# 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.

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
Homework Helper
2. Take advantage of $E^2 - (pc)^2 = (mc^2)^2$.