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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My New Chronograph

I have just bought myself a chronograph.  Specifically, a Chrony F1 with lighting set.  The lights let you use it indoors but I have no intention of so doing.  It’s just that it was on sale and getting it with lights worked out cheaper than getting it without.

For those who don’t know, a chronograph is a device for measuring the speed of an arrow (or bullet, pellet, paintball etc).  It’s basically a long box with a laser at each end that beams upwards.  It starts an internal stopwatch as the front of the arrow goes through the first beam and stops it as it goes through the second beam.  Using the recorded time and the distance between the beams, it works out the speed of the arrow.  This is a brilliantly useful tool: every time you change something – fletchings, brace height, string material, arrow shaft etc – you can shoot before and after through the chrono and see what effect the change has on your arrow speed (and therefore on your trajectory and point of aim: see earlier posts on arrow speed).  I explained all this to Claire, my loving and long-suffering wife.  She replied “you want it because you’re an archery geek and it’s a cool new toy”.  She was, of course, entirely right; but it is also a really useful piece of kit.

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The chronograph is famous among archers for its disappointments.  We all expect our favourite bow to be whipping arrows out at 200 fps or faster.  Then we get to a chronograph and discover that we are barely touching 180.  Like all good science, it can be a cruel destroyer of our cherished illusions, but one that should lead us to make informed changes to improve our situation.

Today the chrono arrived and I took it down the woods for a play, together with my Border Ghillie Dhu and a variety of arrows to test.  I ended up comparing the following (all made by Easton and listed: type; spine; mass in grains; grains per pound of draw weight (gpp)):
XX75 Platinum Plus; 2016; 431gn; 11.65gpp
XX75 Tribute; 2016; 429gn; 11.59gpp
X7 Eclipse; 2014; 372gn; 10.05gpp
X7 Eclipse; 2114; 390gn; 10.54gpp
A/C/C; 3-39; 384gn; 10.38gpp

Note: I have assumed for these purposes that I was drawing 37lbs.  This is the marked weight of the bow at 28″.  I did not measure the draw weight or my draw length for these purposes, since the aim was to compare arrows rather than necessarily give accurate gpp readings.

Judging purely by arrow mass, therefore, we would expect the Platinum Plus and the Tribute to be of similar speed, with the Tributes maybe a tiny bit faster.  We would then expect the 2114 Eclipse to be faster than those two, with the 2014 faster still.  We would expect the A/C/C to be somewhere between the two sizes of Eclipse.  Sounding a note of caution here, I will say that I had limited numbers of the X7 Eclipses and had a few error readings (basically I missed the beam).  This means that the speeds for the X7s may be less reliable than the others.

There are other variables beside mass.  One is the nocks.  Nocks can be tremendously important to arrow speed.  The nocks on all of these arrows were the same, except for the A/C/C, which has a rather better (and more expensive) nock.  The remainder all used the same nocks.  The situation is complicated a little by the fact that the nocks on the aluminium arrows (i.e. all except the A/C/C) have been splayed and this can mean that they vary slightly in fit.  We shall keep this at the back of our minds while remembering that I shot at least 3 of each type of arrow and then took an average, which should mean that differences in nock splaying cancel each other out.

Air resistance can be a big factor in arrow speed.  It is determined by a variety of factors including thickness or shaft and size/setup of fletchings.  The aluminium shafts all have the same fletching but in any event we would not expect resistance to make a significant difference at the short range here (I was standing about 3ft from the chronograph).

The average speed of the arrows is noted below (in feet per second):
Platinum Plus: 173.53 fps
Tribute: 177.59 fps
2014 Eclipse : 188.16 fps
2114 Eclipse : 182.38 fps
A/C/C: 194.42 fps

As we expected, then, the Tribute marginally outshoot the PP.  Both are outshot by the  Eclipses, with the lighter Eclipse faster than the heavier one.  The anomaly is the A/C/C.  This went faster than expected from the simple masses.  One explanation may be the nocks.  Another can be seen when the raw data is examined.  On one shot I really went for it, nailing the release, pushing the bow hand forward and probably drawing quite a bit further than usual.  The result was an arrow that went at 208 fps!  If you remove that arrow and stick with a fair comparison of regular draws, the A/C/Cs drop to 187.65, which is in the expected range, slightly higher than we might expect, but that’s likely to be the nocks.  This removal of the faster arrow is not some form of special pleading, by the way: it is perfectly sound to remove an anomalous result from a set of statistics.  This result varied from the mean by more than twice what any other arrow did and produced a result at odds with all of the rest of the data.  Keeping it in would be poor science.

