Inelastic collisions Knowledge-Based Study

Study notes

1. What is an Inelastic Collision?

   In physics, a collision between two or more bodies is classified based on whether mechanical kinetic energy is conserved during the interaction.

   For any isolated system with no net external forces ($\sum \vec{F}_{\text{ext}} = 0$), total linear momentum is always conserved:

   $$\vec{p}_{\text{initial}} = \vec{p}_{\text{final}}$$

   An inelastic collision is defined as a collision in which the total linear momentum is conserved, but the total kinetic energy is not conserved ($K_f < K_i$).

   The lost kinetic energy does not vanish; rather, it is transformed into other energy forms such as:

   • Thermal energy (heat generated during the collision)
   • Acoustic energy (sound waves)
   • Work done in permanent structural or plastic deformation of the colliding objects

   Objects Sticking Together

   A completely (or perfectly) inelastic collision represents the extreme case of an inelastic interaction. In this scenario, the colliding bodies stick together upon impact and move forward with a single, shared final velocity ($\vec{v}_f$).

   Consider two masses, $m_1$ and $m_2$, moving along a straight line with initial velocities $v_{1i}$ and $v_{2i}$:

   $$m_1 v_{1i} + m_2 v_{2i} = (m_1 + m_2) v_f$$

   Solving directly for the common final velocity yields:

   $$v_f = \frac{m_1 v_{1i} + m_2 v_{2i}}{m_1 + m_2}$$

   Note: Velocities are directional vector quantities. Motion in the chosen negative direction must be entered with a negative sign.

   Quantifying Energy Dissipation

   Although total energy is universally conserved, mechanical kinetic energy decreases in an inelastic event. The change in kinetic energy ($\Delta K$) is given by:

   $$\Delta K = K_f - K_i$$

   For a completely inelastic collision in one dimension:

   • Initial Kinetic Energy: $$K_i = \frac{1}{2}m_1 v_{1i}^2 + \frac{1}{2}m_2 v_{2i}^2$$
   • Final Kinetic Energy: $$K_f = \frac{1}{2}(m_1 + m_2) v_f^2$$

   Because $K_f < K_i$, $\Delta K$ is negative. The absolute value $|\Delta K|$ corresponds precisely to the internal energy increase, deformation work, and thermal energy produced by the impact.

   Solving Collision Problems

   1. Identify the system: Confirm that external forces are negligible over the brief duration of the collision so that momentum conservation holds.
   2. Establish a coordinate axis: Assign positive and negative signs consistently according to direction of motion.
   3. Set up momentum conservation: Write $m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f}$. If the objects stick together, set $v_{1f} = v_{2f} = v_f$.
   4. Analyze energy if required: Calculate $K_i$ and $K_f$ separately to determine the amount of dissipated kinetic energy. Never equate initial and final kinetic energies when solving for velocities in an inelastic collision.