Definitions

# Inelastic collision

An inelastic collision is a collision in which kinetic energy is not conserved (see elastic collision).

In collisions of macroscopic bodies, some kinetic energy is turned into vibrational energy of the atoms, causing a heating effect.

The molecules of a gas or liquid rarely experience perfectly elastic collisions because kinetic energy is exchanged between the molecules' translational motion and their internal degrees of freedom with each collision. At any one instant, half the collisions are – to a varying extent – inelastic (the pair possesses less kinetic energy after the collision than before), and half could be described as “super-elastic” (possessing more kinetic energy after the collision than before). Averaged across an entire sample, molecular collisions are elastic.

Inelastic collisions may not conserve kinetic energy, but they do obey conservation of momentum. Simple ballistic pendulum problems obey the conservation of kinetic energy only when the block swings to its largest angle.

In nuclear physics, an inelastic collision is one in which the incoming particle causes the nucleus it strikes to become excited or to break up. Deep inelastic scattering is a method of probing the structure of subatomic particles in much the same way as Rutherford probed the inside of the atom (see Rutherford scattering). Such experiments were performed on protons in the late 1960s using high-energy electrons at the Stanford Linear Accelerator (SLAC). As in Rutherford scattering, deep inelastic scattering of electrons by proton targets revealed that most of the incident electrons interact very little and pass straight through, with only a small number bouncing back. This indicates that the charge in the proton is concentrated in small lumps, reminiscent of Rutherford's discovery that the positive charge in an atom is concentrated at the nucleus. However, in the case of the proton, the evidence suggested three distinct concentrations of charge (quarks) and not one.

## Formulas

The formulas for the velocities after a one-dimensional collision are:

$V_\left\{1f\right\}=frac\left\{\left(C_R + 1\right)M_\left\{2\right\}V_2+V_\left\{1\right\}\left(M_1-C_R M_2\right)\right\}\left\{M_1+M_2\right\}$
$V_\left\{2f\right\}=frac\left\{\left(C_R + 1\right)M_\left\{1\right\}V_1+V_\left\{2\right\}\left(M_2-C_R M_1\right)\right\}\left\{M_1+ M_2\right\}$

where

$V_\left\{1f\right\}$ is the final velocity of the first object after impact
$V_\left\{2f\right\}$ is the final velocity of the second object after impact
$V_\left\{1\right\}$ is the initial velocity of the first object before impact
$V_\left\{2\right\}$ is the initial velocity of the second object before impact
$M_\left\{1\right\}$ is the mass of the first object
$M_\left\{2\right\}$ is the mass of the second object
$C_\left\{R\right\}$ is the coefficient of restitution; if it is 1 we have an elastic collision; if it is 0 we have a perfectly inelastic collision, see below.

In a center of momentum frame the formulas reduce to:

$V_\left\{1f\right\}=-C_R V_\left\{1\right\}$
$V_\left\{2f\right\}=-C_R V_\left\{2\right\}$

For two- and three-dimensional collisions the velocities in these formulas are the components perpendicular to the tangent line/plane at the point of contact.

## Perfectly inelastic collision

In a perfectly inelastic collision , i.e., a zero coefficientcient of restitution, the colliding particles stick together. We have due to conservation of momentum:

$m_1 mathbf v_\left\{1,i\right\} + m_2 mathbf v_\left\{2,i\right\} = left\left(m_1 + m_2 right\right) mathbf v_f ,$
hence the final velocity is
$mathbf v_f=frac\left\{m_1 mathbf v_\left\{1,i\right\} + m_2 mathbf v_\left\{2,i\right\}\right\}\left\{m_1 + m_2\right\}$

The reduction of total kinetic energy is equal to the total kinetic energy before the collision in a center of momentum frame with respect to the system of two particles, because in such a frame the kinetic energy after the collision is zero. In this frame most of the kinetic energy before the collision is that of the particle with the smaller mass. In another frame, in addition to the reduction of kinetic energy there may be a transfer of kinetic energy from one particle to the other; the fact that this depends on the frame shows how relative this is.

With time reversed we have the situation of two objects pushed away from each other, e.g. shooting a projectile, or a rocket applying thrust (compare the derivation of the Tsiolkovsky rocket equation).