Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Velocity with respect to acceleration

  1. Jul 3, 2009 #1
    Is it possible to differentiate a function with respect to acceleration where the function is expressed in terms of velocity?

    [tex]\frac{dy}{da} = \frac{d}{da}{\frac{1}{\sqrt{1 - v^2/c^2}}}[/tex]
     
  2. jcsd
  3. Jul 3, 2009 #2

    arildno

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    Dearly Missed

    Well, if say the acceleration/time-relationship is invertible, so that time may be regarded as a function of the acceleration, we would have:
    [tex]\frac{dv}{da}=\frac{dv}{dt}\frac{dt}{da}=a\frac{dt}{da}=\frac{a}{\frac{da}{dt}}[/tex]

    Thus, the derivative of velocity wrt. to acceleration is the fraction between the acceleration itself and its rate of change.
     
  4. Jul 3, 2009 #3
    Thanks for your reply.
    I tried to keep my question simple but I think that that was a mistake. My maths is extremely rusty and I definitely feel uncomfortable with it.

    The situation that I am dealing with is the relationship between the energy of a body and its acceleration.

    I want to determine the relationship dE/da (E is energy, a is acceleration)
    I have arrived at the expression;
    [tex]\frac{dE}{da} = \frac{d}{da}{mc^2\frac{1}{\sqrt{1 - v^2/c^2}}}[/tex]

    and I am not sure how to proceed from this point.
     
  5. Jul 3, 2009 #4

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    [tex]mc^2(1- v^2/c^2)^{-1/2}[/tex]
    Now use the chain rule.
     
  6. Jul 3, 2009 #5
    The velocity v is the only variable in the equation. Surely I need to express v in terms of acceleration ‘a’ before I can differentiate the expression using the chain rule?

    If I was resolving dE/dv, I believe that I could go ahead and differentiate the expression using the chain rule but I am trying to resolve dE/da.
     
  7. Jul 3, 2009 #6

    Office_Shredder

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    dE/da = dE/dv*dv/da

    aka the chain rule. Go for it
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook