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I'm searching for a function that could represent the acceleration of a car. The function has to rise fast from $(0,0)$ to a maximum and then fall slower. For $x\to \infty$ it should go to $0$.


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Welcome to math.SE: since you are new, I wanted to let you know a few things about the site. In order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are; this will prevent people from telling you things you already know, and help them give their answers at the right level. If this is homework, please add the homework tag; people will still help, so don't worry. – Noah Snyder Nov 20 '12 at 13:20
This looks like a problem for Physics.SE, but there is very little background or motivation given, so it is hard to tell. Consider flagging a moderator to migrate the question to Physics.SE. – robjohn Nov 20 '12 at 14:09
@robjohn: I don't think it's really a physics question. It's a standard first chapter of a Calculus "review of functions" question where you're supposed to sketch something roughly right not actually work out any physics in detail. – Noah Snyder Nov 20 '12 at 19:49
@NoahSnyder: Fitting a mathematical model to a physical system is one of the main goals of physics. The first sentence of the question asks just for that. The remainder of the question asks for a function that has particular constraints. That is a mathematical question. More background or motivation would help to determine where the best answer might be found. – robjohn Nov 20 '12 at 22:05

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from here: $a(t)= t\cdot e^{-k(t-t_0)}$

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you can take any $\frac{f(t)}{g(t)}$ where $g(t),f(t)$ polynomials and $deg(g(t) \geq deg(f(t))$ just make sure $f(0)=0$ and $g(0)\ne0$. For example look at $\frac{t}{t^2 + 1}$

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