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**Speed of an arrow from a bow (Latex format)**

I am trying to determine the function for the velocity of an arrow out of a bow (leaf spring).

I am talking without considering factors such as air resistance and friction. I came up with:

[tex]

V=\sqrt{ \frac {2F_{av}S_f10}{m_d+ \frac {1}{2}m_fR^{-1}}}

[/tex]

[tex]S_f[/tex]= Spring tip movement in meters;

[tex]S_d[/tex]= Arrow's run in meters;

[tex]m_f[/tex]= Spring weight in Kg;

[tex]m_d[/tex]= Arrow weight in Kg;

[tex]F_{av}[/tex]= Average force exerted by the spring along [tex]S_f[/tex] in Newtons;

[tex]

R=\frac {S_d}{S_f}

[/tex]

Using this formula I got results which appear right:

Overall [tex]V[/tex] appears to be directly proportional to arrow to spring ratios in mass and distance.

Still I am unsure about it. Is it right?

If so, is there a more elegant formulation?

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