The Reality of the Crash
While "elastic" collisions (where objects bounce perfectly and retain all energy) are useful theoretical models, almost all macroscopic collisions in the real world are inelastic.
In an inelastic collision, the objects collide, and kinetic energy is permanently lost from the system. Where does the energy go? It is transformed. A car crash converts massive kinetic energy into the horrifying sound of the impact, the thermal heat of grinding metal, and the permanent physical deformation of the chassis.
A Perfectly Inelastic Collision is the extreme end of this spectrum. This occurs when two objects collide, completely fail to bounce, and instead lock together to move as a single combined mass (like a bullet embedding in a block of wood, or two train cars coupling). This type of collision results in the maximum possible loss of kinetic energy.
The Rule of Momentum
It is critical to remember that while kinetic energy is lost in an inelastic collision, momentum is always perfectly conserved. The total momentum ($mv$) before the crash mathematically dictates exactly how fast the combined wreckage will move after the crash.
The Formulas
To fully understand a perfectly inelastic collision, we must calculate two things: the final velocity of the combined wreckage, and exactly how much kinetic energy was destroyed in the impact.
Example Calculation
A $1,000 , \text{kg}$ car running a red light at $20 , \text{m/s}$ T-bones a stationary $1,500 , \text{kg}$ SUV. They crumple and lock together.
- Total Initial Momentum: $(1000 \cdot 20) + (1500 \cdot 0) = 20,000 , \text{kg}\cdot\text{m/s}$.
- Final Velocity ($v_f$): $20,000 / (1000 + 1500) = \mathbf{8.0 , \text{m/s}}$. The locked wreckage slides across the intersection at $8 , \text{m/s}$.
- Initial Kinetic Energy: $0.5 \cdot 1000 \cdot (20^2) = \mathbf{200,000 , \text{Joules}}$.
- Final Kinetic Energy: $0.5 \cdot 2500 \cdot (8^2) = 0.5 \cdot 2500 \cdot 64 = \mathbf{80,000 , \text{Joules}}$.
- Energy Lost: $200,000 - 80,000 = \mathbf{120,000 , \text{Joules}}$.
In that split second, 120,000 Joules of pure kinetic energy was violently transformed into heat, sound, and crushed steel.