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Let's say that a point particle moves along a line? We can come with a function describing the speed of the particle. It is not hard to imagine what would the motion of the particle look like if our speed function is given by a linear function.

My question is: What would the motion look like if our speed was given by Dirichlet function? In particular the speed at irrational times would be 0 meters per second and the speed at rational times would be 1 meter per second. How far would the particle travel in 10 seconds?

Adam
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This would make no sense. No object could move and not move so rapidly and exactly (because of conservation of momentum). Even if it could, the Dirichlet function is not Riemann integrable. It is however Lebesgue integral and from negative infinity to infinity has integral $0$. So that would mean the particle did move at all. However, it had some velocity so it had to have moved. This is where the Math doesn't make physical sense. That is because of the inherent problem I started with in describing a particles motion/velocity with the Dirichlet function.

But in general, yes, we can come up with the equations of motion (including velocity) of a particle--if even only numerically. Often in Physics, we use Calculus of Variations and techiniques such as Euler-Lagrange to solve for the equations of motions. See this for a bit more on that.

mathematics2x2life
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  • What if the position of a particle is described by a non-differentiable function? How would we determine its speed at each point? – Adam Dec 03 '13 at 23:32
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    You wouldnt, it wouldnt have a velocity. v=dx/dt – Abdulh Khazzak Gustav ElFakiri Dec 03 '13 at 23:35
  • Exactly, for our laws of Physics to make sense, your motion needs to be differentiable. As shown by example in my answer. If you wanted such things to be possible, you'd have to create a new 'Physics' and start from scratch because the normal rules wouldn't applied. When motion can be non-differentiable, then for the most part, everything goes out the window. – mathematics2x2life Dec 03 '13 at 23:37