Question 1. Two (fictional) heavy particles are fired towards each other in a li

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Question 1.

Two (fictional) heavy particles are fired towards each other in a linear accelerator. Particle 1, mov- ing towards the right, has a rest mass m1 = 16mp (mp is the proton mass) and velocity v1 = 3c/5 in the laboratory frame. Particle 2, moving towards the left, has a rest mass m2 = 9mp. The particles collide elastically and form a metastable particle, stationary in the laboratory frame.

(a) Show, using conservation of total momentum before and after the collision, that the velocity of particle 2 in the laboratory frame is given by v2 = 4c/5.

(b) What is the mass of the metastable particle after the collision, in units of proton mass? Is mass conserved in the collision?

Now consider a reference frame co-moving with particle 1.

(c) In this reference frame, what is the velocity of particle 2?

(d) The rest mass is Lorentz invariant, and therefore independent of the choice of reference frame. Use conservation of energy and momentum in this new reference frame to prove that the collision will result in a particle of the same mass given by your answer to part (b) above, and moving to the left with velocity v1.

Question 1. Two (fictional) heavy particles are fired towards each other in a linear accelerator. Particle 1, mov- ing towards the right, has a rest mass m1 = 16mp (mp is the proton mass) and velocity v1 = 3c/5 in the laboratory frame. Particle 2, moving towards the left, has a rest mass m2 = 9mp. The particles collide elastically and form a metastable particle, stationary in the laboratory frame. (a) Show, using conservation of total momentum before and after the collision, that the velocity of particle 2 in the laboratory frame is given by v2 = 4c/5. (b) What is the mass of the metastable particle after the collision, in units of proton mass? Is mass conserved in the collision? Now consider a reference frame co-moving with particle 1. (c) In this reference frame, what is the velocity of particle 2? (d) The rest mass is Lorentz invariant, and therefore independent of the choice of reference frame. Use conservation of energy and momentum in this new reference frame to prove that the collision will result in a particle of the same mass given by your answer to part (b) above, and 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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