How Heavy Should an Arrow Be? Flight Arrows, Crossbows, and the Physics Behind Them

This is a fairly good reconstruction of a medieval longbow war arrow. It's got the feathery flights at the back end for stability, and it's got the iron warhead at the front. And in between the two it's got a long wooden shaft which is quite thick. It has to be quite thick, it's much thicker than a target arrow or a hunting arrow, because the warbows used were very, very powerful. And the only way you could make an arrow stronger was to make it thicker. So this is a thick shaft to withstand the very substantial launch stresses. So the archer would draw back his arrow, and at this point it would be quite content and stationary. And then he would release the string, which would put a massive stress on one end of the arrow. And this pushing would cause the arrow to bend. And why would it bend? Well, it would bend because of something called inertia. Things that have mass are reluctant to move, and the more mass they have the more reluctant they are to move. So you've got a length of wood that would rather stay still, frankly, and you suddenly push it that way. And because there's a reluctance to move, which I'll represent with as a resistance in my hand pressing that way, as you put a massive stress on one end it bends. So, they had to be quite thick to withstand the launch pressures. But actually there is an inaccuracy in this reconstruction, because you'll see that it's equally thick all the way down its length. This is a modern piece of dowling, I think it's cedar wood that's been produced by a machine I imagine. And it's very easy for a machine to produce something today with even thickness all the way down its length. But, in fact, longbow arrows weren't perfectly even in thickness all the way down their length. And there's a reason for that. Now, I'm going to demonstrate with this piece of balsa wood, which fell out of the sky during a firework display. Now, where, when I press on the ends of this piece of wood, where will it break, do you think? Will it break at some random point? Well, it might, because there may be some unpredicted weakness in the wood. But assuming the wood is of equal strength all the way down its length, it actually will break in a predictable way. Now, if I take this adjustable spanner, yes, that's right, I called it an adjustable spanner because I am British. And I clamp this into the spanner, whoops, wrong way. And what you've got then is, let's imagine I'm infinitely strong, this is totally immobile. If I press down somewhere along the length of this, it will break. But where? Well, I put it to you, it's going to break here. And the reason is because this acts as a big lever when I push down. So where the leverage, yes, that's right, I said leverage, I'm British, is greatest, it will cause the arrow or bit of wood to break. OK, it split a bit because it went down the grain, but you can see it split up towards this end. So, if you get another one of these, it was a big firework display, and press in from either end, where will it break? Will it break where the stress is greatest because that's where the stress is being applied? Or at the other end? Well, it's going to break in the middle, isn't it? This broke actually surprisingly easily. It's going to break in the middle because this end is acting as a lever on the middle, and this end is acting as a lever on the middle. So the part of the arrow where the leverage of both ends is greatest in effect is actually the middle. So, yeah, by and large they will break in the middle. So, if you want to make an arrow that's strong enough to withstand the launch stresses, you need to make it thickest in the middle. But actually the ends won't be subject to so much stress. You could make those thinner and still the arrow would stay intact, would survive launch. And why would you want to do that? Well, you might want to make the arrow lighter. But there's another part to this equation. You see, as soon as you make the stick shorter, you then have to put more stress on it to get it to break. And, oh, that took a lot more. And you can see it still broke somewhere near the middle. And if we then take the stick that's shorter again, and now you'll understand why I'm wearing gloves because this is going to hurt, you have to put an awful lot of stress on it. Okay, it didn't break the arse because of where it split earlier. But you can imagine, every time I break this, it becomes harder to break by pushing it inwards from the ends. And, oh, that really hurts now, even through the gloves. So, what that means is that the crossbow bolt, or quarrel, by the way, as far as I understand it, the meaning of quarrel derives from the quar, as in quad, quadrilateral, quarter, four. A quarrel has a four-sided warhead on it, a sort of square section thing. And the bolt doesn't necessarily. So that, I think, is why quarrels are called quarrels. A quarrel in a crossbow, it's only about that long. But the launch stresses on it can be huge because crossbows are often very powerful. But because it's short, it doesn't have to be very thick to withstand the launch stresses. So a crossbow bolt can be the same thickness as a longbow arrow, even though it may be subject to enormously greater poundage of bow and launch stress. Now, where is a longbow arrow thickest, therefore? I said that at the point of greatest stress, so you might say in the middle. But a longbow arrow isn't actually thickest in the middle. And I think the main reason for this is the warhead. Now if I show you where the balance point on this is, it's about there, you see. It's way off the centre because this is the point in the arrow with the greatest inertia. Metal is much denser than wood. So just as the centre of gravity is shifted forwards, the inertia of the head is so much greater than the wood behind it, that when you launch the arrow, the point at which it bends greatest is also shifted forwards from the centre. So that is why when you look at a medieval arrow, they tend to be barrelled, yes, but they tend to be at their thickest somewhere towards the front. Okay, so that explains the shape of medieval arrows. It also explains why crossbow