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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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  • I asked a vaguely formatted version of this question here: http://math.stackexchange.com/q/1752363/ and ended up with some extremely vague connection with non analytic function which I can't figure out. –  Apr 22 '16 at 06:31
  • Thus isn't a physics problem at all. It's a problem of understanding epsilontics, i.e. the precursor of modern calculus. That, of course, is something you have to take to the mathematicians, indeed, and they probably gave you the correct answers. I correct myself... they didn't care much... – CuriousOne Apr 22 '16 at 06:33
  • Related: http://physics.stackexchange.com/q/111251/2451 , http://physics.stackexchange.com/q/60480/2451 and links therein. – Qmechanic Apr 22 '16 at 06:34
  • As a side note: a "particle" is not a physical system but a physical fiction. It's the approximation of the motion of an extended object to the motion of its center of mass, neglecting all internal degrees of freedom. In reality the world is "smooth" because particles don't exist. – CuriousOne Apr 22 '16 at 06:38
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    it is the instantaneous force, which is dp/dt. Why is there a problem? the "suddenly" means a impulse https://en.wikipedia.org/wiki/Impulse_%28physics%29 . all these are ok in classical mechanics problems – anna v Apr 22 '16 at 09:30
  • @CuriousOne: The motion represented by $e^{-\frac{1}{t^2}}$ is smooth (and zero for all derivatives at 0; this is a great example I learnt yesterday), even if you include particles. I don't understand why you mention the world is 'smooth' ? –  Apr 23 '16 at 02:48
  • I just mentioned it because a lot of the approximation we use in physics (like the delta function) are implying that physical interactions happen in zero time, are concentrated on a point etc. , but these are simplifications we make to avoid having to solve complex differential equations. Examples like yours, which is a beautiful mathematical function, on the other hand, are irrelevant because of thermodynamics. In reality there is always some non-trivial motion at some non-zero frequency, which makes the effective interaction always finite. – CuriousOne Apr 23 '16 at 05:31
  • @CuriousOne: I don't know any physics except basic mechanics (up-to Feynman Vol 1 Chapter 9), so I can't speak of thermodynamics now. Except the approximation you used in your penultimate comment and last comment, i. Which other approximations are used in basic mechanics (I couldn't find them in the books) ? ii. (Dumb question) How my question is consistent with newton's second law ? –  Apr 23 '16 at 12:00
  • I think you should look at my comment from the viewpoint of ontology/interpretation. There is theory, in which we can determine the functional dependence of physical variables (at least locally) with arbitrary precision, and then there is the experiment, which always leaves some non-trivial statistical errors (stemming at least from the thermodynamics of the measurement). In well behaved examples (like a parabolic stone throw) a slight variation of the initial condition will lead to a small variation of the outcome, in examples like yours it won't be that simple. – CuriousOne Apr 23 '16 at 18:00

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,

1

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.

  • I don't buy your first portion of the answer, but the remaining are (although bit out of context) good and should be used as an warning. What is the cow here: '... assume the cow is spherical ...' ? –  Apr 22 '16 at 10:39
  • @ArkaKarmakar Spherical cows are a famous physics joke. I'm not sure why you don't agree with the first part, because it is obviously true: objects really are made of charged particles surrounded by fields, and it is the fields which interact, and the fields are smooth. –  Apr 22 '16 at 11:20
  • I don't agree means (in this context) I didn't learn that yet. –  Apr 22 '16 at 11:50