# Simplifying KE/Momentum Relativistic Problem

## 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: $\gamma$K0mK0c2+$\gamma$$\pi$0m$\pi$0c2 = 1.86 GeV/c^2 (The total energy of the two Mesons must equal the first meson)
Without going through all the horrible steps me and my friend did, eventually we got a equation that solved: VK0=f(V$\pi$0) (I don't have the final equation we got on me)

When then plugged that back into: $\gamma$K0mK0c2+$\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.

## Answers and Replies

jtbell
Mentor
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
Staff Emeritus
Science Advisor
Homework Helper
Education Advisor
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.

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.