Introduction

Relevance of the Topic

Our study, Impulse and Momentum: Coefficient of Restitution, is one of the fundamental pillars of Classical Physics. Understanding the behavior of bodies and their interactions in motion is a crucial intellectual feat and is a basis for many practical applications, from sports to industry. This topic will be especially relevant to understand the conservation of energy and linear momentum, concepts that will transcend our future discussions.

Contextualization

Situated within the Broad World of Motion, impulse and momentum play vital roles. These concepts are key to understanding how objects move and interact with each other. Our current exploration, the Coefficient of Restitution, is just one of the many aspects of this vast universe. However, it is a fundamental aspect that creates the perfect bridge to our next topic, energy conservation. This lecture note prepares us for an exciting and relevant journey in understanding our physical world and beyond!

Theoretical Development

Components

• Impulse: It is a physical quantity that represents the change in the momentum of a body over a period of time. When a force acts on a body over a period of time, the effect produced is measured by the impulse of the force. The impulse formula is the product of the force (F) by the time of action of the force (Δt) in the same direction and sense.

  • Impulse and change in linear momentum: The change in the linear momentum of a body is directly proportional to the applied impulse. Mathematically, the change in linear momentum (Δp) is equal to the impulse (J). If the force is constant, we can express the impulse as the product of this force by the time interval of force application (J = FΔt).

• Momentum: This is a measure of an object's motion, calculated by multiplying the object's mass by its velocity. Linear momentum is a vector quantity and therefore has direction and magnitude. If its direction remains constant, the momentum vector is conserved.

  • Conservation of Linear Momentum: The momentum (linear momentum) of an isolated system is conserved, meaning the sum of the linear momenta before and after the collision is the same. Thus, we can explore the final state of the system, using conservation laws to determine the final velocities.

• Coefficient of Restitution (e): It is a measure of the elasticity of a collision between two bodies. Mathematically, it is the ratio of the relative velocity of the bodies after the collision to the relative velocity before the collision. The coefficient of restitution is a dimensionless number and is an indication of how much kinetic energy of translational motion is conserved in a collision.

  • Distinguishing Between Elastic and Inelastic Collisions: In an elastic collision, the coefficient of restitution is 1, which means that kinetic energy is completely preserved. In inelastic collisions, the coefficient of restitution is less than 1, indicating energy loss during the collision. Therefore, the coefficient of restitution is an important tool to distinguish between different types of collisions.

Key Terms

• Collision: This is the event in which two or more bodies interact for a period of time. During a collision, the momentum of the bodies involved can change.
• Kinetic Energy: It is the energy that a moving body possesses due to its velocity. The conservation of kinetic energy in a collision is governed by the coefficient of restitution.

Examples and Cases

• Billiard Ball Case: When a billiard ball hits another billiard ball, we observe that after the collision both balls are in motion, meaning there was a transfer of motion (momentum) between them. The coefficient of restitution can be used to measure the elasticity of the collision.
• Car Collision Case: If two cars collide, the total momentum of the cars before the collision is equal to the total momentum of the cars after the collision. However, the coefficient of restitution will tell us how much kinetic energy was lost due to the deformation of the cars in the collision.
• Solid Spheres Case: If a solid sphere in motion collides with another fixed solid sphere, the first sphere stops in the collision and the second sphere starts moving. In this case, the coefficient of restitution is zero, as all the kinetic energy of motion was transferred to the second sphere.

Detailed Summary

Key Points:

• Relevance of Studying Coefficients of Restitution: This tool is essential to understand energy transformations during collisions. Through the coefficient of restitution, we can identify whether a collision is elastic (fully energetic) or inelastic (with energy loss).

• Impulse and Change in Linear Momentum: The connection between impulse and change in linear momentum is essential in solving problems involving the coefficient of restitution. When the force is constant, the impulse is responsible for the variation in linear momentum.

• Conservation of Linear Momentum: Understanding this principle is vital for collision analysis. The total amount of momentum (linear momentum) before the collision must be equal to the total amount of momentum after the collision (as long as there are no external forces).

• Definition of Coefficient of Restitution: This term is explained appropriately, as the ratio between the relative velocities of two bodies before and after the collision.

• Differentiation between Elastic and Inelastic Collisions: Only when the coefficient of restitution is equal to 1 (or 100%), the collision is considered elastic, implying total conservation of kinetic energy. Otherwise, the collision is inelastic, and there is a loss of kinetic energy in the form of heat and deformation.

Conclusions:

• Implementation of the Coefficient of Restitution: Coefficients of restitution are used in various areas such as sports (e.g., in golf or tennis) and in industry (in machinery and equipment) to understand the type and efficiency of collisions.

• Importance of Conservation of Linear Momentum: The conservation of linear momentum is a powerful conceptual and mathematical tool in physics. Its application in the analysis of coefficients of restitution reinforces its relevance.

• Problem Solving: The examples and problems presented are useful to solidify the understanding of the coefficient of restitution concept and its applicability.

• Link to General Principles of Physics: The study of the coefficient of restitution reinforces and deepens the understanding of general principles of physics, such as the conservation of linear momentum and energy conservation.

Exercises:

1. A ball weighing 0.5 kg in motion with a velocity of 4 m/s collides head-on and elastically with another ball weighing 0.8 kg that is initially at rest. What will be the velocity of each ball after the collision?

2. Two steel balls, A and B, are moving in the same direction. Ball A has a mass of 0.2 kg and a velocity of 5 m/s. Ball B has a mass of 0.5 kg and a velocity of 2 m/s. After the collision, ball A has a velocity of 3 m/s. Calculate the final velocity of ball B and the coefficient of restitution of the collision.

3. A baseball player hits a ball with a racket. If the ball's velocity is 20 m/s before the collision and 15 m/s after the collision, what is the coefficient of restitution? Is the collision elastic or inelastic? Justify your answer.