The final adjusted results, therefore, are:
XX75 Platinum Plus; 11.65 gpp: 173.53 fps
XX75 Tribute; 11.59 gpp: 177.59 fps
2114 X7 Eclipse; 10.54 gpp: 182.38 fps
4/40 A/C/C; 10.38 gpp: 187.65 fps
2014 X7 Eclipse; 10.05 gpp: 188.16 fps

This is in accordance with our expectations of reduced mass bringing greater speed.  What lessons have I learned?  Well, for starters I shall stop buying Platinum Plus and go back to Tributes as my basic arrow.  I shall also look at investing in another set of X7s.  Quicks Archery very kindly sponsored me and Claire by giving us reduced rates on X7s before Korea last year.  These have gradually vanished into the undergrowth or been bent around trees or target frames and the time has clearly come to replace them!

I do not claim that this is anything approaching a perfect experiment.  There are various things that should be (and will be) addressed.  One is shooting more arrows so as to get a better idea of the average speed.  Another is getting somebody else to shoot it, to try to obviate any unconscious bias (although I did this to some extent by not weighing the arrows until after I had shot).  My draw length is not as consistent as it should be, although this should have evened out between the different arrows.

This is the first in what will be an ongoing series of articles where I get to play with the chronograph.  I shall set out more detailed and scientific findings in future posts, in which I hope to deal not just with the effect of changing other variables such as brace height but also to try a variety of bows and do some kind of comparison (I am intrigued by the number of relatively inexpensive bows out there that advertise speeds over 200 fps without giving a draw length, draw weight, arrow weight etc.  After reading this post, I hope that you will share my cynicism of such claims).

Relatively Fast Horses

I have written elsewhere about the balance between speed and accuracy when devising a scoring system for horseback archery.  I have also written about the application of Einstein’s special theory of relativity on the sport.  It’s time to put them together.

Lightspeed and Spacetime

Those who have read my previous articles on relativity (search for “Einstein” to read these articles) will recall that physics has shown through repeated experiment that c, the speed of light, is the same to all observers.  The usual demonstration of this is that if you stand on a train travelling at 100mph and throw a ball forwards at 100mph then the ball will appear to you to be going at 100mph but to somebody on the platform it is going at 200mph.  If you shine a beam of light, however, you and the person on the platform will both measure the light as travelling at the same speed: c.  .

Even in these ill-educated times I feel justified in saying that every schoolboy knows that speed = distance/time.  Since the two observers register the same speed but different distances (the person on the train only sees the movement along the train but the person on the platform sees that plus the movement of the train), it follows that they must measure different times for the ball’s travel.  It turns out that the faster you travel relative to somebody else, the slower time will pass for you than for them.  This fact has also been demonstrated repeatedly.  Methods include flying atomic clocks around the world and registering the fact that they measure different times.  More prosaically, the GPS system relies on special relativity.  The clocks in the satellites are set to run at a different rate from those on the ground to allow for the time-altering effects of relativity.  Without this adjustment satnavs would be hopeless: the tiny alteration in clock speed amounts to a change of about 10km per day.  (Most of this adjustment is due to the fact that time also runs differently under different strengths of gravity, but the speed of the satellites is also factored in.)

The corollary of the above effect is that space contracts at high speed.  If a moving and a stationary observer both see a photon (particle of light) travel from A to B then they will age on the speed (c) but disagree on the time (because their speeds are different relative to the photon).  They must therefore disagree about the distance traveled.  Experiment and theory agree: as you move, space becomes shorter in the direction of your travel.  Like time dilation, this effect is basically non-existent at everyday speeds but gets bigger close to the speed of light.  In a calculation that seems to make a mockery of the very concept of lightyears as a unit of distance, it can be shown that if we were to go at 99.9999999% of the speed of light we could travel 3 million lightyears (the distance to the Andromeda galaxy) in 50 years.  This is because the distance between us contracts at those high speed.

The general theory of relativity combined space and time into a single entity called (not very imaginatively) spacetime.  The predictions made by this theory have been confirmed so often and in so many different contexts that it must be considered the closest thing to “fact” that physics currently has about reality.

Can’t We All Just Agree?

