Velocity/Talk

Question: If a particle has a defined position at every time, must it necessarily also have a defined velocity? Consider a particle moving along a line, so its position along the line at time t is x(t). Suppose we define x(t) as follows:

          { 1   if t > 0
  x(t) =  {
          { 0   if t <= 0

If I remember my calculus correctly, x'(0) is undefined, while x(t) = 0. Does it therefore follow that at time t=0, the particle has a position and is moving but has no velocity? Would it be physically possible (i.e. compatible with the laws of physics as we currently understand them) for a particle with that behaviour to actually exist? -- SJK

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In classical (non-quantum) mechanics a particle with mass cannot make such an instantaneous jump in position. It implies infinite acceleration which implies infinite force. So this case is not physically possible in classical mechanics (assuming zero-mass particles are not physically possible). -- Eob

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I am in browse mode tonight... but at some point mention will have to made of tangent spaces and tie the discussion back to differential geometry.

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Eob: What about in quantum mechanics? IIRC, quantum mechanics predicts instantaneous jumps in position (consider e.g. the Bohr model of the atom). And it has a zero-mass particle, the photon... -- SJK