section_path: Definition in classical mechanics/Application to collisions/Inelastic collisions
section_name: Inelastic collisions
paragraph_text: Inelastic collisions
  
  In an inelastic collision, some of the kinetic energy of the colliding bodies is converted into other forms of energy (such as heat or sound). Examples include traffic collisions, in which the effect of loss of kinetic energy can be seen in the damage to the vehicles; electrons losing some of their energy to atoms (as in the Franck–Hertz experiment); and particle accelerators in which the kinetic energy is converted into mass in the form of new particles.
  
  In a perfectly inelastic collision (such as a bug hitting a windshield), both bodies have the same motion afterwards. A head-on inelastic collision between two bodies can be represented by velocities in one dimension, along a line passing through the bodies. If the velocities are v and v before the collision then in a perfectly inelastic collision both bodies will be travelling with velocity v after the collision. The equation expressing conservation of momentum is:
  
  m A v A 1 + m B v B 1 = (m A + m B) v 2. {\displaystyle {\begin{aligned}m_{A}v_{A1}+m_{B}v_{B1}&=\left(m_{A}+m_{B}\right)v_{2}\,.\end{aligned}}}
  
  If one body is motionless to begin with (e.g. u 2 = 0 {\displaystyle u_{2}=0}), the equation for conservation of momentum is
  
  m A v A 1 = (m A + m B) v 2, {\displaystyle m_{A}v_{A1}=\left(m_{A}+m_{B}\right)v_{2}\,,}
  
  so
  
  v 2 = m A m A + m B v A 1. {\displaystyle v_{2}={\frac {m_{A}}{m_{A}+m_{B}}}v_{A1}\,.}
  
  In a different situation, if the frame of reference is moving at the final velocity such that v 2 = 0 {\displaystyle v_{2}=0}, the objects would be brought to rest by a perfectly inelastic collision and 100% of the kinetic energy is converted to other forms of energy. In this instance the initial velocities of the bodies would be non-zero, or the bodies would have to be massless.
  
  One measure of the inelasticity of the collision is the coefficient of restitution C, defined as the ratio of relative velocity of separation to relative velocity of approach. In applying this measure to a ball bouncing from a solid surface, this can be easily measured using the following formula:
  
  C R = bounce height drop height. {\displaystyle C_{\text{R}}={\sqrt {\frac {\text{bounce height}}{\text{drop height}}}}\,.}
  
  The momentum and energy equations also apply to the motions of objects that begin together and then move apart. For example, an explosion is the result of a chain reaction that transforms potential energy stored in chemical, mechanical, or nuclear form into kinetic energy, acoustic energy, and electromagnetic radiation. Rockets also make use of conservation of momentum: propellant is thrust outward, gaining momentum, and an equal and opposite momentum is imparted to the rocket.