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conservation of linear momentum

Impulse and Conservation of Linear Momentum

center of mass frame

It is best to transform the inertial frame of reference of conservation of linear momentum such that it's at rest ( i.e., v2=0.png ) and accordingly adjust the relative velocity of conservation of linear momentum, which is conservation of linear momentum, in order to accommodate this transformation.

transform center of mass frame

§ Impulse - Momentum Theorem:

impulse-momentum theorem

§ Corollaries: Where there is no net external force acting upon the "Center of Mass Frame" system, S_prime.png, the following laws of conservation are relevant:

impulse-momentum corollary

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, conservation of linear momentum.

§ Derivation of Coefficient of Restitution ( or Elasticity ):

conservation of linear momentum

§ 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:

restitution coefficient

2). Case of no energy lost ( perfectly elastic ):

imperfectly elastic or inelastic

3). Case of some energy lost ( imperfectly elastic or inelastic ):

perfectly elastic

§ Definitions:

(i). System Center of Mass ( conservation of linear momentum ):

definition center of mass cm

In general:

inertial frame center of mass

(ii). Center of Mass velocity:

definition cm center of mass velocity

(iii). Momentum and Center of Mass Velocity:

definition total momentum

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 conservation of linear momentum

2). total momentum is always conserved; therefore, center of mass velocity conservation of linear momentum will always be the same, notwithstanding any conservation of linear momentum internal collisions:

momentum conserved

notice: both mass and momentum are conserved.

(iv). Galilean velocity transformation:


It can be shown that for collisions in the Center of Mass Frame 

newton's 3rd law,

where Newton's 3rd Law of Motion of opposite but not necessarily equal velocities is modified owing to  elastic restitution coefficient, coefficient of restitution.


conservation of linear momentum

isaac newton laws

conservation of linear momentum

newton's 3rd law of action-reaction

(v). Kinetic energy in the center of mass:

conserved k.e. kinetic energy at cm center of conserved mass

(vi). Maximum kinetic energy loss of a system of particles:

maximum conserved k.e. kinetic energy

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.

§ Derivations:

(i). Equivalent velocities conserving energy and momentum ( see: elasticity derivation above ):

imperfect collision elasticity

conservation of linear momentum

perfect collision elasticity

conservation of mass

conservation of momentum

(ii). System energy loss:

k.e. kinetic energy loss at cm center of mass collision

(iii). System energy loss and Coefficient of Restitution:

1). Case of no energy lost ( perfectly elastic collision ):

perfectly elastic collision

2). Case of all energy lost ( perfectly inelastic collision ):

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,

k.e. maximum 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 center of mass velocity = 0:

transformed cm center of mass velocity zero

It is best to transform the center of mass inertial frame of reference so that it's at rest ( i.e., velocity cm center of mass = 0 ) and accordingly adjust the relative velocity of conservation of linear momentum which is cm center of mass velocity to accommodate this transformation while maintaining conservation impulse-momentum.

kinetic energy k.e.

(v). final determination of mass:

conservation of linear momentum

conservation of linear momentum

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