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I really can't understand what happens during the time $t(0)$ to $t(0+dt)$ in the following crackpot arguement:

A particle is at rest (in an ideal frictionless world) until $t(0)$. So every order of the temporal derivative of the position is zero. Then suddenly I hit the particle. In the interval $t(0) \; to \; t(0+dt)$, position is changing, so velocity is non zero. Velocity is changing (zero to nonzero), so accleration is nonzero. (Now I can't understand what's going on) The acleration is changing (zero to nonzero), so the jerk is nonzero and so on.

Now I can't understand

  1. What's wrong with this argument (Please pinpoint it to the place where it breaks down)

  2. How it is consistent with Newton's second law.

Qmechanic
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2 Answers2

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The equation of motion of the particle is $$m \ddot{x}(t) = F(t)$$ where $x(t)$ is the position and $F(t)$ is the force. In the situation you describe, ("suddenly I hit the particle"), the force as a function of time can be written as $F(t) \propto \delta (t)$, with $\delta$ the Dirac distribution. Integrating once, you obtain that $$\dot{x}(t) \propto \theta(t)$$ where $\theta(t)$ is $0$ for $t<0$ and $1$ for $t>0$ (the integration constant vanishes because the particle is at rest for $t<0$). In this modelization, indeed the velocity is discontinuous. This is because the "sudden hit" is represented by the $\delta$ distribution.

The situation described above is an idealization of the real physical situation. In real life, there is no "sudden hit", and $F(t)$ is a regular function which spans a short interval of time. For example,

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In this case, there is no problem of regularity.

Antoine
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What's wrong with this is the phrase 'suddenly I hit the particle'. What you are assuming by this is that you have some hard-edged and rigid object which you crash into the particle, which is also hard-edged. But you don't have either of those things: what you have is something which is both not rigid and whose surface is actually a little bit of EM field which interacts with some EM field surrounding the particle, both of which are analytic (or at least smooth, but actually they are analytic I think).

This is an example of something that frequently leads people astray. Physicists like to make a lot of convenient assumptions about things, because it makes the maths easier to do: '... a perfectly rigid ...', '... a square-well potential ...', '... friction is zero ...' '... assume the cow is spherical ...'. None of these assumptions are true: they are merely useful simplifications we make to avoid having to do difficult maths.

About half of being a physicist is then understanding when these assumptions are safe. A good (but far from perfect) heuristic for this is: if you get really physically surprising answers (quantities being infinite, failures of determinism in classical mechanics and many other bad things) look very closely at the assumptions being made. In particular, in this case, words like 'suddenly' are a big red flag.