InterviewSolution
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InterviewSolution
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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). |
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Answer» (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)` |
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