Find a reference frame where momenta of electron and proton are equal

You still need to find the new gamma and beta to do the Lorentz transform, but that should be straightforward. And don't forget to use the correct mass for each particle.
  • #1
khfrekek92
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Homework Statement


The electron is traveling at a speed of β=.9999999, γ=1957, with mass mc^2=.51099 MeV.
The Proton is traveling at a speed of β=.9, γ=2.29, with mass mc^2=938.27 MeV.
They are heading in opposite directions, directly towards each other on the x-axis

Find the reference frame where their momenta are equivalent, and find the total energy in the frame, and the total kinetic energy.


Homework Equations


see below


The Attempt at a Solution


I'm not exactly sure if I'm going about this right but I have started a lorentz transform with the momentum 4-vectors of both the electron and proton. Then I took the second component of the resultant vectors which would be equal to γp, then I set them equal to each other and tried to solve for a velocity. Am I going about his right? All the work is shown in the attached jpeg.
 

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  • #2
I don't think your working is correct. For the electron, you've got its original momentum 4-vector in terms of [itex]\gamma_e[/itex] and ue, and then you do a Lorentz transform on that, but you use the same [itex]\gamma_e[/itex], so the momentum 4-vector you get on the right hand side is just the 4-vector for an electron at rest. (You can check by cancelling down).

You should be able to answer the question by using the same Lorentz transform on the electron and proton (to get them both into the new reference frame). But you need to use a new gamma and beta. Also, you mentioned making the spatial momenta of the electron and proton equal. But I don't think this is what the question is asking for. I think the question is asking for the spatial momenta to be equal and opposite.

One last thing: you should be able to work out the answer by using a Lorentz transform of the electron and proton 4-momenta. But you can make things easier on yourself if you think about what you are trying to find in the new reference frame, you'll see that although transforming the individual 4-momenta is the most straightforward thing to do, it is not the easiest calculation to get the answer.
 
  • #3
Oh! Yeah I have no idea what I was doing.. But I fixed my mistake and now no matter what I do I get that the speed of the reference frame is 1.000116893*c! That is definitely not possible...
And I'm not sure what you mean about making it easier, I have to find the frame where the mass*velocity is the same for the proton as it is for the electron.. hmmm..
 
  • #4
a speed greater than c? yeah, there must be something going wrong... maybe you're trying to do the transform of reference frame in the wrong direction or something like that?
About making it easier: (hint) you can do addition or subtraction on 4-vectors you already have, to get another 4-vector.
 
  • #5
I didn't even think of that! So all I have to do is sum the fourvectors and do the lorentz transform on that vector to get the the zero momentum frame?! Then use the invariant to get the speed?!
 
  • #6
Yep, just add the two fourvectors. Its easy to forget this 'trick', but it often makes questions easier.
 

1. How can you find a reference frame where the momenta of an electron and proton are equal?

To find a reference frame where the momenta of an electron and proton are equal, you can use the concept of relativistic velocity addition. This involves finding the velocity of the electron and proton in their respective rest frames, and then adding these velocities to find the velocity of the electron relative to the proton.

2. Why is it important to find a reference frame where the momenta of an electron and proton are equal?

It is important to find a reference frame where the momenta of an electron and proton are equal because this frame allows for a simplified understanding of the interaction between the two particles. In this frame, the electron and proton will have the same momentum and will appear to be moving with the same speed, making it easier to study their interactions.

3. What does it mean when the momenta of an electron and proton are equal in a reference frame?

In a reference frame where the momenta of an electron and proton are equal, it means that the two particles are moving at the same speed in opposite directions. This also implies that the magnitude of their momenta, or their momentum vectors, are equal.

4. Can the momenta of an electron and proton ever be equal in all reference frames?

No, the momenta of an electron and proton cannot be equal in all reference frames. This is because the concept of relativity states that the laws of physics should be the same in all inertial reference frames. Therefore, the momenta of the two particles can only be equal in a specific reference frame, which is typically chosen for convenience or to simplify calculations.

5. How does finding a reference frame where the momenta of an electron and proton are equal relate to the principles of relativity?

The concept of finding a reference frame where the momenta of an electron and proton are equal is an application of the principles of relativity. The principles of relativity state that the laws of physics should be the same in all inertial reference frames. Therefore, the choice of reference frame where the momenta of the two particles are equal is arbitrary, as long as all physical laws are consistent in that frame.

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