The mathematics of spacetime provides a solution to the seeming chaos of time and distance being so malleable.  The solution is this: although different observers will disagree about time and space, they will agree about distances in spacetime.  This being the case, the week also agree about speed in spacetime.

Through a piece of mathematics that is a little beyond what I want to call with while typing on a touchscreen tablet after dinner, it can be shown that the relationship of time dilation and observed speed fit together in spacetime to demonstrate that everything, from photons to the stars that emit them (and everything in between, including you and me and, crucially to the title of this piece (which you have probably forgotten, given how loosely it connects to the subject matter), horses) travels at the same speed.  Thus we get to possibly my favourite physics fact: we are all traveling at the speed of light.

WTF?

You read that right.  We are all moving at the speed of light.   More correctly, we are all moving at speed c.  The reason we don’t notice that is that we are moving through spacetime and we use our allocation of speed through space and time differently.  If you sit still, i.e. you don’t move in space, them you move through time at the speed c.  If you are a massless particle such as a photon then you move entirely in space and time does not pass for you at all. If you move at any lesser speed through space then your movement through time is such that your total speed in spacetime is c.

Fast Women and Slow Horses

George Best, a successful footballer and notorious drinker, once said “I blew a lot of my money on slow horses and fast women.  The rest I just squandered”.  Many horseback archers would sympathise about slow horses, having missed out on speed points.  I shall make no comment about fast women and horseback archers…

Technically speaking, there are no slow horses.  There are just horses that travel more in the time dimension than others.  A fast woman is presumably one who will go further while time seems to pass more slowly. 

I’ll leave it there…

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Technically, all three of these horses are traveling at the same speed…

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!

What Else Can Einstein Tell Us About Horseback Archery?

Ce article est a propos de le theorie de la relativite.  Ce n’a pas de utilite pratique, mais si tu le comprends en Anglais, Fred, je serais tres impressionne!

This article is about arrow speed really.  It’s just not going to help you to improve yours.  I promise I’ll get there!  Speaking of getting places…

Where Are You Going?

As I type this I am sitting on a chair in my kitchen.  I may think of myself as not moving.  This would be quite spectacularly wrong: the Earth is rotating at about 1000mph.  It is also orbiting the sun at about 66,000mph.  The solar system orbits the centre of the galaxy at nearly 500,000mph.  Our galaxy is approaching the constellation Hydra at a speed of about 1.3 million mph.  If there is one thing I am definitely doing, it is moving.  (Since I am in the UK, I am also on a tectonic plate that is moving away from North America at about 1” per year.  I shall not make any political quips whatsoever.)

The point of all this, apart from being generally awesome facts, is that everything and everybody is moving.  This is the basis of the theory of relativity: since everybody is moving you can treat any given thing as being stationary and everything else as moving relative to it.

When I flew to the USA this summer I basically sat in my seat for a several hours while America got closer.  From my point of view the Earth moved beneath my seat.  An observer on the ground would say I flew 5,000 miles.  An observer on the sun, however, might say that owing to the rotation of the Earth, I stayed pretty much still while America travelled to me and the Earth moved in space as well, which neither I nor the ground-based observer would see.  We see the sun moving through the sky, but in a different way from the way that an observer in the middle of the galaxy would see it, and so on.

All motion is relative and we are all free to consider ourselves as standing still while everything else moves around us (except for things that do not move relative to us, such as me and the aeroplane).  This is sensible in and of itself but it gets weird fairly soon…

Relativity and Mogu

Imagine a mogu match.  For those who don’t know, this involves one person towing a cloth-covered wicker ball while one or more others chase him and try to shoot the ball with blunt arrows dipped in ink.  It’s brilliant fun and quite spectacularly dangerous.

Your friend is towing the ball and you are 10m behind it, both of you riding at 10m/s. You shoot an arrow at 50m/s directly in front of you (at the mogu ball, but let’s ignore up and down).  You observe the arrow’s flight from your horse and I observe it from my seat in the stands.

From my point of view you are moving at 10m/s and your arrow at 60m/s (50m/s from the bow and 10m/s from the speed of the horse.  From your point of view, you and the ball can be thought of as both standing still while the world moves.  You therefore see the arrow as travelling 10m.  It takes 1/5s (10m @ 50m/s).  I, on the other hand, see the arrow travel the 10m between you but as far as I can see it has also travelled another 2m because you and the ball are moving as well.  I therefore observe the arrow travel 12m at 60m/s.  Like you, I see it taking 1/5s.

