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Match the following: $$ \begin{array}{|c|c|c|} \hline {\text { Column I }} & {\text { Column II }} \\ \hline \text { i. } \text { Inelastic collisions } & \text { a. } \begin{array}{l} \text { kinetic energy of the systen } \\ \text { may decrease } \end{array} \\ \hline \text { ii. Elastic collisions } & \begin{array}{l} \text { b. } \\ \text { may increase } \end{array} \\ \hline \begin{array}{l} \text { iii. Total work done is } \\ \text { zero } \end{array} & \text { c. } \begin{array}{l} \text { kinetic energy of the systen } \\ \text { may remain constant } \end{array} \\ \hline \begin{array}{l} \text { iv. } \\ \text { fon-conservative } \\ \text { forces are not } \\ \text { present } \end{array} & \begin{array}{l} \text { d. } \text { just before and after } \\ \text { collision, momentum } \\ \text { remains constant } \end{array} \\ \hline \end{array} $$

Step by step solution

01

Understanding Inelastic Collisions

In inelastic collisions, some of the kinetic energy is transformed into other forms of energy, like heat or sound, which means the kinetic energy of the system may decrease. Therefore, match i. Inelastic collisions with a. kinetic energy of the system may decrease.

02

Analyzing Elastic Collisions

In elastic collisions, no kinetic energy is lost; it remains constant before and after the collision. Thus, match ii. Elastic collisions with c. kinetic energy of the system may remain constant.

03

Examining Total Work Done is Zero

When the total work done is zero, it typically implies that non-conservative forces are absent and mechanical energy is conserved, which means momentum remains constant. Thus, match iii. Total work done is zero with d. just before and after collision, momentum remains constant.

04

Identifying Scenario without Non-Conservative Forces

In scenarios where non-conservative forces are not present, energy transformations do not involve any external work done. This is similar to the condition in step 3, where momentum remains constant as well. Therefore, iv. non-conservative forces are not present should also match with d. just before and after collision, momentum remains constant.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Inelastic Collisions

Inelastic collisions occur when two objects collide, and part of their kinetic energy is transformed into other forms of energy, such as heat or sound. This means that the total kinetic energy after the collision is less than before the collision. An everyday example of an inelastic collision is a car crash where the vehicles stick together.

The equation for momentum in inelastic collisions can be expressed as: \[ m_1 u_1 + m_2 u_2 = (m_1 + m_2)v \] Where:

• \(m_1\) and \(m_2\) are the masses of the objects.
• \(u_1\) and \(u_2\) are their initial velocities.
• \(v\) is the final velocity of the combined mass.

In this situation, while the total momentum is conserved, kinetic energy is not, which is why it's not recovered fully and decreases.

Elastic Collisions

Elastic collisions are an idealized type of collision where both momentum and kinetic energy are conserved. During such collisions, the total kinetic energy of the system remains constant before and after the interaction. These types of collisions rarely occur in everyday life but can be demonstrated by atoms or perfectly hard spheres bouncing off each other.

The conditions for elastic collisions can be captured with the following equations: Momentum conservation:
\[ m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2 \]
Kinetic energy conservation:
\[ \frac{1}{2}m_1 u_1^2 + \frac{1}{2}m_2 u_2^2 = \frac{1}{2}m_1 v_1^2 + \frac{1}{2}m_2 v_2^2 \]
Where:

• \(v_1\) and \(v_2\) are the final velocities of the two objects.

Elastic collisions are important in physics because they help us understand the fundamental principles of particle interactions.

Conservation of Momentum

The concept of the conservation of momentum is a fundamental principle in physics, stating that within a closed system, the total momentum remains constant if no external forces are acting upon it. This principle applies to all types of collisions, whether they are elastic or inelastic.

The mathematical expression for the conservation of momentum is: \[ \sum m_i u_i = \sum m_i v_i \] Where the initial sum of all momenta \( \sum m_i u_i \) equals the final sum of all momenta \( \sum m_i v_i \).

The conservation of momentum helps predict the final velocities of objects after a collision if their masses and initial velocities are known, providing insights into the nature of various collision types.

Non-Conservative Forces

Non-conservative forces, such as friction or air resistance, are forces that cause energy to be transformed into non-recoverable forms like heat or sound. Unlike conservative forces, non-conservative forces cause a system to lose mechanical energy.

An example of a non-conservative force is air resistance acting on a moving object. These forces result in energy losses that affect the overall conservation principles in the system.

In an idealized physics environment where non-conservative forces are absent, such as certain vacuum-sealed labs, the total mechanical energy of a system remains constant. This means that just before and after a collision, momentum and energy remain effectively conserved without external losses.

Understanding the role of non-conservative forces is crucial for accurately predicting real-world physical interactions and solving complex mechanical problems.