Relativity Science Calculator - Impulse and Linear Momentum

Impulse and Conservation of Linear Momentum

center_mass_frame.png

It is best to transform the inertial frame of reference of m2.png such that it's at rest ( i.e., v2=0.png ) and accordingly adjust the relative velocity of m1.png, which is v1.png, in order to accommodate this transformation.

center_mass_frame_transform.png

§ Impulse - Momentum Theorem:

impulse_momentum_theorem.png

§ 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:

corollary.png

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, S_prime.png.

§ 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:

coefficient_restitution.png

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

case1.png

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

case2.png

§ Definitions:

(i). System Center of Mass ( ):

definition_center_mass.png

In general:

center_mass_generalized.png

(ii). Center of Mass velocity:

definition_cm_velocity.png

(iii). Momentum and Center of Mass Velocity:

definition_total_momentum.png

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 mass

2). total momentum is always conserved; therefore, center of mass velocity will always be the same,

notwithstanding any internal collisions:

momentum_conserved.png

notice: both mass and momentum are conserved.

(iv). Galilean velocity transformation:

galilean_velocity_transform.png.

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

modified_newton_3rd_law.png,

where Newton's 3rd Law of Motion of opposite but not necessarily equal velocities is modified owing to coefficient_restitution2.png, coefficient of restitution.

Proof:

i.png

newton_modified1.png

ii.png

newton_modified2.png

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

k.e._in_cm.png

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

k.e._maximum.png

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 ):

derivation1.png

i.png

derivation1a.png

ii.png

derivation1b.png

(ii). System energy loss:

k.e._loss.png

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

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

perfectly_elastic.png

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

perfectly_inelastic.png

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.png

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 ):

imperfectly_elastic.png

4). Summary:

Types of Collisions and their Respective Kinetic Energies
Type Kinetic Energy Coefficient of Restitution
Perfectly Elastic conserved ε=1
Partially Elastic not conserved 0< ε <1
Perfectly Inelastic maximum possible k.e. loss ε=0
Hyperelastic k.e. gained ε > 1

(iv). Assume v_cm.png = 0:

v_cm_transformed.png

It is best to transform the center of mass inertial frame of reference so that it's at rest ( i.e., v_cm.png = 0 ) and accordingly adjust the relative velocity of m1.png which is v1.png to accommodate this transformation while maintaining v2=0.png.

kinetic_energy2.png

(v). final determination of m2.png:

m_2_final_derivation.png