Question

(3) A radioactive atom in a beam produced by an accelerator has a speed v with respect to the laboratory. The atom decays and ejects an electron of speed c/2 relative to itself. What is the speed of the electron relative to the laboratory, if ejected in the forward direction? If ejected in the backward direction? [10 pts]

Answer 1

Let A represents laboratory, B represents the atom, and C represents the ejected electron.

Velocity of the atom (B) relative to laboratory (A) is v_BA=v

If the electron is ejected in the forward direction, the velocity of the electron (C) relative to atom (B) is v_CB=+c/2

We want to find velocity of electron (C) relative to laboratory (A) v_CA=?

Using Einstein velocity addition rule

$v_{CA}=\frac{v_{CV}+v_{BA}}{1+v_{CV}v_{BA}/c^2}$

$v_{CA}=\frac{v+c/2}{1+vc/2c^2}$

$\boxed{v_{CA}=\frac{v+0.5c}{1+0.5v/c}}$

If the electron is ejected in the backward direction, the velocity of the electron (C) relative to atom (B) is

v_CB=-c/2

We want to find velocity of electron (C) relative to laboratory (A) v_CA=?

Using Einstein velocity addition rule

$v_{CA}=\frac{v_{CV}+v_{BA}}{1+v_{CV}v_{BA}/c^2}$

$v_{CA}=\frac{v-c/2}{1-vc/2c^2}$

$\boxed{v_{CA}=\frac{v-0.5c}{1-0.5v/c}}$