A certain radioactive (parent) nucleus transforms to a different (daughter) nucleus by emitting an electron and a neutrino. The parent was at rest at the origin of an xy coordinate system. The electron moves away from the origin with linear momentum ?

Question

Amelia Adams · Accepted Answer

(a) #~~2.3049xx10^-22# #kgm/s#
(b) #~~3.7^o#
(c) #~~2.919xx 10^-19J# Initially Given that linear momentum of the parent nucleus is 0 and it is situated at #(0,0)#. After beta decay linear momentum (#m.v#) due to movement of electron #=-2.3 xx 10^-22 hati# #kg.m/s# Also linear momentum due to movement of neutrino #=-1.5 xx 10^-23 hatj# #kg.m/s# Since x and y axes are orthogonal to each other, therefore, by the law of conservation of momentum, momentum of daughter nucleus is equal and opposite to the momenta in both axes. #:.# momentum of daughter nucleus #=2.3 xx 10^-22 hati+1.5 xx 10^-23 hatj# # kg.m //s# Magnitude of Resultant momentum vector of daughter nucleus #=sqrt[(2.3 xx 10^-22)^2 +(1.5 xx 10^-23)^2]# #~~2.3049xx10^-22# #kgm/s# ........(1) Angle #theta# with the x-axis #=sin^-1[(1.5 xx 10^-23)/(2.3 xx 10^-22)]# #~~3.7^o# Velocity of the daughter nucleus, from calculated momentum in (1) #v=(2.3049xx10^-22)/(9.1 × 10^-26)m//s# Kinetic energy of the daughter nucleus #=1/2mv^2=1/2(9.1 × 10^-26)((2.3049xx10^-22)/(9.1 × 10^-26))^2# #~~2.919xx 10^-19J#

Leo Abeyta · Answer

undefined The linear momentum of the electron can be determined using conservation of momentum. Since the parent nucleus was initially at rest, the total initial momentum is zero. Therefore, the momentum of the daughter nucleus and the momentum of the emitted electron must be equal in magnitude but opposite in direction to ensure conservation of momentum. The direction of the momentum of the electron will be away from the origin, as it moves away from the point where the parent nucleus was located.