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Definition/Summary
For a collision between two objects, the coefficient of restitution is the ratio of the relative speed after to the relative speed before the collision.
The coefficient of restitution is a number between 0 (perfectly inelastic collision) and 1 (elastic collision) inclusive.
Equations
The coefficient of restitution is
\textrm{C.O.R.} = <br /> \frac{|\vec{v_{2f}} - \vec{v_{1f}}|}<br /> {|\vec{v_{2i}} - \vec{v_{1i}}|}
where \vec{v_{1i}} and \vec{v_{1f}} are the initial and final velocities, respectively, of object #1. A similar definition holds for the velocities of object #2.
While this is a useful definition for studying collisions of particles in physics, there is an alternative used to define the C.O.R. of everyday objects. In this definition, the velocities are replaced with the components perpendicular to the plane or line of impact. In the case of a 1-d collision, the two definitions are equivalent.
Be sure you know which definition of C.O.R. is the accepted practice in a given situation. For the remainder of this discussion, we use the definition in the equation shown above.
For an object colliding with a fixed object or surface, v_{2i} and v_{2f} are zero, and the C.O.R reduces to:
\textrm{C.O.R.} = \frac{|\vec{v_{1f}}|}{|\vec{v_{1i}}|}
In the center-of-mass reference frame of two objects of mass m_1 and m_2 -- and only in that frame -- the initial and final total kinetic energies are related to the C.O.R. by
\frac{KE_f}{KE_i} = \textrm{C.O.R}^2
where
KE_i \ = \ \frac{1}{2} m_1 v_{1i}^2 \ + \ <br /> \frac{1}{2} m_2 v_{2i}^2
and
KE_f \ = \ \frac{1}{2} m_1 v_{1f}^2 \ + \ <br /> \frac{1}{2} m_2 v_{2f}^2
Extended explanation
Elastic and perfectly inelastic collisions
The coefficient of restitution describes the inelasticity of collisions. If the C.O.R. is 1, the collision is elastic and kinetic energy is conserved. A C.O.R of zero represents a perfectly inelastic collision; after the collision the objects stick together and, in the center-of-mass frame, have zero velocity.
Simplifying 1-d collision problems
In a 1-dimensional elastic collision (C.O.R. = 1), the conservation-of-energy equation may be replaced with
v_{2f} - v_{1f}<br /> = v_{1i} - v_{2i}
In other words, the relative velocity of the two particles has the same magnitude, but is reversed in direction, before and after the collision. By using this equation instead of the conservation-of-energy equation directly, the work of solving a collision problem is simplified as there are no squared velocity terms to deal with.
* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
For a collision between two objects, the coefficient of restitution is the ratio of the relative speed after to the relative speed before the collision.
The coefficient of restitution is a number between 0 (perfectly inelastic collision) and 1 (elastic collision) inclusive.
Equations
The coefficient of restitution is
\textrm{C.O.R.} = <br /> \frac{|\vec{v_{2f}} - \vec{v_{1f}}|}<br /> {|\vec{v_{2i}} - \vec{v_{1i}}|}
where \vec{v_{1i}} and \vec{v_{1f}} are the initial and final velocities, respectively, of object #1. A similar definition holds for the velocities of object #2.
While this is a useful definition for studying collisions of particles in physics, there is an alternative used to define the C.O.R. of everyday objects. In this definition, the velocities are replaced with the components perpendicular to the plane or line of impact. In the case of a 1-d collision, the two definitions are equivalent.
Be sure you know which definition of C.O.R. is the accepted practice in a given situation. For the remainder of this discussion, we use the definition in the equation shown above.
For an object colliding with a fixed object or surface, v_{2i} and v_{2f} are zero, and the C.O.R reduces to:
\textrm{C.O.R.} = \frac{|\vec{v_{1f}}|}{|\vec{v_{1i}}|}
In the center-of-mass reference frame of two objects of mass m_1 and m_2 -- and only in that frame -- the initial and final total kinetic energies are related to the C.O.R. by
\frac{KE_f}{KE_i} = \textrm{C.O.R}^2
where
KE_i \ = \ \frac{1}{2} m_1 v_{1i}^2 \ + \ <br /> \frac{1}{2} m_2 v_{2i}^2
and
KE_f \ = \ \frac{1}{2} m_1 v_{1f}^2 \ + \ <br /> \frac{1}{2} m_2 v_{2f}^2
Extended explanation
Elastic and perfectly inelastic collisions
The coefficient of restitution describes the inelasticity of collisions. If the C.O.R. is 1, the collision is elastic and kinetic energy is conserved. A C.O.R of zero represents a perfectly inelastic collision; after the collision the objects stick together and, in the center-of-mass frame, have zero velocity.
Simplifying 1-d collision problems
In a 1-dimensional elastic collision (C.O.R. = 1), the conservation-of-energy equation may be replaced with
v_{2f} - v_{1f}<br /> = v_{1i} - v_{2i}
In other words, the relative velocity of the two particles has the same magnitude, but is reversed in direction, before and after the collision. By using this equation instead of the conservation-of-energy equation directly, the work of solving a collision problem is simplified as there are no squared velocity terms to deal with.
* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!