Kaon Decay -- Special relativity

[Sophia Su]

Homework Statement

A K0 particle is unstable and has a mass of 8.87x10-28 kg. It can decay into π+ and π- particles, each of mass 2.49x10-28 kg. Suppose that a K0 is moving in the +x direction and decays by this process, with the π+ moving off at 0.9c and the π- moving off at 0.8c.
a) What was the speed of the K0 before the decay?
b) In what directions do each of the π particles move after the decay?

Homework Equations

Pinitial=P1+P2

The Attempt at a Solution

I assume that the momentum is conserved thus Pinital=P1+P2. I also know that P=gamma*mass*velocity
So with that information, I can solve for the momentum of the two particles. P1+P2=2.12*10^-19
Now I know my total momentum, thus I isolate for velocity. HOWEVER, the original mass of the kaon is 8.87*10^-28, but the masses of the two particles do not add up to the total mass, where did the lost mass go? Energy? If so, how do I solve th problem?

Also, for part b) do I just say, they are in opposite directions?

 

[BvU]

Hello Sophia, welcome to PF !

You'll have to do momentum conservation in two directions (longitudinal and transverse).
And (b) is right in the Kaon rest frame, but not in the lab frame !
In that rest frame the leftover energy from the decay is converted into kinetic energy of the pions. Equally - from momentum conservation

 

[jtbell]

the masses of the two particles do not add up to the total mass, where did the lost mass go? Energy?

Correct. More precisely, some of the rest-energy (mc²) of the kaon becomes part of the kinetic energy of the pions. Total energy is conserved. Therefore you have three conservation equations:

• Conservation of energy
• Conservation of x-momentum
• Conservation of y-momentum

Set them up and identify your unknown quantities.

 

[Sophia Su]

Hello Sophia, welcome to PF !

You'll have to do momentum conservation in two directions (longitudinal and transverse).
And (b) is right in the Kaon rest frame, but not in the lab frame !
In that rest frame the leftover energy from the decay is converted into kinetic energy of the pions. Equally - from momentum conservation

Thank you!

 

[Sophia Su]

Correct. More precisely, some of the rest-energy (mc²) of the kaon becomes part of the kinetic energy of the pions. Total energy is conserved. Therefore you have three conservation equations:

• Conservation of energy
• Conservation of x-momentum
• Conservation of y-momentum

Set them up and identify your unknown quantities.

That was very helpful! :)