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In classical mechanics, collision problems are concerned with the relation of the magnitudes and direction of the velocities of colliding bodies after collision to the velocity vectors of the bodies before collision. When the only forces on the colliding bodies are those exerted by the bodies themselves, the principle of conservation of momentum states that the total momentum of the system is unchanged in the collision process. This result is particularly useful when the forces between the colliding bodies act only during the instant of collision. The velocities can then change only during the collision process, which takes place in a short time interval. Under these conditions the forces can be treated as impulsive forces, the effects of which can be expressed in terms of an experimental parameter known as the coefficient of restitution, which is discussed later. See also: Conservation of momentum; Impact

The study of collisions of molecules, atoms, and nuclear particles is an important field of physics. Here the object is usually to obtain information about the forces acting between the particles. The velocities of the particles are measured before and after collision. Although quantum mechanics instead of classical mechanics should be used to describe the motion of the particles, many of the conclusions of classical collision theory are valid. See also: Physics; Quantum mechanics; Scattering experiments (atoms and molecules); Scattering experiments (nuclei)

Classification

Collisions can be classed as elastic and inelastic. In an elastic collision, mechanical energy is conserved; that is, the total kinetic energy of the system of particles after collision equals the total kinetic energy before collision. For inelastic collisions, however, the total kinetic energy after collision is different from the initial total kinetic energy. See also: Deep inelastic collisions; Energy

In classical mechanics, the total mechanical energy after an inelastic collision is ordinarily less than the initial total mechanical energy, and the mechanical energy which is lost is converted into heat. However, an inelastic collision in which the total energy after collision is greater than the initial total energy sometimes can occur in classical mechanics. For example, a collision can cause an explosion which converts chemical energy into mechanical energy. In molecular, atomic, and nuclear systems, which are governed by quantum mechanics, the energy levels of the particles can be changed during collisions. Thus these inelastic collisions can involve either a gain or loss in mechanical energy. See also: Explosive; Heat

Consider a one-dimensional collision of two particles in which the particles have masses m₁ and m₂ and initial velocities u₁ and u₂. If they interact only during the collision, an application of the principle of conservation of momentum yields Eq. (1),

(1)

where ν₁ and ν₂ are the velocities of m₁ and m₂, respectively, after collision.

Coefficient of restitution

It has been found experimentally that in collision processes, the following Eq. (2) holds:

(2)

In the equation, e is a constant known as the coefficient of restitution, the value of which depends on the properties of the colliding bodies. The magnitude of e varies from 0 to 1. A coefficient of restitution equal to 1 indicates a perfectly elastic collision, a theoretically possible, yet impractical-in-reality collision where no net conversion of kinetic energy into other forms occurs. A coefficient of restitution equal to zero indicates a perfectly inelastic collision, where the colliding bodies stick together after collision. Examples of perfectly inelastic collisions include two colliding balls of putty or a bullet fired into a wooden block. Equations (1) and (2) can be solved for the unknown velocities ν₂ and ν₁ in the one-dimensional collision of two particles.

The concept of coefficient of restitution can be generalized to treat collisions involving the plane motion of smooth bodies—both of particles and larger bodies for which rotation effects must be considered. For these collisions, experiments show that the velocity components to be used in Eq. (2) for e are the components along the common normal to the surfaces of the bodies at the point where they make contact in the collision. For smooth bodies, the velocity components perpendicular to this direction are unchanged. Use of this result and the principle of conservation of momentum is sufficient to solve two-dimensional collision problems of smooth bodies. For collisions of smooth spheres the velocity components to be used in Eq. (2) for e are those on the line joining the centers of the spheres. Velocity components perpendicular to this direction are unchanged.

Center-of-mass coordinates

A simplification of the description of both classical and quantum me­chanical collisions can be obtained by using a co­ordinate system which moves with the velocity of the center of mass before collision. (Since for an iso­lated system the center of mass of the system can be shown to be unaccelerated at all times, the velocity of the center of mass of the system of particles does not change during collision.) The coordinate system which moves with the center of mass is called the center-of-mass system, while the stationary system is the laboratory system.

The description of a collision in the center-of-mass system is simplified because in this coordinate system the total momentum is equal to zero, both before and after collision. In the case of a two-particle collision the particles therefore must be oppositely directed after collision, and the magnitude of one of the velocities in the center-of-mass system can be determined if the other magnitude is known. See also: Center of mass

Paul W. Schmidt