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# Physics/pressure on impact

Question
If I were to shoot a 50 lb draw weight bow from 10 yards at a target, about how much pressure would the arrow put on the target when it hits?
P.S. A 30 lb draw weight will send an arrow at about 150 ft per second and a 70 lb draw weight will shoot at about 230 ft per second, so I think a 50 lb would probably shoot about 190 ft per second, but I'm not totally sure.
Any answer will be greatly appreciated. Thank you!

Hello Jordan,

I can't calculate that without some additional data. But I can show you how. You probably can measure the required data.

Pressure is force/area. So I would need the force that stops the arrow and the area of the contact with the target. Regarding the area, is the arrow's tip bigger than the shaft of the arrow? Does it make a round hole? The simplest situation would be that the tip is just a pointed end with the same diameter as the shaft. A fancy arrowhead would make it hard to determine a contact area. But I'll assume you can come up with a value of area in the units m^2.

Oh wait, it looks like you would probably prefer to work with feet and pounds and such. I'd rather convert the speed and other data to SI units, show you how to calculate pressure in N/m^2, and change back to pressure in pound/in^2. So I'll assume you can come up with a value of area, A, in the units inches^2. To get the area in m^2 you'll need to multiply the area in inch^2 by the conversion factor
0.00064516 m^2/1 inch^2

The depth of penetration in the target, and some other data, can be used to determine the force. The arrow's speed changes from 190 ft/s to 0 in that distance. To convert to meters/sec, I'll multiply by the conversion factor
(0.3048 m/s)/(1 ft/s)
190 ft/s * (0.3048 m/s)/(1 ft/s) = 57.9 m/s
I'll assume you know or can determine the weight of an arrow in ounces. We need to convert to kilograms. The conversion factor is
0.02834952 kg/1 oz
We can use the mass of the arrow and its speed to calculate the kinetic energy.
KE = (1/2)*m*v^2
where m is the result of converting the weight in ounces to kg and v = 57.9 m/s. The result has the unit Joules.

With the depth of penetration and the original kinetic energy, we can calculate the force applied to the arrow. Measure the depth in inches and convert to meters using the conversion factor
0.3048 m/1 ft
So the work done on the arrow is given by
W = F*d
where d is the result of the conversion of the penetration depth. Work is also measured in Joules.

Set the result of the kinetic energy calculation equal to F*d and solve for F. That result will be in Newtons. Divide by the area in m^2 and the result is pressure in N/m^2.

Then the conversion factor to get pressure in pounds/in^2 is
(0.001422334 pounds/in^2)/(1 N/m^2)

I hope this helps,
Steve

Physics

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#### Steve Johnson

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I would be delighted to help with questions up through the first year of college Physics. Particularly Electricity, Electronics and Newtonian Mechanics (motion, acceleration etc.). I decline questions on relativity and Atomic Physics. I also could discuss the Space Shuttle and space flight in general.

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I have a BS in Physics and an MS in Electrical Engineering. I am retired now. My professional career was in Electrical Engineering with considerable time spent working with accelerometers, gyroscopes and flight dynamics (Physics related topics) while working on the Space Shuttle. I gave formal classroom lessons to technical co-workers periodically over a several year period.

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BS Physics, North Dakota State University
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