Another Special Relativity Question

[Chip90]

Homework Statement

A meson is traveling with a rest mass of 500 MeV/c^2. It splits into two meson, each with rest mass of 140MeV/c^2. One Meson stops and the other one continues. Find the KE of the meson that is moving.

Homework Equations

Pi=Pf and Ei=Ef

The Attempt at a Solution

So this is another meson question I am stuck on. This is what I have

E^2=(mc^2)^2 + (pc)^2

Ei= (500^2*c^4 + c^2*Pi^2)^0.5

Ef= (140^2*c^4 + c^2*Pf^2)^0.5 + (140^2*c^4)^0.5

or do the C^2 in the denominator of the rest mass cancel with with the C^2s in the equations to give:Ei= (500^2 + c^2*Pi^2)^0.5

Ef= (140^2 + c^2*Pf^2)^0.5 + (140^2)^0.5

I think it is the second way. Once I do Pi=Pf and solve for momentum, I get P=2.46e-6 MeV/C = 2.47 eV/C

I don't think momentum should be that low correct? Also, how do I proceed to find the KE?

 

[fzero]

Momentum has units of eV/c, so you missed a factor of c in your result. Kinetic energy, T, is energy minus the mass, so

$$T= E - mc^2.$$

 

[Chip90]

right, I just forgot to type that in, I have it on my paper. But, so do the ev/C^2 and the c^2 in the equations cancel? If so, why would that momentum be so low?

also how did you get that equation in there lol? it would make typing these equations look a lot better

 

[fzero]

right, I just forgot to type that in, I have it on my paper. But, so do the ev/C^2 and the c^2 in the equations cancel? If so, why would that momentum be so low?

The c^2s that you were talking about in the energy equation do cancel (i.e. your 2nd set of equations are correct), but I think you lost a factor of c when you solved for the momentum. I find p = 740 MeV/c, which is just about c times your answer.

also how did you get that equation in there lol? it would make typing these equations look a lot better

You can click on the image to get the code for that. Some references for learning LaTex are here: https://www.physicsforums.com/showthread.php?t=386951

 

[Chip90]

Thanks for the guide; let me get back to once I resolve that equation.

 

[Chip90]

ok i figured out what i was doing wrong, I was accidently dividing by C somewhere. I am lost about that equation though. I understand it, but I only have momentum, not total energy. What did I find by solving for P?

I already knew the mc^2 part correct? (which is 140). I need to find E. So do I do:

E^2=(mc^2)^2 + (pc)^2

then find E and then use that eq?

That would give me E^2=140^2 + 740^2 = 753 MeV and KE=613 MeV. That doesn't seem correct.. How can it be more than the rest mass *c^2 for the original meson?

 

[fzero]

ok i figured out what i was doing wrong, I was accidently dividing by C somewhere. I am lost about that equation though. I understand it, but I only have momentum, not total energy. What did I find by solving for P?

I already knew the mc^2 part correct? (which is 140). I need to find E. So do I do:

E^2=(mc^2)^2 + (pc)^2

then find E and then use that eq?

That's correct.

That would give me E^2=140^2 + 740^2 = 753 MeV and KE=613 MeV. That doesn't seem correct.. How can it be more than the rest mass *c^2 for the original meson?

The original meson also had momentum 740 MeV/c, so you can check that the kinetic energy of the final state meson is not more than the energy of the original meson.

 

[Chip90]

oh ok, that makes sense, thank you!