Zero Acceleration

  1. 1. The problem statement, all variables and given/known data
    Let c be a path in R^3 with zero acceleration. Prove that c is a straight line or a point.


    2. Relevant equations
    F(c(t)) = ma(t)
    a(t) = c''(t)


    3. The attempt at a solution
    so i know that since the acceleration is zero, the velocity must be constant, and when you integrate a constant, you get a straight line...but how to I prove mathematically that the velocity is constant, because you can't integrate 0dt, as far as I know?
     
  2. jcsd
  3. arildno

    arildno 12,015
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    Gold Member

    The indefinite integral, i.e, the anti-derivative of 0 is, indeed, a constant; that is we have:
    [tex]\int{0}dx=C[/tex]
     
  4. oh ok, so if I integrate that again I get that c(t) = Ct + D, which fits the general equation for a line

    but then, does that also prove that c(t) could just be a single point?
     
  5. arildno

    arildno 12,015
    Science Advisor
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    Indeed, since big C could be..0!
     
  6. oh. duh!
    gratzie
     
  7. HallsofIvy

    HallsofIvy 40,925
    Staff Emeritus
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    Damn, I hate mixed "physics" and "mathematics" problems! You or whoever set this problem, should know that a "path" DOES NOT HAVE an "acceleration". I expect this problem should be "find the equation of motion of a particle whose trajectory is a given path in R3 with acceleration 0. Show that the path is either a straight line or a point". Then you would begin with [itex]\vec{a}= d\vec{v}/dt=[/itex] and go from there.
     
    Last edited: Oct 22, 2007
  8. Gib Z

    Gib Z 3,348
    Homework Helper

    The easiest way would have been to recognise that acceleration is a vector quantity, it is affected both by direction or magnitude. No acceleration, no change in direction, which means constant gradient. Simple as that.
     
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