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Don't let the word 'physics' put you off - read the article for an insight into how modern science can help us understand the history of the weapon we now use for sport (and ignore the formulae if you must!) It is generally believed that the main factor responsible for the English victory at the battle the Agincourt in 1415 was the longbow. Gareth Rees describes from a physicist's point of view why we believe this simple weapon was so devastatingly effective.

Battle of Najera

In 1415 King Henry V of England took a small army to France to try to enforce the English claim to the French throne. By late autumn things were not going well for the English. The weather was poor, and Henry's army was short of provisions, exhausted, and badly stricken with dysentery. Henry decided to make for his stronghold at Calais for the winter, but the French saw an opportunity to annihilate the English forces and advanced with a huge army to do battle.

The two armies met at the little village of Agincourt on the evening of 24 October, after the English forces had marched 260 miles in 17 days. King Henry's offer to buy peace was rejected, and on the following afternoon one of the decisive battles of the Hundred Years' War took place.

The battle of Agincourt has entered English folklore - and, indeed, popular culture as a result of the Laurence Olivier and Kenneth Branah film versions of Shakespeare's Henry V. No more than 6000 soldiers in the service of the English king faced about 50,000 French soldiers. Apart from the gross disparity in numbers, the other substantial difference between the two armies was in their use of the longbow. The English army was composed largely of bowmen (about 80%), whereas the French used virtually none.

The massive French cavalry charge was met by a storm of English arrows, as a result of which the cavalry fled back through the front columns of the French infantry. The English soldiers waded into the chaos armed with hatchets and billhooks and, backed up by their own small cavalry and the threat of their longbows, succeeded in dispersing the whole French army.


The bow - any bow - is basically a spring. The archer does work on this spring as he draws the bow, storing potential energy in the elastically deformed bowstave. When he releases the string, some of this potential energy is converted into kinetic energy of the arrow, through the action of the tension in the bowstring accelerating the arrow, the arrow leaves the bow at high speed and wings its way towards its target. Its orientation is stabilised by three fletchings at the rear of the arrow.


If we draw a graph of the force F needed to draw the arrow back through a distance x, the area under the graph represents the work done on the system and hence the potential energy stored in the bow. If the graph is a straight line through the origin (i.e. the bow behaves like a spring that obeys Hooke's law), this energy will be equal to Fx/2 (see diagram ).

energy curve

In fact, the graph of F against x is usually a curve, because of the complicated shape of the bowstave (it is thicker in the middle and thinner at the ends) and the fact that the tension in the bowstring does not always pull in the same direction relative to the ends of the bow. We deal with this by introducing an efficiency term e, and writing the total energy stored as eFx/2. While a modern bow made of composite materials can have an efficiency greater than 1, a medieval longbow would have had an efficiency of about 0.9.


The simplest assumption we can make is that all of the potential energy eFx/2 is converted to kinetic energy of the arrow. Writing m for the mass of the arrow and v for its initial speed, we would then have

½mv2 = ½ eFx
v = (eFx/m)-2

In fact, this is always an overestimate of the initial speed of the arrow. The main reason for this is that, at the instant when the arrow leaves the bowstring, parts of the bow itself are moving. These parts will thus have some kinetic energy which, like the kinetic energy of the arrow, has been supplied by the potential energy stored in the bow. Exact calculations of this effect are extremely difficult, and can only really be done by computer modelling. However, we can get a rough idea when we realise that the speed of a particular part of the bow must be proportional to the speed of the arrow. We can thus write the kinetic energy of the bow as:

k ½ Mv2

where M is the mass of the bow, and k is a factor which represents the sum of the kinetic energies of all the parts of the bow. Experiments and computer models show that, for a medieval longbow, k is typically between 0.03 and 0.07, depending on the precise design of the bow. Thus, we should really write

½ mv2 + k ½ Mv2 = ½ eFx

which we can rearrange to give a formula for v:

v = ( eFx / (m + kM))-2


The formula we worked out above can actually tell us something about the ideal material for a bow. Obviously, the initial speed v of the arrow should be as large as possible, and this can be achieved by making the term eFx as large as we can manage, and the mass of the bow M as small as possible (we can't do anything much about the constant k, and, as we shall see below, there are good reasons why we can't make m, the mass of the arrow, too small). Since eFx is twice the elastic potential energy stored in the bow, we need to make the elastic energy stored per unit mass of the bow as large as possible. This is achieved by choosing a material with a large elastic modulus, a low density and a large value of the maximum allowable strain before permanent deformation occurs. We can say, in effect, that the ideal material is light, tough and springy.

