1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Kinetic energy of daughter nucleus and alpha particle from alpha decay

  1. Aug 31, 2011 #1
    1. The problem statement, all variables and given/known data
    In [itex]\alpha[/itex] decay a nucleus X at rest decays to a daughter nucleus Y and an [itex]\alpha[/itex] particle. Conservation of momentum and kinetic energy gives:
    [itex]M_{\alpha}v_{\alpha}+M_{Y}v_{Y}=0[/itex]

    [itex]\frac{1}{2}M_{\alpha}v_{\alpha}^{2}+\frac{1}{2}M_{Y}v_{Y}^{2}=Q[/itex]

    Where the Q value is the available energy found through [itex]Q=(M(X)-M(Y)-M(\alpha))c^{2}[/itex]

    Show the kinetic energy of the two decay products are given by

    [itex]E^{\alpha}_{k}=\frac{M_{Y}}{(M_{Y}+M_{\alpha})}Q[/itex]

    [itex]E^{Y}_{k}=\frac{M_{\alpha}}{(M_{Y}+M_{\alpha})}Q[/itex]


    2. Relevant equations
    [itex]\frac{1}{2}M_{\alpha}v_{\alpha}^{2}+\frac{1}{2}M_{Y}v_{Y}^{2}=Q[/itex]

    [itex]M_{\alpha}v_{\alpha}+M_{Y}v_{Y}=0[/itex]

    [itex]Q=(M(X)-M(Y)-M(\alpha))c^{2}[/itex]

    [itex]E^{\alpha}_{k}=\frac{M_{Y}}{(M_{Y}+M_{\alpha})}Q[/itex]

    [itex]E^{Y}_{k}=\frac{M_{\alpha}}{(M_{Y}+M_{\alpha})}Q[/itex]

    3. The attempt at a solution
    I have tried rearranging the energy equation to get [itex]E^{\alpha}_{k}=Q-\frac{1}{2}M_{Y}v_{Y}^{2}[/itex]
    since
    [itex]E^{\alpha}_{k}=\frac{1}{2}M_{\alpha}v_{\alpha}^{2}[/itex]

    Then rearranging the conservation of momentum equation and substituting in for various variables but I can't get anything that looks like the required expressions. I know this is a fairly simple algebra exercise but I just can't figure out what to do so any advice or suggestions would be appreciated.
     
  2. jcsd
  3. Sep 2, 2011 #2
    I managed to figure this out. I rearranged the expression for Q to get [itex]E^{\alpha}_{K}[/itex]like in my initial attempt but then I took Q out as a common factor which was the key. It was then just a matter of doing about 3 pages of algebra on what was inside the brackets.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Kinetic energy of daughter nucleus and alpha particle from alpha decay
Loading...