Consider the following inelastic collision. Two lumps of matter are moving direc

Question

Consider the following inelastic collision. Two lumps of matter are moving directly toward each other. Each lump has a mass of 0.500 kg, and is moving at a speed of 0.910c. The two lumps collide and stick together. Answer the following questions, keeping in mind that relativistic effects cannot be neglected in this case.

What is the final speed, in terms of c, of the combined lump? ____________c

What is the final mass of the combined lump immediately after the collision, assuming that there has not yet been significant energy loss due to radiation or fragmentation? ______________kg

Explanation / Answer

Call the moving particle ‘M’, and the particle at rest ‘R’. (The composite particle is defined to be ‘C’.)

The momentum and energy of the particle at rest are

PR = 0 and ER = m

The momentum of the moving particle is

PM = mv/sqrt(1-v2) = 0.910m/sqrt(1-0.910^2) = 2.195m

And it’s energy is

EM = PM/v = (2.195m)/0.910 = 2.41m

For the composite particle, the conservation of energy implies that

EC= EM + ER = 2.41m + m = 3.41m

while the conservation of momentum implies that    

PC=PM = 2.195m

The speed of the composite particle is

vC = PC/EC = (2.195m)/( 3.41m) = 0.6437

Thus vC = 0.6437c

The mass of the composite particle is given by the (positive) solution to

mC2 = EC2 - PC2 = (3.41m)^2 – (2.195m)^2

mC = 2.601m

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