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!

1 thought on “Arrow Speed – Graphic Predictions

  1. Ah, Grasshopper, I am sorry to hear of your maladies… and also your physics! F=ma is a fine equation but not the one you want to use (the force changes from max to 0 through the power stroke, so is not a good variable to work from). Since the same bow is being used at the same draw length, then the same amount of energy is being loaded into the bow and KE = 0.5 m v^2 (kay eee equals one half em vee squared). Each shot therefore could be expected to have the same KE, so you end up with the speeds being affected by the square root of the inverse ratio of the two masses (s2 = s1 x (m1/m2)^0.5). Since the masses are fairly close together in size, the mass ratio is close to 1 so the speeds will be fairly close together (but different).

    The only problem with this analysis is the KE(bow) = a KE (arrow) assumption where “a” is a simple fraction. Typically the fraction of KE that the arrow receives from the bow is about 3/4 (0.75 or 75%) of the energy loaded into the bow. The rest (that doesn’t come out as arrow speed / kinetic energy of the arrow) comes out as physical vibrations in the bow, vibrations in the air (sound), and heat. Now the problem is that “a”, a measure of the efficiency of the energy transfer, is not constant. It, too, depends on factors like arrow spine.

    I remember a friend shooting with me at our practice butts (compound bow) and who took one half turn off of each limb and his arrows hit the target slightly higher(!) as a result. He was mightily perplexed. The loss in draw weight (force) should have resulted in the arrows hitting lower, but the adjustment of draw weight also gave him a better match with the spine of his arrows, so his efficiency ended up compensating for the loss of draw force and actually made his arrows faster.

    Such complexities all serve to make much of these topics counterintuitive, which is why we are often perplexed.

    I look forward to seeing your experimental results and how you get better soon!


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