Elastic and Inelastic Collisions §

Introduction §

Collisions are fundamental concepts in physics, particularly in the study of mechanics. They are categorized into two types: elastic and inelastic collisions. Understanding these collisions is crucial for grasping the principles of momentum and energy conservation.

Elastic Collisions §

An elastic collision is an event where both momentum and kinetic energy are conserved. These collisions are idealized scenarios, often used as approximations because, in the real world, some energy is always converted into other forms.

Key Characteristics §

1. Conservation of Kinetic Energy: The total kinetic energy of the system remains constant before and after the collision.
2. Conservation of Momentum: The total momentum of the system is conserved, owing to Newton’s third law.

Example: Billiard Balls §

When two billiard balls collide, they exhibit behaviour close to an elastic collision. The kinetic energy and momentum are transferred from one ball to the other.

Inelastic Collisions §

In contrast, an inelastic collision is where the kinetic energy is not conserved, although the momentum is. Some of the kinetic energy is converted into other forms of energy, such as heat, sound, or deformation energy.

Key Characteristics §

1. Non-conservation of Kinetic Energy: The total kinetic energy after the collision is less than that before the collision.
2. Conservation of Momentum: Despite the loss in kinetic energy, the total momentum of the system is still conserved.

Example: Car Crash §

A car crash is a classic example of an inelastic collision, where kinetic energy is partly converted into other forms of energy, leading to the deformation of the cars.

Mathematical Representation §

Elastic Collisions §

In an elastic collision between two objects, the following equations are used:

1. $m_1{v}_{1i}+m_2{v}_{2i}=m_1{v}_{1f}+m_2{v}_{2f}$ (Momentum Conservation)
2. $\frac{1}{2}{m}_1{v}_{1i}^2+\frac{1}{2}{m}_2{v}_{2i}^2=\frac{1}{2}{m}_1{v}_{1f}^2+\frac{1}{2}{m}_2{v}_{2f}^2$ (Kinetic Energy Conservation)

Inelastic Collisions §

For inelastic collisions, only the momentum conservation equation is used:

1. $m_1{v}_{1i}+m_2{v}_{2i}=m_1{v}_{1f}+m_2{v}_{2f}$

In completely inelastic collisions, the two objects stick together after the collision, leading to a single final velocity for both objects.

Test Questions §

1. Define an elastic collision and an inelastic collision.
2. In a perfectly inelastic collision, if two objects stick together after colliding, how does one calculate their final velocity?
3. Consider two objects with masses $m_1$ and $m_2$ moving with velocities $v_1$ and $v_2$ respectively. Derive the equation for conservation of momentum in an elastic collision.

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This note provides a fundamental understanding of elastic and inelastic collisions, integrating historical context and practical examples. For deeper exploration, consider analyzing real-world scenarios where these principles apply.