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?
Best Answer
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.