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Let think I am studying some simple system (this is a thought experiment), where I have two classic objects, and the position versus time plots follows two curves (I will left out physics constants, to keep focus in the math I am having trouble to understand): one parabolic path as a reference given by: $$g(t) = t\,(\sqrt{e}-t)$$ and another path that is less steep, grows slowly at the beginning and falls later like the parabolic path, and I believe is always below the parabolic example, given this behavior by: $$f(t) =\begin{cases}0,\quad t=0 \\ \dfrac{t^2}{2}\left(1-2\ln(t)\right),\, t>0\end{cases}$$

At first, as you could see in the plot both looks quite "innocent": comparing both position vs time path

But if I go to their speed profiles, then something weird could be noticed: while $g'(x) = \sqrt(e)-2t$ so is always linearly losing speed, but instead for the other curve: $$f'(t) = -t\,\ln(t^2)$$ which is also losing speed, but at the beginning, instead of showing a less steep start, it has an INFINITE SPEED!!!, so it doesn't matter how many constants I am avoiding, there will be always an interval near $t=0$ when it is starting to move faster the $c$ the speed of light!!

Here the speed profile:

speed profiles

Here the acceleration profile, showing $f(t)$ diverges at $t=0$:

acceleration profiles

I think I have some conceptual misunderstanding here (so I hope you could explain what I am understanding wrong - but if the maths' problem still holds when corrected, please focus in explain it also), but this is kind of saying that a smooth start from rest position is impossible, which I really doubt, maybe I am mistaken, but daily experience intuition tells that not everything start moving violently, like when I slightly blow over a ping-pong ball.

Or maybe just the behavior of $f(t)$ is aphysical, in this case hope you explain why with detail (but hopefully, in simple terms). I think is quite counter-intuitive that a smoothest start leads to unachievable speeds, I quite lost here.

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    $\begingroup$ But $\lim_{t\to 0} t\ln(t^2) = 0$, so the speed is zero at $t=0$. The fact that the acceleration is infinite at $t=0$ is a problem, but that just means that this function is somewhat unphysical at $t=0$, which, I don't know, that doesn't seem like a problem in general, because this function didn't come from any particular physical situation. Even if it did, that would just mean that the physical model that generated this solution isn't a good approximation at short times. $\endgroup$
    – march
    Commented Sep 26 at 15:34
  • $\begingroup$ Also: note that this issue also applies to the parabolic solution! If the speed is zero for $t<0$, then there is a "kink" in the derivative at $t=0$, indicating an infinite acceleration. Again, this means that an instantaneous finite change in velocity is unphysical, but again, we're just modeling the situation using a simplified mathematical model that doesn't exactly quantitatively match reality, which is okay. If we need more precision, we'll build it into the model by smoothly "turning on" the force "around" $t=0$. $\endgroup$
    – march
    Commented Sep 26 at 15:37
  • $\begingroup$ @march thanks for commenting. which is weird is that is always below the parabolic curve at first, so how it become faster if is traveling less distance? I am very lost in the math, and I am trying to understand if there is some issues on how we describe physics (math tools), I didn't find the system accidentally. I am trying to understand how it start moving and how things stop moving in accurate ways.. if you could elaborate in an answer it could be awesome. $\endgroup$
    – Joako
    Commented Sep 26 at 15:39
  • $\begingroup$ The answer to "how it become faster if is traveling less distance" is told in the parable of the Tortoise and the Hare. The Hare might have a larger top speed, but if most of the time it's moving slower than the Tortoise (because the Hare is taking a nap), it will travel a smaller distance in the same amount of time as the Tortoise. Note that the speed of the log-function particle is smaller at the beginning and for a large amount of time near the end of the trajectories. Top speed doesn't matter; it's average speed that matters for how far it ends up going. $\endgroup$
    – march
    Commented Sep 26 at 15:43
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    $\begingroup$ All of that said, I'm still not sure exactly what your question is. At first, it seems to be about infinite velocity at $t=0$, but that's a misconception in the question, because the speed is never infinite: it's zero at $t=0$. The acceleration is infinite at $t=0$, but that's true for both trajectories provided we say that $v=0$ for $t<0$. Now you say that your question is about the fact that one particle moves farther even though its top speed is larger than the other one. So I'm still not sure how to answer your question. $\endgroup$
    – march
    Commented Sep 26 at 15:46

1 Answer 1

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You're misinterpreting the plots. In the velocity diagram, the points of the lines representing the function give you velocity; if you look at slope there, you're looking at acceleration.

At $t = 0$, velocity is zero, and acceleration positive (going to infinite following your model) for the "orange function"; velocity is positive and acceleration negative for the "blue function"

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