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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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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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