Recently, I noticed a debate in the archery
community over whether heavier arrows would improve penetration depth.
The people interested were primarily bow hunters asking if heavier arrows would get them cleaner
kills. This topic has a lot of misinformation going
around about it and it isn’t much better on physics forums.
I noticed a basic trend. The light arrows camp would claim that light
arrows had more energy because they were faster. They also claimed that energy was all that
mattered. Thus, they concluded that light arrows were
better or, at the very least, as good as heavy ones.
The heavy arrows camp claimed that momentum was what really mattered.
They also pointed to experimental evidence for heavy arrows penetrating deeper.
Looking into this from an historical perspective, the debate is actually quite old.
There are arguably heavy arrow cultures and light arrow cultures.
The heavy arrow cultures include the English, with their longbowmen, The Japanese, and the
Manchu. Even against armored opponents, their bows
had a fearsome reputation. On the other side of the spectrum, the Saracens
and turks used lighter arrows. These archers were renowned for their range,
speed of their arrows, and the sheer volume of fire they put out.
The difference in the approaches was visible at Arsuf during the third crusade.
There, Saracen archers peppered the crusaders with arrows from a distance.
Despite giving many the appearance of pincussions, the arrows did not make much of an impression
on the crusader’s padded armor. This led to a situation where crusades were
shrugging off several arrow strikes, but could kill their adversary with a single crossbow
bolt. So enough about this, let’s discuss bow physics
and arrow peneration. Let’s start by examining bow operation.
First, the bow stores a certain amount of energy in the draw.
Looking at the force to draw distance plot of a simple bow, the force increases linearly
with draw distance. The total energy stored during the draw is
given by the area underneath our curve. As a side note, a considerable amount of work
has been done to make the force curves on modern bows nonlinear.
This is to maximize the stored energy for a given draw weight.
After we store energy in the bow, we transfer that energy to the arrow.
This is done by pulling the ends of the bowstring apart to move the middle forward.
The ratio of arrow movement to bow tip movement spikes at the release, resulting in most of
the energy being transferred to the arrow. In other words, this is a very efficient transfer
of energy. What’s the takeaway from this?
Well, except in extreme cases, the energy of the arrow is mainly determined by the bow.
Sure, energy transfer is more efficient for heavy arrows, but it doesn’t change massively.
This means that our model should give heavy arrows and light arrows the same starting
energy. Great. Now that’s out of the way, let’s talk
about penetration depth. Let’s see what models might favor large mass
per cross sectional area. Time to do some spherical cow stuff.
I think I’ll do something close to Newton’s approximation.
I’ll model the target as a bunch of weights on springs and hit it with an arrow.
When the arrow hits a weight, it will knock it out of the way in an elastic collision.
It will then hit another weight and the process will repeat itself many times…
This can be simplified further by ignoring the side components and doing a series of
1D elastic collisions. Simulating two arrows of equal energy, we
find that the lighter one slows much more quickly than the heavier one.
A quick plot of velocity and force shows that this is just a derivation of drag, but it
shows that mass matters in at least one regime. Of course, if you are willing to accept this
without any evidence, I have a bridge to sell you.
We are going to need experimental evidence, which most people aren’t set up to do.
Thankfully, in the modern age of the internet, anyone can look at a slow motion video and
plot the projectile position frame by frame. It’s amazing how much information you can
extract from this method. Using this technique with bullets in ballistics
gel, I found that force was actually linear with velocity.
This suggests the gel is behaving as a viscous fluid.
A little calculus and we find that momentum linearly decreases with distance.
Thus, starting momentum determines the maximum penetration.
Does this work for arrows? Yeah. More or less.
There’s a little weirdness at the end, but the rest works out fine.
So, both bullets and arrows are in the viscous drag regime where momentum determines penetration
depth. When fired from the same bow, heavier arrows
get more penetration, provided cross sectional area remains the same.
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