Arrow Speed and Mass

I am now going to turn my attention to the arrow.  In this post I shall look at some of the the physics of the arrow’s flight.  In my next post I shall look in more detail at how to select your arrows and tune your equipment for the best arrow flight.

The Need For Speed

I spent a lot of time talking about how a bow achieves high arrow speed.  Why is this so desirable?  There are two reasons.  The main reason given in standard archery is flat trajectory.

Once an arrow leaves the bow it is influenced by only two forces: gravity and air resistance, also called drag.  We shall look at drag in a moment but first let’s consider gravity (and here you can be thankful that we are ignoring Einstein, because gravity in Einstein’s theory is really weird).

Galileo’s Arrow

Gravity is, for these purposes, a force that pulls all objects towards the ground.  Since it is a force it should have the effect of accelerating objects in line with Newton’s law F=ma (force = mass x acceleration).  We should therefore expect the acceleration due to gravity to be equal to the force divided by the arrow’s mass.  This is wrong because the strength of gravity on the arrow is proportionate to the mass of the Earth and the mass of the arrow.  For any given object, therefore, the effect of mass in resisting the force is exactly cancelled out by the effect of the same mass in generating the gravitational force in the first place.  This is a long way of saying that acceleration due to gravity is not affected by the mass of the arrow.  This was first demonstrated by Galileo Galilei.  He did it by rolling balls of different masses down inclined slopes.  Alas he did not drop things from the Leaning Tower of Pisa.

As an aside, this point was beautifully demonstrated by Apollo 15 astronaut David Scott in 1971.  He dropped a hammer and a feather on the moon and they hit the surface together.  The only reason this doesn’t happen on Earth is because of air resistance.  I am going to ignore air resistance when discussing gravity.  This will make no practical difference.

An arrow will fall to the ground at a fixed rate.  As well as not being affected by mass, the rate of descent is not affected by horizontal speed.  An arrow shot horizontally that misses the target (probably one of mine) will hit the ground at the same moment as an arrow that is simply dropped (I’ve done that as well).  The longer an arrow takes to reach the target, therefore, the more it will lose height on its way and hence the more you will have to aim high to hit the target.

The way to avoid having to aim high is clearly to get the arrow to the target quickly, and this means shooting it at a high speed.  The higher the arrow speed, the less you will have to aim high.  This, incidentally, is the origin of the phrase “point blank range”, which simply means a range at which you do not have to aim high to allow for gravity.  The faster the arrow/bullet/cannonball is travelling, the further point blank range is.

That is the reason given by regular archers for having high arrow speeds.  Of course, regular archers are a bunch of sissies who stand still while they shoot.  Those of us who are shooting from the back of a running horse have an extra reason for wanting high arrow speeds: you don’t have to aim off as much to allow for the speed of the horse either, for analogous reasons.

Air Resistance Is Such A Drag

Sorry, couldn’t resist.  I shall use the term drag because it is quicker.  It refers to the way that air slows things down.  I mentioned that it is why a feather falls more slowly than a hammer on Earth but not on the moon (where there is no air).  We are here ignoring vertical drag and looking only at the drag that slows the arrow’s forward speed.  I am also going to ignore the effect of the fletchings.  This is because you can put any size fletching on any size and mass arrow, so it is not so important to arrow selection.  All else being equal, use small fletchings to get high arrow speeds.

Drag has some relatively complicated physics associated with it and, since we all know that I hate complicated physics, I shall simplify it.  The first point is that drag is higher on things with a large cross section.  In archery terms, this means that a thick arrow has more drag than a thin arrow.  If all else is equal, go for a thinner rather than a thicker arrow.  Note that “if all else is equal” does not mean that you should compromise on getting the correct arrow mass and spine.  They are more important than the width of the arrow.

The other main factor that will affect the drag force on your arrow is its speed.  The faster something is going, the greater the force of drag.  Think about air as being like water: you can swirl your hand lazily through the water in a bath or pool but if you try a sudden fast hand movement through the water then you will feel the water resistance increase immediately.  Air is the same.

Now, drag is a force and like all forces it obeys F=ma.  A given amount of drag will have a greater effect on a lighter than a heavier (more massive) arrow.

Speed, Mass and Deceleration

The last two paragraphs show that a heavier, slower arrow will maintain its speed through the air more than a lighter, faster arrow will.  This is a major concern for hunters, because it affects the kinetic energy at impact, which will affect penetration.  For the same reason, the different deceleration rates would have concerned our warrior forebears.  We are not concerned with kinetic energy: we only need enough to penetrate a foam target.  We want the arrow to reach the target as quickly as possible and stick in.  This means that speed is everything.

If the deceleration rates vary, however, then isn’t there a risk that the light arrow, despite starting out quicker than the heavy one, will still end up reaching the target later because of the difference in drag?  Should we consider sacrificing speed for mass?

The short answer is no.  In the kind of situations we will encounter, studies suggest that the heavier arrow will not catch up with the lighter one, provided that each has been shot from the same bow, to which each arrow is properly matched.

If you have a bow with much greater energy storage then maybe your heavier arrow will catch my lighter one from my faster bow that stores less energy but for a given bow the increase in arrow mass will be matched by a decrease in arrow speed that outweighs in terms of time taken to hit the target.

Maximum Dry Fire Mass

I touched on this is an earlier post.  As you decrease arrow mass you will reach a point where arrow speeds do not increase any further.  This is the maximum dry fire mass.  Flight archers, in their quest for maximum distance, will often have arrows at this mass.  There is no point in going lighter because you will not increase your speed but you will increase the effect of drag.  Outside of flight shooting you will want to be heavier than maximum dry fire mass anyway, to save yourself and your bow from excess vibration energy.

How Heavy Should My Arrows Be?

This is the all-important question and like all such questions there is no easy answer.  It will depend on your bow, your string, draw length, fletchings and a host of other things.  There is, however, a generalised rule of thumb.  Archers sometimes measure arrow mass in terms of grains per pound of draw weight (gpp).  As we know, draw weight is not the whole story but it is a decent approximation of energy storage and speed.

Hunters will often use arrows of 12-15gpp.  They like to have heavy arrows for greater punch at the target.  Flight shooters may well be using arrows around 5gpp or lighter.  Either extreme is probably not a good idea for a competitive mounted archer.

Ideally you should consult your bowyer and experiment with various weights within the bowyer’s suggested range.  As a general rule, however, you should probably be aiming at somewhere around 7-10gpp.  That should give you a nice speed without damaging the bow or your arm.

ALWAYS CONSULT YOUR BOWYERIF YOU BREAK YOUR BOW THROUGH USING ARROWS THAT ARE TOO LIGHT FOR IT THEN THAT IS YOUR PROBLEM, NOT MINE!

And on that cheery, note, I shall leave you and start on my next article: arrow spine.  As always, please do comment – it makes me feel loved!

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