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Physics

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    $\begingroup$ +1 for the picture. How did you do it? $\endgroup$
    – a06e
    Commented Mar 31, 2013 at 2:42
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    $\begingroup$ @becko: I drew it in GeoGebra. Here's the worksheet file, if you want to play with it. (Try dragging the $x_1$ point around!) $\endgroup$
    – Ilmari Karonen
    Commented Mar 31, 2013 at 2:45
  • $\begingroup$ I'm not sure if I missed anything in your explanation, but are you saying that the moon speeds up its tangential velocity? And why should a1 affect the velocity v1 if it hasn't fallen by a1 yet? $\endgroup$
    – mtanti
    Commented Mar 31, 2013 at 10:16
  • $\begingroup$ @mtanti: The tangential velocity stays constant, as you can see from the picture. As for why $v_1$ should depend on $a_1$, remember that the acceleration vector is changing continuously from $a_0$ to $a_1$ as the moon moves from $x_0$ to $x_1$. Taking the mean of $a_0$ and $a_1$ in the approximate formula for $v_1$ is just a linear approximation of this gradual change. $\endgroup$
    – Ilmari Karonen
    Commented Mar 31, 2013 at 18:05
  • $\begingroup$ I think I finally got it when I wrote a program which simulates this. The tangential velocity does speed up as it falls since the new tangential velocity will be the resultant vector (adding the previous tangential velocity to the falling velocity). This makes the moon able to speed past the earth, although not in a circular orbit.In my simulations the moon would always crash into the earth eventually but maybe its because I didn't take into account other factors like centrifugal force. $\endgroup$
    – mtanti
    Commented May 4, 2013 at 16:49