1. Not finding help here? Sign up for a free 30min 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!

Need help with Special Relativity: Force and Energy

  1. Jun 20, 2010 #1
    1. The problem statement, all variables and given/known data

    I'm trying to show that

    [tex]\frac{dE}{dT}[/tex]= F x V where F is force and V is velocity

    Can someone please help me out?


    2. Relevant equations

    F=[tex]\frac{dP}{dT}[/tex]=M x V x [tex]\gamma[/tex]


    All of the other equations from special relativity (contraction, dilation, energy) are probably important as well.

    3. The attempt at a solution

    Honestly, I have no idea where to start. Can someone please point me in the right direction. I'm a high school junior and I have very little experience with relativity. I understand that I should show some work, but I truly don't have any but I have been trying for a while.

    Please help me get started!
     
  2. jcsd
  3. Jun 20, 2010 #2

    vela

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    You should know the formula for the relativistic energy E of an object. Try differentiating that with respect to time and see what you get.
     
  4. Jun 20, 2010 #3
    The formulas for relativistic energy don't involve time (t) as far as I can tell.
    I'm not sure how taking the derivative of that would help unless I'm
    overlooking something.
     
  5. Jun 20, 2010 #4

    vela

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    It doesn't depend explicitly on time, but the energy depends on [itex]\gamma[/itex], which, in turn, depends on velocity, which does change with time.
     
  6. Jun 20, 2010 #5
    gosh, now I'm even more stuck, thanks anyway.

    so you meant that I should use [tex]\frac{x}{t}[/tex] instead of v?

    well after doing that, and taking the derivative, I can see no way for it to manipulate in F x V.
     
  7. Jun 20, 2010 #6

    vela

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    No, I think that'll make it more complicated. Keep it in terms of just the velocity v.

    Start with this first. You know that

    [tex]\gamma = \frac{1}{\sqrt{1-(v/c)^2}}[/tex]

    What do you get if you differentiate it with respect to time, remembering that v is a function of time?
     
    Last edited: Jun 20, 2010
  8. Jun 20, 2010 #7
    ok, this is what I have

    [tex]\frac{dE}{dt}[/tex]=[tex]mc^{2}\frac{-\gamma^{3}v}{c^{2}}\frac{dv}{dt}[/tex]= F x V

    am I doing this correctly? when equate the equation above with F x V (where F = gamma*mass*acceleration*v)
    i get that -[tex]\gamma^{3}=\gamma[/tex] so I get the feeling I messed up somewhere...

    I really appreciate your help!
     
    Last edited: Jun 20, 2010
  9. Jun 20, 2010 #8

    vela

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    Hmm, not sure how you came up with that. Can you show more steps?
     
  10. Jun 20, 2010 #9
    Sure,

    So here's the derivative of gamma
    [tex] \dot {\gamma} = \frac{d}{dt} \left( 1- \frac{v^2}{c^2} \right) ^{-1/2} = \left( \frac{-1}{2} \right) \left( \frac{-2v \dot {v} }{c^2} \right) \left( 1- \frac{v^2}{c^2} \right) ^{-3/2} = \gamma ^3 \left( \frac{v \dot {v}}{c^2} \right) [/tex]

    hence
    [tex]\frac{dE}{dt}=mc^{2}\frac{-\gamma^{3}v}{c^{2}}\frac{dv}{dt}[/tex]

    because E = mc[tex]^{2}\gamma[/tex].

    [tex]F\bulletV=m\gamma\frac{dv}{dt}v = \frac{dE}{dt}=mc^{2}\frac{\gamma^{3}v}{c^{2}}\frac{dv}{dt}[/tex]

    and then I simplified this
     
    Last edited: Jun 20, 2010
  11. Jun 20, 2010 #10

    vela

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    You're getting there. You need to be a bit more careful when calculating F. You know that the momentum is given by [itex]p=\gamma mv[/itex]. When you differentiate it, you need to use the product rule.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Need help with Special Relativity: Force and Energy
Loading...