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There is a new sport called "Archery Tag" that involves shooting opponents with foam-tipped arrows fired out of a real bow. The official Archery Tag web site presents data that claims to show the safety of these arrows:

http://www.archerytag.com/safety

The site claims that the arrows, when fired have a kinetic energy of 20.9 foot-pounds (28.3 J). (The site actually says "20.9 pounds per foot", which I assume is an editing error, because "pounds per foot" is not a unit of energy.) It also says that when it impacts its target, the pressure transferred to the target is 5.6 psi (38 kPa).

Since the heads of the arrows are circlss 2 1/8 inches (5.4 cm) in diameter, the surface area is 3.5 square inches (23 sq cm) and thus the total force the arrow exerts on the target is 20 lb (87 N). By Newton's Third Law, this is the same amount of force that the target exerts back on the arrow to bring it to a stop. Since the amount of work needed to bring the arrow to a stop is equal to the kinetic energy of the arrow, we can solve for distance in the formula "work equals force times distance" and get that the distance taken to bring the arrow to a stop is 20.9 foot-pounds / 20 lb = 1.05 ft (28.3 J / 87 N = 32 cm). But this seems absurd because it would mean that the foam would have to compress over a distance of over a foot (about 30 cm), since that is how far the arrow would have to move during the time it is in contact with the target, but the entire length of the foam part looks to be at most a few inches (under 10 cm) as you can see from the pictures such as the ones on the front page (http://www.archerytag.com). So it seems like the figures given on the web site cannot possibly be correct.

Is my analysis correct?

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  • $\begingroup$ Hi Alex, and welcome to Physics Stack Exchange! Is there some reason you believe your analysis may be incorrect - perhaps a particular step you're not sure if you've done correctly? Generally we discourage questions that just ask us to check someone's work. It's better to identify a specific concept that is giving you trouble and ask about that (still in the context of the actual problem you're trying to solve). $\endgroup$
    – David Z
    Commented Mar 26, 2012 at 22:26
  • $\begingroup$ I think your calculation is correct. But hopefully the arrows will quickly slow down due to their large cross section. And my guess for the real danger is what happens when someone is right next to your bow when you release the arrow (my prediction is that you will impale them), and also, what happens when the foam falls off. $\endgroup$
    – Carl Brannen
    Commented Mar 27, 2012 at 0:19

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The foam pad seems to be about as long as it is wide i.e. 2.125". Assuming all this compresses on impact I get a pressure of about 35 psi, which is about a factor of six higher than they quote, so yes there's an error on the web site though probably not a fatal (literally :-) error.

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Sirs,

Thank You for your insight concerning the information presented on the archery tag website. In reviewing the wording and terminology, we agree it was not stated as accurately as intended. They are revising that section of the page to represent test results more accurately.

Test results: test distance: 10 feet Mass: 650 grains velocity: 92 fps(feet per second) draw weight: 28lb draw length: 26"

Your additional comments are most welcome.

Matt Lennon Shryke Engineering

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