Arrows of Light

Now imagine that you shoot an arrow made of light, which necessarily travels at c: the speed of light.  This is roughly 300,000,000 m/s.  One of the weirdest things about physics is this: everyone observes the same speed for light, regardless of your motion relative to the source of the light.  We will both observe the light arrow travelling at speed c.

Now rerun the mogu game.  You still see the arrow travel the 10m between you and the ball.  I still see it travel slightly further because the whole scene is moving past me.  This is disturbing because we both see the same speed of the light arrow but I see it travel further than you do.

Speed = distance / time.  Therefore time = distance / speed.  Since we both see the same speed but since I see it travel further than you do, I must see it take more time to hit the ball (you hit!  Well done!).  The speed of light is constant; time is not.

The faster you and the ball are travelling, the further the light arrow will seem to travel from my viewpoint, and therefore the slower I see time running for you and the ball.  This is called time dilation: the faster you move, the slower time passes for you as seen by somebody else.   This is very odd indeed but it is a real effect that can be measured in the real world.  It is not to do with light, it is just easiest to explain using light.  It affects all objects at all speeds.  But at everyday speeds it is a really small effect.

What’s Your Draw Length?

Let’s leave mogu for a nice simple Korean or Hungarian track.  We will agree on the moment that you pass the start and finish lines.  We will also agree on our relative movement, although we may disagree as to who is moving.  We will disagree on the time you take to travel the distance (or the distance takes to travel past you), because of time dilation.  If we agree about the speed but disagree about the time then we must disagree about the distance (speed = distance / time, so distance = speed x time).  This is the phenomenon of length contraction.  Things in motion relative to you get shorter from your point of view.  Since you consider yourself to be sitting still and the track to be moving, the track gets fractionally shorter for you.  This is why you can travel it in 10s of your time but still take slightly longer form my point of view: from my standpoint the track has not contracted so you have travelled further.

Conversely, as I sit in the stands (why do we sit in stands?  Shouldn’t they be called sits?) the track is stationary for me but you are moving.  You become slightly shorter in the direction of your travel.  The same is true of anything else that is moving relative to me.  This includes your arrows (real arrows now, not arrows of light).  When somebody asks you how long your arrows are, the correct answer is to ask them whether they mean before or after you release it, since an arrow shot at 180fps will be shorter by about 0.000000000000015m, which is about the length of a proton.  And no matter how consistent you think your draw length is, it is shorter when you are riding than when you are standing still.  Just not by much.

The Judge’s Decision is Final

The net result of all this is that if you measure yourself completing a 90m track in 9s then sufficiently accurate timing gates would record your time as 9.0000000000000005s.  The difference, 5 femtoseconds, is about one hundred-million-millionth of the time it takes you to blink.  In other words, it is an inconceivably small difference.  But it is really there.

All competitors are reminded that scores are awarded on the basis of the time shown by the timing gates and that we only go to two decimal places…

Einstein’s Horsebow

Every archer knows that a bow’s draw weight increases as you pull it back (I am ignoring compound bows, with their pulleys and cams. I advise everybody else to ignore them as well). Perhaps my favourite fact of all time is that as you pull the string back you also increase the physical mass of the bow.

As every schoolboy once knew (I feel really old…), Albert Einstein proved in 1905 that the energy of an object at rest is equal to its mass multiplied by the speed of light squared. In mathematical notation, this is written as E=mc2, the most famous equation in history, known even to people who have no idea what it means. Since the speed of light, c, stays the same (roughly 300,000,000 metres/second), any increase in energy must lead to an increase in mass.

A bow is a device for storing energy and then releasing it suddenly into an arrow. As you draw a bow it stores the energy that goes into drawing it (some of the energy, anyway). This means that a bow at full draw has more energy than a bow that is braced. Since energy equals mass x a constant (c2), a drawn bow has a greater mass than an undrawn one. As you pull your bow, you make it heavier.

Of course, you don’t notice this effect. That’s because if E=mc2 then m=E/c2. No matter how strong you are, the energy you put into your bow is tiny compared to c2 (90,000,000,000,000,000). The increase in mass when you pull back an average draw weight horsebow is about the same as adding a single bacterium to the bow. That’s about a millionth of a grain of sand.

This is of no relevance whatsoever to the practical workings of bows and arrow, which is what I am going to write about next, but it is a fantastic fact.

And no, Einstein did not have a horsebow.