A neutron with kinetic energy `T=2m_0c^2`, where `m_0` is its rest mass, strikes another, stationary, neutron. Determine:
(a) the combined kinetic energy `overset~T` of both neutrons in the frame of their centre of inertia and the momentum `overset~p` of each neutron in that frame,
(b) the velocity of the centre of inertia of this system of particles.
Instruction. Make use of the invariant `E^2-p^2c^2` remaining constant on transition from one inertial reference frame to another (E is the total energy of the system, p is its composite momentum).

Explanation

(b) & (a) In the CM frame, the total momentum is zero. Thus
`V/c=(cp_(1x))/(E_1+E_2)=(sqrt(T(T+2m_0c^2)))/(T+2m_0c^2)=sqrt((T)/(T+2m_0c^2))`
where we have used the result of problem
Then
`(1)/(sqrt(1-V^2//c^2))=(1)/(sqrt(1-(T)/(T+2m_0c^2)))=sqrt((T+2m_0c^2)/(2m_0c^2))`
Total energy in the CM frame is
`(2m_0c^2)/(sqrt(1-V^2//c^2))=2m_0c^2sqrt((T+2m_0c^2)/(2m_0c^2))=sqrt(2m_0c^2(T+2m_0c^2))=overset~T+2m_0c^2`
So `overset~T=2m_0c^2(sqrt(1+(T)/(2m_0c^2))-1)`
Also `2sqrt(c^2overset~p^2+m_0^2c^4)=sqrt(2m_0c^2(T+2m_0c^2)), 4c^2overset~p^2=2m_0c^2T`, or `overset~p=sqrt(1/2m_0T)`