Arrow Speed – Graphic Predictions

In my last post I wrote about the testing of my various arrow through my shiny new chronograph.  Since then I have bought two different types of all carbon shaft to try: Easton Apollo 560 and Easton PowerFlight 500.  For the PowerFlights I have a choice of points: 60gn or 100gn points (the Apollos have 100gn points).  Is it possible to predict the speed of these arrows?  Since I have had a flare-up of an old back injury and for the last 48 hours haven’t been able to stand upright or walk without a limp (and ideally a stick) I am going to attempt to do exactly that.

The first big caveat is the one pointed out by Steve Ruis in his comment to my last post: correct arrow spine is critical to speed.  In particular, dynamic spine is important.  As regular readers will know, this is the extent to which the arrow bends as it is shot, as opposed to static spine, which is how much it bends when a weight is suspended from it at rest.  These three variants (Apollo, PF with 60gn and PF with 100gn) are going to have different dynamic spines.  The Apollo, as a .560 spine, is weaker in static spine than the .500 PFs (I am going by the marked spine for these purposes).  With their 100gn points they are likely to remain weaker than the PFs.  The PFs have the same static spine (being identical shafts) but the 100gn points make the dynamic spine weaker than the 60gn points do.  This may well have an effect on speed.  I shall be able to correct for this factor to a limited extent by shooting each variant as a bareshaft first (I am also going to do this with the other arrow types I tested in the previous article).  Having shot the Apollo bareshaft already, I can say that they are a touch weak but not too bad.  They are stiffer than most of the aluminium arrows I tested in the last article (though not than the A/C/Cs, which might explain that arrow’s slightly high performance).

Leaving spine hypothetically to one side, what predictions can we make?  Well, the new arrows weigh as follows (+/- 0.5gn):

Apollo: 360gn

PF w/100gn: 350gn

PF w/60gn: 310gn

We would therefore predict that they would all fly faster than even the fastest (and lightest) of the arrows in other test (which weighed 372gn).  We would expect the PowerFlights to be faster still, with the 60gn tips being fastest (subject, as I say, to the effect of spine).

What is the relationship between mass and speed?  Applying our favourite formula, F=ma, we should expect a linear relationship.  That is to say, since a=F/m, where F is a constant (the stored energy in the bow), we should expect speed to rise in inverse proportion to the drop in arrow mass.  Note that this does not mean that if we half arrow mass we double arrow speed.  The mass that is being propelled by the bow includes the mass of the bow’s limbs and the string.  With that clarification in mind, however, we should expect to see a straight line if anybody were to be sad and geeky enough to draw a graph of arrow mass against arrow speed.  Like the one below, for example.

Arrow speed graph

As you can see, the arrows I tested last time form a straight line, subject to some pretty sizable error bars caused by poor shooting form, variations in spine etc.

I have added dotted lines to represent the three new arrow variants that I intend to shoot in the next few days.  The prediction from the graph (apologies for the unclear numbers on the y-axis: it was late when I drew this graph) is that the Apollos will fly at 192fps, the PF with 100gn points will go at just under 195fps and the PFs with 60gn points will be around 204fps.

As you will have gathered from the various caveats (variations in spine; less than perfect consistency in my shooting technique; differing nocks; drawing a graph at midnight in a childrens’ drawing pad etc) mean that this is not exactly perfect science.  I am not, as one should do, isolating one variable.  My prediction, however, is that factors such as spine difference will not affect the speeds to an extent that trumps weight.  I expect to see the order of speeds as predicted and I do not think that the actual speeds will be out by more than about 5fps.

And as soon as my back heals, I shall test it and let you know!

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