Question

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 ?

Answer 1

(a) `~~2.3049xx10^-22` `kgm/s`
(b) `~~3.7^o`
(c) `~~2.919xx 10^-19J`

Explanation:

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`