What Are the Threshold Kinetic Energies in Particle Decay Reactions?

[nv125]

Electrodynamics and Relativity. Please Help!

Homework Statement

Consider the decay of a particle.

(a) Find the threshold kinetic energies of the incident particles in each of
the following reactions:

(i) ∏⁺ + p →∏⁺+ ∏° + p
(ii) p + n →p + n + ∏° + ∏°

The incident particle is the first particle listed in the reactions; the target particle is at rest. The masses of the ∏⁺ and ∏° ions are 139.567MeV/c2 and 134.963 MeV/c2 respectively while the masses of the neutron and proton are respectively 939.566 MeV/c2 and 938.272 MeV/c2.

Homework Equations

The teacher suggested that I will need to use the following equations to get the solution
(Here P is the momentum and Px is the momentum of the incident particle ∏⁺in the X direction, assuming it hits the target particle p(proton) horizontally, so no P(momentum) in y or z direction)

ƩP_i^u=0

P_uP^u=m²c²= E²/C²-P²

Also said that using this notations will be useful to solve the problem

For ∏⁺, the incident particle:
P_u=(E/C, Px, 0, 0)
P^u=(E/C, -Px, 0, 0) where Px is the momentum of the incident particle, and no need to worry about y and z direction so just put zeros.

The Attempt at a Solution

This is what I have done so far to solve for part i, but I have no clue how to continue.
Since we are solving for the threshold kinetic energy, so assuming that ∏⁺, ∏° and p are at rest after the reaction.

(E/C, Px, 0, 0) +(m_pC², 0, 0, 0) - (m_∏⁺C²,0, 0, 0 ) -(m_∏°C²,0, 0, 0 ) - (m_pC²,0, 0, 0 ) = 0

m_p, m_∏⁺, m_∏° are just the rest mass.

PLEASE PLEASE help me to do this problem! thank you so much!

 

[fzero]

Since we are solving for the threshold kinetic energy, so assuming that ∏⁺, ∏° and p are at rest after the reaction.

(E/C, Px, 0, 0) +(m_pC², 0, 0, 0) - (m_∏⁺C²,0, 0, 0 ) -(m_∏°C²,0, 0, 0 ) - (m_pC²,0, 0, 0 ) = 0

As you see, trying to have all 3 final state particles at rest is impossible due to conservation of momentum. You need to allow for some final momenta and then think about what being at threshold implies.