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

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In summary, the conversation discusses the determination of the function for the velocity of an arrow out of a bow, without considering factors such as air resistance and friction. The formula for this velocity is V=\sqrt{ \frac {2F_{av}S_f10}{m_d+ \frac {1}{2}m_fR^{-1}}}, where S_f is the spring tip movement, S_d is the arrow's run, m_f is the spring weight, m_d is the arrow weight, F_{av} is the average force exerted by the spring, and R is the ratio between the arrow and spring masses. The conversation also mentions the importance of considering the mass of the spring in the calculation.
  • #1
npu
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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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  • #2
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.
 
  • #3
It would help if you provided a derivation of this proof. Can't help you otherwise.
 
  • #4
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.
 

What factors affect the speed of an arrow from a bow?

The speed of an arrow from a bow is affected by several factors, including the draw weight of the bow, the draw length of the arrow, the weight of the arrow, and the type of bow used. The draw weight of the bow is the amount of force required to draw the string back, and a higher draw weight will result in a faster arrow speed. The draw length of the arrow refers to how far back the string is pulled before releasing, and a longer draw length will result in a faster arrow speed. The weight of the arrow also plays a role, as a lighter arrow will travel faster than a heavier one. Finally, the type of bow used can impact the speed of an arrow, with compound bows typically being faster than recurve or traditional bows.

What is the maximum speed an arrow can reach?

The maximum speed an arrow can reach depends on various factors, including the bow used, the weight and type of arrow, and the archer's strength and technique. In general, arrows can reach speeds of over 300 feet per second, with some compound bows capable of launching arrows at speeds of 400 feet per second or more.

How does arrow speed affect accuracy?

Arrow speed can have a significant impact on accuracy. A faster arrow will typically have a flatter trajectory, meaning it will travel in a straighter line and be less affected by factors like wind and gravity. This can result in more accurate shots, especially at longer distances. However, if the arrow is traveling too fast, it may also be more difficult to control and may result in less accuracy.

Can the speed of an arrow be increased?

Yes, the speed of an arrow can be increased through various means. One way is by increasing the draw weight of the bow, which will result in a more powerful shot. Another way is to use a lighter arrow, which will be easier for the bow to propel at a faster speed. Additionally, some archers use special techniques or equipment like release aids to help increase arrow speed.

Is a faster arrow always better?

Not necessarily. While a faster arrow may have advantages in terms of trajectory and potential accuracy, it may also be more difficult to control and may result in a less accurate shot. Additionally, a faster arrow may have more force upon impact, which can result in a shorter effective range and potentially cause damage to the target or the arrow itself. Ultimately, the best arrow speed will depend on the specific needs and preferences of the archer.

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