bolts don't have to be quite so thick. But there's more to this. How far will an arrow go? Well, it's based partly upon its weight. Think of a stone. How far could you throw a stone? Well, if I gave you a pebble like that, or a little stone, that's right, I said stone or pebble, not rock because I'm British, you couldn't throw it anything like as far as a great big rock like this. A big rock like this you could probably just throw far enough so it didn't land on your toe. But a pebble that you could hold in your hand like that you could throw way further. And why is that? Well, it's because of the inertia. The inertia of a really big thing is really difficult to overcome. So if you are of constant strength, then you won't be able to throw it very far. But you, being the same strength as you were when you tried to throw the large thing, trying to throw a smaller thing will be able to overcome its inertia and accelerate it up to a much higher speed, and so it'll go much further. So the lighter the thing, the further it goes, right? No, you know that's not true. Now if I take this stone here, and this ping-pong ball, you will see two objects of roughly the same size. So which of these do you think I could throw further? You know, don't you? I could throw this even though it's much heavier, much further than this. Why can't I throw a ping-pong ball very far? I can't throw it very far. Even though I can accelerate it up to very high speeds really easily because it's so light, I can't throw it very far because as soon as it leaves my hand it will start slowing down very, very quickly because of air resistance. Now if I'm on the moon where there's no atmosphere, I know this is different. But assuming I'm on Earth and I'm trying to throw it through air, this will not go very far. Because inertia is greater when something has more mass, and that's the reluctance to be accelerated. But momentum is also greater when something has more mass, and that's its reluctance to be decelerated. So the forces of air resistance trying to slow this down will have a very easy time of it because it has very little momentum. So in other words, the wind can affect a ping-pong ball quite a lot. Whereas when I throw this, it leaves my hand perhaps not going as fast, but air resistance will slow this down very slowly because it's got a lot more mass and a lot more momentum. And if I try to affect this with wind, Oh, I did actually get it to move, which I wasn't expecting. But it doesn't get affected by wind anything like so much. Now, just think how strong the wind is on something that's been launched from a high-powered warbow or crossbow. The air resistance when it leaves that weapon is going to be very, very high. Crossbow bolts, you may remember, are a lot lighter than longbow arrows. Which is good because that means they'll have very low inertia, which means you'll be able to accelerate them really quickly up to very high speeds. Which is handy because you've only got that distance of span to accelerate it from stationary to maximum speed. So having something very light is very useful. And those of you who've taken part in science experiments at school where you've got a little car which you have to accelerate using a rubber band or a gas canister or something, you'll know that the lighter the vehicle, the easier it is to get it up to very high speeds quickly. So, if you want to get something a long distance, you want a missile with a lot of range, you want it an ideal weight for the power of whatever your launch method is. If it's the power of your throwing arm for a stone, say for instance, there's an ideal weight of stone for a given thrower. If it's too heavy, then you won't be able to accelerate it very fast, and it won't go very far for that reason. And if it's too light, then air resistance will affect it too greatly and it won't go so far. So there's a perfect weight of stone for you to throw. And similarly, there is a perfect weight of arrow for a given power of bow or crossbow. I realised whilst editing this that perhaps I've been a little bit misleading, so I should clarify. Not all longbow arrows by any means were barrelled, but many were. If you want to shoot a Frenchman who's not very far away, you might want to clonk him with a really big heavy stick. So you just use a simple cylinder of wood for an arrow shaft and thwack him. But if he's a very long way away, you might select one of your specialist shooting Frenchmen who are a long way away arrows, which would be perhaps barrelled, and it's called a flight arrow. Arrows designed to go a very long way. So they're barrelled, so they're strong enough to survive launch, but they minimise weight by tapering to either end. And so they're light enough to go a long way, because light things go a long way because they're light. But of course these arrows would still be heavy enough not to be so light that the air resistance ... Remember the ping-pong ball? You were listening, good. So a light bolt with a thin bit of wood that's still, because it's short, strong enough to withstand the launch pressure, will survive the launch of a very, very high poundage, but short span and go up to a very, very high speed really, really quickly. Which is great. But remember the ping-pong ball. The quarrel is light. It doesn't have the momentum of something much heavier. So air resistance will have an easy time slowing it down, which is why crossbows were really amazingly deadly up to about 60 yards. Because you try a crossbow and, poosh, that bolt leaps off the weapon at very high velocity. But after about 60 yards it slows down really quite significantly. And if you twang one into the air to drop onto the enemy, they often report it as being unstable and doing this in flight as they come down. And so at extreme range, a man in armour being hit with a crossbow bolt, probably fine, doesn't bother him that much. But one of these dropping down out of the sky would still be stable, because they were long enough and had better flights than a quarrel. And when they hit you they would have the weight of the hull of that piece of wood pressing into you. So at very long range you really don't want to be hit by a longbow. Lindybeige!