PH253: Exam note #3

Through the first three problems, I am guessing I will have to scale the exam a bit. Number 2 was the best of the three, probably owing to the very similar practice problems.

One thing to note: the practice problem most relevant (#10) had one mass-less decay product, on the exam question that was not true ... similar problems, but not quite identical. The main difference is that E=pc only works for massless particles.

On the exam problem, since the decay products were identical in mass, and the parent started at rest, the two decay products must have equal and opposite momentum. Thus, their velocities must be equal and opposite, and their gamma factors the same. If that is true, the two decay products must have the same energy. Energy conservation then equates the rest energy of the parent (Mc^2) with the total energy of the two decay products. Since the two products have the same mass and gamma factor:

Mc^2 = \gamma_1 m_1c^2 + \gamma_2 m_2 c^2\\M = m \left(\gamma_1+\gamma_2\right) \qquad (\text{since } m_1=m_2\equiv m)\\
M = 2m\gamma \qquad (\text{since } \gamma_1=\gamma_2\equiv\gamma)\\
\frac{v}{c} = \sqrt{1-4m^2/M^2}