Head on head collision fraction of kE retained. How do we derrive the formulae in case of elastic , inelastic, or stable equilibrium conditions?

When we discuss head-on collisions, particularly in the context of kinetic energy (KE) retention, we need to differentiate between elastic and inelastic collisions, as well as stable equilibrium conditions. Each type of collision has distinct characteristics that affect how kinetic energy is conserved or transformed. Let's break down the derivation of the formulas for each scenario.

Elastic Collisions

In an elastic collision, both momentum and kinetic energy are conserved. This means that the total kinetic energy before the collision equals the total kinetic energy after the collision. To derive the fraction of kinetic energy retained, we start with the following principles:

• Conservation of Momentum: For two objects with masses \( m_1 \) and \( m_2 \) and initial velocities \( u_1 \) and \( u_2 \), the equation is:

  \( m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2 \)

• Conservation of Kinetic Energy: The total kinetic energy before and after the collision is given by:

  \( \frac{1}{2} m_1 u_1^2 + \frac{1}{2} m_2 u_2^2 = \frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2 \)

By solving these two equations simultaneously, we can find the final velocities \( v_1 \) and \( v_2 \). The fraction of kinetic energy retained can then be expressed as:

\( \text{Fraction of KE retained} = \frac{\text{KE after}}{\text{KE before}} = \frac{\frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2}{\frac{1}{2} m_1 u_1^2 + \frac{1}{2} m_2 u_2^2} \)

Inelastic Collisions

In an inelastic collision, momentum is conserved, but kinetic energy is not. Some of the kinetic energy is transformed into other forms of energy, such as heat or sound. The derivation follows a similar approach:

• Conservation of Momentum:

  \( m_1 u_1 + m_2 u_2 = (m_1 + m_2) v \)

• Kinetic Energy Loss: The kinetic energy before the collision is greater than after:

  \( \text{KE before} = \frac{1}{2} m_1 u_1^2 + \frac{1}{2} m_2 u_2^2 \)

  \( \text{KE after} = \frac{1}{2} (m_1 + m_2) v^2 \)

To find the fraction of kinetic energy retained, we can express it as:

\( \text{Fraction of KE retained} = \frac{\text{KE after}}{\text{KE before}} = \frac{\frac{1}{2} (m_1 + m_2) v^2}{\frac{1}{2} m_1 u_1^2 + \frac{1}{2} m_2 u_2^2} \)

Stable Equilibrium Conditions

In stable equilibrium, we often consider systems where the forces are balanced, and any small displacement results in a restoring force. While this concept is more about forces than collisions, it can relate to how objects behave post-collision. In such cases, the kinetic energy might not be a primary focus, but rather how the system returns to equilibrium.

For a system in stable equilibrium, the energy considerations might involve potential energy rather than kinetic energy. However, if we consider a collision that leads to a stable configuration, we can analyze the energy states before and after the collision, similar to the previous cases.

Summary

In summary, the fraction of kinetic energy retained in collisions depends on whether the collision is elastic or inelastic. In elastic collisions, both momentum and kinetic energy are conserved, while inelastic collisions conserve momentum but not kinetic energy. Understanding these principles allows us to derive the relevant formulas and analyze the outcomes of different collision scenarios effectively.