2 heavy particles are fired towards each other in an accelerator. Particle 1 mov

Question

2 heavy particles are fired towards each other in an accelerator. Particle 1 moving to the right has a rest mass m1 of 16 proton masses, and a velocity v1 of 3c/5 in the laboratory frame. Particle 2 moving to the left has a rest mass m2 of 9 protons. They collide elastically and form a metastable particle stationary in the laboratory frame.

1. Show with conservation of total momentum before and after the collision that the velocity v2 of particle 2 is -4c/5 in the laboratory frame.

2. What is the mass of the metastable particle after the collision in units of proton mass and is mass conserved.

3. In the reference frame moving with particle 1 what is the velocity of particle 2?

4. Use conservation of mass and momentum to prove that the collision will give a particle of the same mass in this reference frame as in question 2 moving to the left with velocity v1.

Explanation / Answer

1)

For elastic Collision,

Momentum Before Collision = Momentum after collision

m1'*v1 + m2'*v2 = (m1'+m2')*V

as after collison the newly formed particle is stationary

so, V = 0

hence

m1'*v1 + m2'*v2 = (m1'+m2')*V = 0

m1'*v1 + m2'*v2 = 0

v2 = -(m1'*v1/ m2')

here we have to apply the concept of relativity as the velocity of particles is comparable to speed of light(c).

so,

m1' = m1/sqrt(1-v1^2/c^2) = 16*mp/sqrt(1-(3/5)^2) = 20*mp

m2' = m2/sqrt(1-v2^2/c^2) = 9*mp/sqrt(1-v2^2/c^2)

so,

v2 = -(20*mp*3*c/5/(9*mp/sqrt(1-v2^2/c^2)) = -(60/45)c/sqrt(1-v2^2/c^2)

solving this eq.

we get

v2 = -4c/5

2)

mass of the metastable particle = m1'+m2'

=m1/sqrt(1-v1^2/c^2)+m2/sqrt(1-v2^2/c^2)

= 16*mp/sqrt(1-(3/5)^2)+9*mp/sqrt(1-(4/5)^2)

= 20*mp + 15*mp

= 35*mp

Yes mass is conserved

3)

Velocity of 2 with respect t 1 = V21

V21 = v2*sqrt(1-v2^2/c^2)-v1*sqrt(1-v1^2/c^2)

= -4c/5*sqrt(1-(4/5)^2)-3c/5*sqrt(1-(3/5)^2)

=-096*c

4)

so in the new reference frame mass will have the same mass as it's rest mass...

so

mass in this reference frame = 15*mp+9*mp/sqrt(1-0.96^2) = 47*mp == 45*mp ..... ( the ans are very close... slight difference is due the fact i used round figure of 0.96c for V21

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