Impulse and Conservation of Linear Momentum
It is best to transform the inertial frame of reference of such that it's at rest ( i.e., ) and accordingly adjust the relative velocity of , which is , in order to accommodate this transformation.
§ Corollaries: Where there is no net external force acting upon the "Center of Mass Frame" system, , the following laws of conservation are relevant:
That is, the "before and after" mass is conserved as well as energy and momentum are conserved for the condition where there is no net external force applied to the "Center of Mass Frame" system, .
§ Derivation of Coefficient of Restitution ( or Elasticity ):
§ Coefficient of Restitution ( or Elasticity ):
However, some energy is generally dissipated or lost in any collision(s) between masses where this lost k.e. ( kinetic energy ) is found in body distortion, internal molecular or atomic motion, heat, sound, radiation and so forth.
1). Define Coefficient of Restitution:
2). Case of no energy lost ( perfectly elastic ):
3). Case of some energy lost ( imperfectly elastic or inelastic ):
§ Definitions:
(i). System Center of Mass ( ):
In general:
(ii). Center of Mass velocity:
(iii). Momentum and Center of Mass Velocity:
What this means is that:
1). total momentum in a system of sundry particles is equal to the momentum of a single equivalent particle of
2). total momentum is always conserved; therefore, center of mass velocity will always be the same, notwithstanding any internal collisions:
notice: both mass and momentum are conserved.
(iv). Galilean velocity transformation:
It can be shown that for collisions in the Center of Mass Frame
,
where Newton's 3rd Law of Motion of opposite but not necessarily equal velocities is modified owing to , coefficient of restitution.
Proof:
(v). Kinetic energy in the center of mass:
(vi). Maximum kinetic energy loss of a system of particles:
What this signifies is that for any system of particles the maximum energy loss will never be more than the kinetic energy of its center of mass frame which is equivalent to the kinetic energy of the particles contained within the center of mass frame itself.
(i). Equivalent velocities conserving energy and momentum ( see: elasticity derivation above ):
(ii). System energy loss:
(iii). System energy loss and Coefficient of Restitution:
1). Case of no energy lost ( perfectly elastic collision ):
2). Case of all energy lost ( perfectly inelastic collision ):
In this latter case of a totally inelastic collision, all of the particles stick together in one cohered lump, so to speak, and the final velocity of this cohered lump is equal to the velocity of the center of mass frame where there is the maximum possible loss of kinetic energy,
note: see above: § Definitions  (vi). Maximum kinetic energy loss of a system of particles.
3). Case of some energy lost ( partially or imperfectly elastic collision ):
4). Summary:
Types of Collisions and their Respective Kinetic Energies  

Type  Kinetic Energy  Coefficient of Restitution  
Perfectly Elastic  ε=1  
Partially Elastic  0< ε <1  
Perfectly Inelastic  ε=0  
Hyperelastic  ε > 1 
(iv). Assume = 0:
It is best to transform the center of mass inertial frame of reference so that it's at rest ( i.e., = 0 ) and accordingly adjust the relative velocity of which is to accommodate this transformation while maintaining .
(v). final determination of :
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