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?
The indefinite integral, i.e, the anti-derivative of 0 is, indeed, a constant; that is we have: [tex]\int{0}dx=C[/tex]
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?
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 R^{3} 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.
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.