Question

please answer 11 and 12 because they are connected to the same information given

The Proton would decay to a neutral Pion and a Positron, there are no additional or missing particles. Both the Pion and the Positron are relativistic even when the initial Proton is at rest. Assume that the Proton starts at rest. The Proton has a rest mass of 938 MeV/c the neutral Pion has a rest mass of 135 MeV/c and the Positron has a rest mass of 0.511 MeV/c.

Answer 1

Solution:

Part (11): Gamma factor of electron or positron in the given reaction means energy of electron or positron. So, for the given reaction, the energy of electron or positron is given by

$E_{e^{+1}}=\frac{c^{2}}{2M_{\pi ^{0}}}\left [ {M_{\pi ^{0}}}^{2}+{M_{e^{+1}}}^{2}- {M_{e^{-1}}}^{2} \right ]\\\\ \Rightarrow E_{e^{+1}}=\frac{c^{2}}{2M_{\pi ^{0}}}\left [ {M_{\pi ^{0}}}^{2} \right ]\\\\ \Rightarrow E_{e^{+1}}=\frac{c^{2}M_{\pi ^{0}}}{2}\\\\ \left ( \because M_{\pi ^{0}}=135MeV/c^{2} \right )\\\\ \Rightarrow E_{e^{+1}}=\frac{135}{2}MeV\\\\ \Rightarrow E_{e^{+1}}=67.5MeV\\\\$ (1)

Part (12): The velocity of electron in the reaction is given by

$v_{e}=c\times \left [ \frac{{M_{\pi ^{0}}}^{2}}{4{M_{e^{-1}}}^{2}+{M_{\pi ^{0}}}^{2}} \right ]^{1/2}\\\\ \left ( \because M_{\pi ^{0}}=135MeV/c^{2} \ and \ M_{e^{-1}}=0.511MeV/c^{2} \right )\\\\ \Rightarrow v_{e}=c\times \left [ \frac{135^{2}}{4\times 0.511^{2}+135^{2}} \right ]^{1/2}\\\\ \Rightarrow v_{e}=0.99997c\\\\$ (2)