fast or slow an object changes its position in relation to a fixed point in a given time. However, there are situations where speed is not constant, but changes uniformly, either increasing or decreasing.

The Average Speed in Uniformly Varied Motion (MUV) is crucial for understanding everyday physical processes, from observing a moving object to predicting the time it takes for a satellite to position itself in orbit. The applications are vast, once again showing the importance of mastering this concept for the formation of scientific thinking.

Understanding MUV also paves the way for more complex concepts, such as acceleration and integration, which will be developed later in the course.

Contextualization

In the vast field of Physics, kinematics plays a fundamental role in understanding the motion of bodies and how they interact in space. Within kinematics, there are different types of motion, among them Uniformly Varied Motion (MUV), which differs from Uniform Motion (MU) by having a speed that varies constantly.

To understand the Average Speed of MUV, it is necessary to first know about the concepts of speed, distance, and time, which are the pillars of kinematics.

The concepts prior to the Average Speed of MUV include:

• What is speed, distance, and time;
• How to calculate the average speed in Uniform Motion (MU);
• What is acceleration and what are its units of measurement.

The Average Speed of MUV is directly linked to the concept of constant acceleration. In this context, MUV becomes a natural consequence of constant acceleration, as the speed is changing uniformly over time.

The study of the Average Speed of MUV is therefore inserted in a broader context, not only as an isolated topic, but as an integral and interconnected part of kinematics and the study of motion in general. It is a bridge that connects these theoretical concepts to real life, where the motion of bodies is rarely uniform.

Thus, a solid understanding of the Average Speed of MUV not only provides tools for solving complex mechanics problems, but also assists in logical reasoning and understanding everyday phenomena.

Theoretical Development

Components

• Speed: In physics, speed is a scalar measure of how fast or how slow a body changes its position in a given time. In MUV, speed is not constant, it is increasing or decreasing uniformly.

• Acceleration: It is the rate of change of speed over time. In MUV, acceleration is constant.

• Time: Being part of the calculation of average speed, time is the duration elapsed from the beginning to the end of the motion.

• Distance: Represents the space traveled by the body during motion. In MUV, the distance traveled is the result of the average speed and time.

Key Terms

• Average Speed (Vm): It is the quotient between the total displacement and the time interval elapsed during that displacement.

• Displacement (ΔS): It is the change in position of a body over time. In MUV, displacement is represented by the formula ΔS = Vm * t.

• Torricelli's Equation: This equation is used to calculate the final speed in MUV when we have the initial speed, acceleration, and displacement, and the equation in its most familiar form is v² = v₀² + 2aΔS.

• Initial Speed (v₀) and Final Speed (v): Respectively, the speed at the beginning and at the end of the motion in MUV.

Examples and Cases

• Case 1: A car starts moving from rest, and after 10 seconds it has a speed of 20 m/s. In this case, we can calculate the average acceleration using the formula for average speed, which in MUV is identical to the average acceleration. Therefore, the car's average acceleration is 2 m/s².

• Case 2: An object is in motion with an initial speed of 30 m/s and a constant acceleration of 5 m/s². If the motion time is 4 seconds, we can use the displacement formula in MUV to calculate the distance traveled, which will be 130 meters.

• Case 3: Of all the moving objects a student observes in their daily life, most are not in UM, but rather in MUV. For example, a cyclist accelerating or braking, a plane taking off or landing. In these cases, speed is not constant, but varies uniformly, and therefore the Average Speed of MUV is an essential tool for understanding the motion of these objects.

Detailed Summary

Relevant Points

• Average Speed (Vm) in MUV: The average speed in Uniformly Varied Motion (MUV) is different from the average speed in Uniform Motion (MU). In MUV, speed is constantly changing, either increasing or decreasing uniformly, due to the influence of a constant acceleration.

• Calculation of Displacement (ΔS) in MUV: The displacement formula in MUV is ΔS = Vm * t, where ΔS represents the change in the body's position, Vm the average speed, and t the time interval considered. This means that displacement in MUV is directly proportional to the average speed and time.

• Relationship between Average Speed and Acceleration in MUV: In MUV, the average speed is equal to the average acceleration. This occurs because, with constant acceleration, the speed variation is linear and, therefore, the average speed and average acceleration are equal.

• Practical Applications of Average Speed in MUV: Understanding the Average Speed in MUV is not just a theoretical exercise. This understanding is essential to comprehend, predict, and calculate real-world phenomena and events that do not follow a uniform motion, such as a car braking, a rocket launch, among others.

Conclusions

• Average Speed is a Representative Measure: Average speed is a measure that represents the rate of change of an object's position over time in MUV in a more significant way than instantaneous speed. It considers the entire time and space interval, which can be crucial for certain applications.

• Average Speed in MUV is Directly Proportional to Time: When the average speed of MUV is constant, that is, the acceleration is zero, the average speed becomes directly proportional to time, showing a unique property of this type of motion.

Exercises

1. Exercise 1: Calculate the average speed of an object in MUV that moves for 4 seconds and has a displacement of 20 meters.

2. Exercise 2: A car brakes with an acceleration of 5 m/s² from an initial speed of 30 m/s. How long does it take for the car to stop?

3. Exercise 3: An airplane takes off and, after 20 seconds, is flying at a speed of 50 m/s. If the airplane's acceleration is constant, what is its acceleration?