Elastic & Inelastic Collisions

• Conservation of Linear Momentum: In every collision, the total initial momentum equals the total final momentum ($m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2$), regardless of whether energy is lost.

• Kinetic Energy ($KE$) Conservation: In perfectly elastic collisions, the sum of kinetic energies before impact equals the sum after impact ($\frac{1}{2} m_1 u_1^2 + \frac{1}{2} m_2 u_2^2 = \frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2$).

• Energy Transformation: In inelastic collisions, kinetic energy is not conserved because work is done to permanently deform the objects or is converted into thermal energy and sound.

• Newton's Law of Restitution: The relative velocity of separation is proportional to the relative velocity of approach, governed by the coefficient of restitution $e = \frac{v_2 - v_1}{u_1 - u_2}$.

• Step 1: Define the System and Coordinates: Establish a positive direction (usually right) and identify all mass and initial velocity values, ensuring consistent units across all variables.

• Step 2: Apply Momentum Conservation: Set up the equation $\sum p_{initial} = \sum p_{final}$. This provides one equation with potentially two unknowns ($v_1$ and $v_2$).

• Step 3: Identify Collision Type: If the collision is elastic, use the relative velocity equation ($u_1 - u_2 = v_2 - v_1$) as a second linear equation to solve for the unknowns without needing squared terms.

• Step 4: Solve for Perfectly Inelastic Cases: If objects stick together, set $v_1 = v_2 = V_{final}$ and solve the single momentum equation: $m_1 u_1 + m_2 u_2 = (m_1 + m_2)V_{final}$.

Comparison of Collision Types

• Elastic vs. Inelastic: The primary difference lies in the conservation of kinetic energy; elastic collisions are idealized scenarios (like subatomic particles), while most macroscopic collisions are inelastic.

• Inelastic vs. Perfectly Inelastic: While both lose kinetic energy, a perfectly inelastic collision results in the objects moving with a common final velocity, representing the maximum possible energy loss consistent with momentum conservation.

• Vector Direction Awareness: Always assign signs to velocities based on direction; a ball moving left must have a negative velocity value in your equations or the momentum sum will be incorrect.

• The Relative Velocity Shortcut: For elastic collisions, using $u_1 + v_1 = u_2 + v_2$ is often much faster and less error-prone than solving the quadratic kinetic energy equation.

• Sanity Check - Energy: In any inelastic collision, the final total kinetic energy MUST be less than the initial total kinetic energy; if your calculated $KE_{final}$ is higher, you have made a calculation error.

• Sanity Check - Velocity: In a 1D collision where objects do not pass through each other, the object on the left cannot have a higher final velocity than the object on the right ($v_1 \le v_2$).