# Speed of an arrow 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?

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#### professor

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

#### Cyrus

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

#### Astronuc

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

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.

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