inelastic_relativistic_collision_sol_1

inelastic_relativistic_collision_sol_1 - inelastic...

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inelastic relativistic collision A particle of mass m , moving at speed v = 4 c /5, collides inelastically with a similar particle at rest. (a) What is the speed v C of the composite particle? (b) What is its mass m C ? Solution by Rudy Arthur: Call the moving particle ‘M’, and the particle at rest ‘R’ (the composite particle is defined to be ‘C’). The momentum of the moving particle is M 2 2 4 3 1 mv p v c = = mc M 2 . (1) And, the square of its energy is M 2 2 2 ( ) ( ) E mc p c = + . (2) The energy of the particle at rest is R 2 ( ) E mc = . (3) The square of the energy of the composite particle is C C C 2 2 2 ( ) ( 2 ) E m c p c = + . (4) By conservation of energy: , or squaring and rearranging, M R E E E + = C 2 M R C M R 2 2 2 E E E E E = (5) Substituting (2) and (4) into (5): ( ) ( ) M R C C M 2 2 2 2 2 2 2 ( ) ( ) 2( ) ( ) E E m c p c mc p c = + + By conservation of momentum, C M p p = , so this reduces to M R C 2 2 2 2 2 ( ) 2( ) E E m c mc = Squaring again: ( M R C 2 2 2 2 2
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