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!

Solving system of equations to find angle and energy of Au atom and α

  1. Mar 12, 2013 #1
    1. The problem statement, all variables and given/known data

    An alpha particle traveling with a kinetic energy of 5.5 MeV and a rest-mass of 3727.8 MeV/c^2 strikes a gold atom with a rest-mass of 183,476 MeV/c^2.

    -The gold atom is initially at rest
    -The alpha particle deflects perpendicular to the horizontal in the after state

    4gkewl.jpg

    2. Relevant equations

    Conservation of Energy:

    Ebefore=Eafter


    Conservation of momentum:

    Pbefore=Pafter


    E^2=P^2c^2+(mc^2)^2


    3. The attempt at a solution

    Total Energy Before Collison:

    Ebefore=E(α)+E(Au)=3727.8MeV+5.5MeV+183476MeV=187,209MeV


    Momentum before (in x-direction):

    E^2=p^2c^2+(mc^2)^2

    ==> (3727.8+5.5)^2=P^2c^2+(3727.8)^2

    ==> P=202.6 MeV/c


    AFTER STATE:

    Conservation of momentum:

    in x: P(Au)cosθ=202.6MeV/c (1)

    in y: P(Au)sinθ-P(α)=0 (2)


    Conservation of Energy:

    E(Au)=√(P(Au)^2+183476^2)
    E(α)=√(P(α)^2+3727.8^2)

    ==> √(P(Au)^2+183476^2)+√(P(α)^2+3727.8^2)=187,209 MeV (3)



    Solve the system of equations to find θ, P(Au), and P(α):

    Solve (3) for P(α):

    P(α)=√(1.36998E9-P(Au)^2)


    Plug back into Eq. (2):

    P(Au)sinθ=√(1.36998E9-P(Au)^2)


    Solve (1) for θ:

    θ=arccos(202.6/P(Au))


    Plug into (2) and solve for Au:

    P(Au)sin[arccos(202.6/P(Au))]=√(1.36998E9-P(Au)^2)

    ==> P(Au)=26,172.7 MeV/c


    Solve for θ:

    26,172.7cosθ=202.6

    ==> θ=89.56°


    Solve for P(α):

    26,172.7sin(89.56°)=P(α)

    ==> P(α)=26,171.9 MeV/c



    However, when I plug these numbers back into equation (3) the solution doesn't come out right. I have been looking over this for hours and can't figure out where my error might be.

    Thanks!
     
  2. jcsd
  3. Mar 13, 2013 #2

    Simon Bridge

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    Possibly algebra problems?
    Go through each step carefully - without going too deep, the following stood out:
    Are you missing a c^2 in there? Typo?
    #3 was:
    ... is there a c^2 missing there too?
    How did you get from this to your conclusion
    Have I read that right:
    ##p_{Au}\sin(\arccos(202.6/p_{Au}))=\sqrt{1.36998\times 10^9 - p_{Au}} \\ \Rightarrow p_{Au}=26172.7\text{MeV/c}##
    ... how did you manage to ull out the ##p_{Au}## ?

    ... I'm not saying these are the mistakes - only that this is what stands out.
     
  4. Mar 13, 2013 #3
    I looked back over each step carefully. I must have made my mistake when solving for Pα in Equation 3. I ended up writing a script on MATLAB that solved for the three unknowns and got the answers that way. Thanks for the tips!
     
  5. Mar 13, 2013 #4

    Simon Bridge

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    That's where I think it first went pear shaped too ... the c^2 thing you can get away with by adopting units so that c=1. Many people do that implicitly.

    A good discipline is to do all the algebra in the symbols, leaving the numbers to last.
    Worst case is, you'll have a clear path to backtrack and troubleshoot - in this case, a lot was hidden when you combined numbers so I couldn't tell what you'd done.

    Still... no worries aye ;)
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Solving system of equations to find angle and energy of Au atom and α
Loading...