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Derive a formula for momentum in terms of kinetic energy

  1. Oct 14, 2010 #1
    1. The problem statement, all variables and given/known data

    Using:
    particle velocity, beta
    particle momentum, p
    total energy, E
    Lorentz factor, gamma
    kinetic energy, KE

    Derive an equation for momentum as a function of kinetic energy. The functions have to depend either on the variable in the bracket, p(KE), or on a constant.

    3. The attempt at a solution

    This is what I've done so far, and I am now stuck, and unsure if the way I am doing it is correct or if there is a different approach.


    [tex]E^{2} = p^{2}c^{2} + m^{2}c^{4}[/tex]

    [tex]KE = E - m_{0}c^{2}[/tex]

    [tex]KE = \sqrt{p^{2}c^{2} + m^{2}c^{4}} - m_{0}c^{2}[/tex]

    [tex]p^{2} = \frac{KE^{2}}{c^{2}} - m^{2}c^{2} - m_{0}^{2}c^{4}[/tex]

    The only thing I could think of doing next is:

    [tex]KE = \frac{p^{2}}{2m_{0}} , m_{0} = \frac{p^{2}}{2KE}[/tex]

    [tex] p^{2} = \frac{KE}{c^{2}} - m^{2}c^{2} - \frac{p^{4}}{4KE^{2}}c^{2}[/tex]

    [tex]p^{2} + \frac{p^{4}}{4KE^{2}}c^{2} = \frac{KE}{c^{2}} - m^{2}c^{2}[/tex]

    [tex]p^{2}(1 + \frac{p^{2}}{4KE^{2}}c^{2}) = \frac{KE}{c^{2}} - m^{2}c^{2}[/tex]

    I'm not sure if this is the best or easiest way to do this, as it seems to be pretty messy, and I also have one more m in the equation that I need to get rid of but am not sure of the best way of doing so.

    Any help will be greatly appreciated :)
     
  2. jcsd
  3. Oct 15, 2010 #2
    Ok, so I've been working on this problem for about 24 hours and I think I'm finally getting somewhere with it. In class we were given a sheet of useful formulae, and this included:

    [tex] p = \gamma \beta m_{0} c = \frac{m_{0} \beta c}{\sqrt{1 - \beta^{2}}} [/tex]

    [tex] = \frac{\sqrt{E_{tot}^{2} - m_{0}^{2}c^{4}}}{c} [/tex]

    From this final equation, I noticed that

    [tex] KE = \sqrt{E_{tot}^{2} - m_{0}^{2}c^{4}} [/tex]

    So this means that I have the relation:

    [tex]p = \frac{KE}{c} [/tex]

    Which is momentum which is only dependent on KE or a constant!

    The only problem I have now is working out where that equation from p comes from, can anybody help?
     
  4. Oct 15, 2010 #3

    kreil

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    Gold Member

  5. Oct 15, 2010 #4
    Great, finally I've figured it all out! Thank you for your help! :)
     
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