Simplifying KE/Momentum Relativistic Problem

[Vorde]

Homework Statement

A D⁰ meson (with a rest mass of 1.86 GeV/c²), initially at rest, decays into a K⁰ meson (with a rest mass of .51 GeV/c²) and a $\pi$⁰ meson (with a rest mass of .12 GeV/c²). What is the Kinetic Energy of the $\pi$⁰ meson?

Homework Equations

E=$\gamma$m₀c²

p=$\gamma$m₀v

The Attempt at a Solution

Subtracting final rest energy from intial energy its easy to see that the KE(K⁰+$\pi$⁰)=1.23 GeV

Also: $\gamma$_K0m_K0c²+$\gamma$_$\pi$0m_$\pi$0c² = 1.86 GeV/c^2 (The total energy of the two Mesons must equal the first meson)
Without going through all the horrible steps me and my friend did, eventually we got a equation that solved: V_K0=f(V_$\pi$0) (I don't have the final equation we got on me)


When then plugged that back into: $\gamma$_K0m_K0c²+$\gamma$_$\pi$0m_$\pi$0c² = 1.86 GeV/c^2


Now at this point we had a solvable equation, the only variable was V_$\pi$0, but it was a hellish equation. After about 20 minutes working on trying to solve it we ran out of time and had to give up- the algebra was just to hard.
Had we had enough time, I'm completely confident we could have solved it, as it was simply a matter of foiling again and again (I think we needed to do it three levels down).
My question is: Is there a way of solving this problem without having to go in and solve for one of the variables in terms of the other, or doing that in a simpler equation?

Thank you, especially if you made it through this.

 

[jtbell]

If you don't need the velocities (i.e. you're not asked for them explicitly), don't take that route to find the energies of the outgoing particles. Instead, use

$$E^2 = (pc)^2 + (m_0 c^2)^2$$

which is true for each particle, together with conservation of energy and conservation of momentum. You never have to see a single "v".

 

[vela]

Two suggestions:

1. Stick with energy (total, not kinetic) and momentum.
2. Take advantage of $E^2 - (pc)^2 = (mc^2)^2$.

If you learn how to use four-vectors, you can get the answer in about three or four lines of simple algebra. Read Griffith's chapter on special relativity in his Introduction to Particle Physics.

 

[Vorde]

Thank you to both of you, while I haven't actually gotten the answer yet, I have gotten numerical values for both the momentum and velocity of one of the particles, I'll be able to get the answer with about 5 more minutes of work.

In addition, I know the momentum energy four vector, but I've never been given a problem requiring it so using it didn't occur to me, I am going to look at its usefulness more carefully.