Introduction to Elastic Collisions
An elastic collision is an encounter between two or more bodies in which the total kinetic energy of the system remains conserved. In a real-world scenario, collisions are rarely perfectly elastic due to energy dissipation through heat, sound, or deformation; however, the model of an elastic collision is essential for understanding fundamental mechanics.
Core Principles
For any collision (assuming an isolated system), the following are conserved:
- Linear Momentum: $\vec{P}_{initial} = \vec{P}_{final}$
- Kinetic Energy (only in elastic collisions): $K_{initial} = K_{final}$
The Derivation
Let two bodies of mass $m_1$ and $m_2$ have initial velocities $u_1$ and $u_2$, and final velocities $v_1$ and $v_2$.
1. Conservation of Momentum
$$m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2$$ Rearrange to group by mass: $$m_1(u_1 - v_1) = m_2(v_2 - u_2) \quad \text{--- (Eq. 1)}$$
2. Conservation of Kinetic Energy
$$\frac{1}{2}m_1u_1^2 + \frac{1}{2}m_2u_2^2 = \frac{1}{2}m_1v_1^2 + \frac{1}{2}m_2v_2^2$$ Rearrange to group by mass: $$m_1(u_1^2 - v_1^2) = m_2(v_2^2 - u_2^2)$$ Using the identity $a^2 - b^2 = (a-b)(a+b)$: $$m_1(u_1 - v_1)(u_1 + v_1) = m_2(v_2 - u_2)(v_2 + u_2) \quad \text{--- (Eq. 2)}$$
3. Combining the Equations
Divide (Eq. 2) by (Eq. 1): $$\frac{m_1(u_1 - v_1)(u_1 + v_1)}{m_1(u_1 - v_1)} = \frac{m_2(v_2 - u_2)(v_2 + u_2)}{m_2(v_2 - u_2)}$$ $$u_1 + v_1 = v_2 + u_2$$
Rearranging to isolate the approach and separation velocities: $$u_1 - u_2 = v_2 - v_1$$
Here, $(u_1 - u_2)$ is the relative velocity of approach and $(v_2 - v_1)$ is the relative velocity of separation. This proves they are equal in a one-dimensional elastic collision.