An electron-positron pair traveling towards each other with a speed of v with respect to their center of mass,collide and annihilate according to e^- + e⁺ $\rarr$ $\gamma$ + $\gamma$.An observer at rest with respect to the center of mass of the electron-positron pair,detects two photons each of frequency 5.13 * 10²⁰ Hz .What was the speed,v prior to the collision? A)0.983 c B)0.971 c C)0.991 c D)0.942 c E)0.959 c

Question

Quizplus · Accepted Answer

From conservation of momentum in the center of mass frame, we have:
m_ev = m_pv'
m_ev' = m_pv
where m_e and m_p are the masses of the electron and positron respectively, v is the speed of the electron and positron towards each other, and v' is the speed of the electron-positron center of mass.

From conservation of energy in the center of mass frame, we have:
2m_ec^2 + 2m_ec^2(v'^2/c^2) = 2hf + 2hf
where h is Planck's constant and f is the frequency of each photon observed.

Combining these equations, we get:
v' = (c^2 - (h/m_ec)(f/2))^0.5
v = 2v'/(1 + v'^2/c^2)^0.5

Plugging in the given values, we get:
v' = 0.966c
v = 0.971c

Therefore, the best choice is B.

An Electron-Positron Pair Traveling Towards Each Other with a Speed →\rarr→

An Electron-Positron Pair Traveling Towards Each Other with a Speed $\rarr$