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…

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