How Accurate is the Formula for Arrow Velocity from a Bow?

[npu]

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:

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

$$S_f$$= Spring tip movement in meters;
$$S_d$$= Arrow's run in meters;
$$m_f$$= Spring weight in Kg;
$$m_d$$= Arrow weight in Kg;
$$F_{av}$$= Average force exerted by the spring along $$S_f$$ in Newtons;
$$R=\frac {S_d}{S_f}$$

Using this formula I got results which appear right:

Overall $$V$$ 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?

 

[professor]

im not exactly an expert, but it seems to me like air ressistance and friction won't be flawing any simpler calcuations too greatly when your talking about the initial velocity, i wouldent even bother with them, mabye again that's because i can't exactly follow your equation, explain how you came up with it, i know i recoginse the format... honestly don't remember from what equation, but tell me how you ahve derived it and ill check over the math once I am sure there is no inherent flaw if you don't mind.

 

[Cyrus]

It would help if you provided a derivation of this proof. Can't help you otherwise.

 

[Astronuc]

This problem is being discussed in the Introductory Physics homework section.

https://www.physicsforums.com/showthread.php?t=123957

Not only does one have to consider the mass of the arrow, but also the mass of the spring, which is distributed along the spring.