Medieval bowyers had no choice of material but wood. However, different species of tree give wood of very different properties, and the best is the wood of the yew tree, which has a maximum elastic energy storage per unit mass of about 700 J kg-1, about as good as spring steel. The best medieval bows were made of yew. In 1571, Roger Ascham wrote in his book Toxophilis: 'As for Brasell, Elme, Wych and Ashe, experience doth prove them to be mean for blows, and so to conclude, Ewe of all other things is that whereof perfect shooting would have a bow made.'


Unfortunately, virtually no bowstaves from the medieval period have survived. So how do we know how powerful the bows would have been? Some evidence can be obtained from the arrows, which have survived. Because the 'archer's paradox' demands that a particular bow needs an arrow of suitable spine (stiffness) then by measuring the properties of a medieval arrow we can estimate the strength of the bow for which it was designed. When these calculations were done, the answers were almost unbelievable. They suggested that the force needed to draw a medieval longbow could have been in the range 110 to 180 pounds (500 to 800 Newtons). Although these figures are astonishing, they have been confirmed by calculations based on the bows found in the wreck of Henry VIII's ship Mary Rose, which sank in 1545. It seems likely that in 1415, when archery was at its peak in England as a technique of warfare, bows would have been no less powerful than in 1545, when archery was already beginning to lose ground to firearms.


In modern competitive archery, arrows are usually aimed at an angle not too far above the horizontal, to give them a short, low and fairly accurately predictable trajectory. In a medieval battle, a completely different strategy was adopted. Massed ranks of archers would aim their arrows high, to achieve a large range, without particularly careful aiming. The maximum range would have been a factor of great importance in deciding the strategy for a battle, and this obviously depends on the initial speed v of the arrow.

If air resistance can be ignored, the maximum range of a projectile is V2/g (where g is the acceleration due to gravity), obtained by aiming at 45° to the horizontal. We can calculate this 'ideal' range, using the information we already have. We will use our formula

v = (eFx / (m + kM))-2

and take e (efficiency) = 0.9; F (force required to draw the bow fully) = 700 N (154 lbs); x (distance through which the arrow is drawn back) = 0.58 m; m (mass of the arrow) = 0.060 kg; k (factor to allow for the kinetic energy of the bow) = 0.05; and M (mass of the bow) = 1 kg. This gives v = 57.6 m s-1 and an 'ideal' range of about 340 m.

However, the air resistance on an arrow is not negligible. Experiments in wind-tunnels show that the drag force is dependent upon the speed of the arrow, so that we can put Fdrag = cu2 where c is a constant for a particular arrow and u is the speed of the arrow. The equation of motion of a body under the influences of gravity and of this type of square-law drag force is difficult to solve exactly, but there is a convenient approximation. The maximum range is given to an accuracy of a few percent by the formula

v2/g ( 1 + cv2/mg)-0.74

where v is the initial speed and m is the mass, as long as the value of (cv2/mg) is less than about 10. (Now we can see, why arrows of low mass are not desirable. The maximum range is reduced if m is decreased.) A typical medieval war-arrow would have had a mass m of about 0.060 kg and a value of c of about 10-4 N S2 m-2, giving a value of (cv2/mg) of 0.56 if the initial speed was 57.6 m s-1. The approximate formula is therefore valid, and we can calculate the maximum range as about 240 m.

Interestingly enough, we can confirm that this is the right sort of value. In 1590, Sir Roger Williams wrote: 'Out of 5000 archers not 500 will make any strong shootes . . . few or none do anie great hurt 12 or 14 score off.' A 'score' is twenty yards (18.3 m), so Sir Roger was complaining that the archers of his day (nearly 200 years after Agincourt) were so feeble that they could barely manage to shoot a distance of 220 to 260 m.


We now have a good basic understanding of the flight of the medieval war-arrow. Shot from an extremely powerful bow, the 60 gram arrow would be given an initial speed of almost 60 m s-1. Aimed high in the air, this arrow would have a maximum range of 240 m, and it would arrive with a speed of between 40 and 45 m s-1 (we have not calculated this figure, because there is no simple approximation for it, but it comes from the same detailed calculations that are used to find the maximum range). The obvious question now is what would such an arrow have been capable of doing? Most of the soldiers at whom these heavy war arrows were directed would have been wearing armour. At the time of Agincourt, a typical suit of armour had a mass of between 30 and 45 kg and was made of wrought iron, which is rather soft. Obviously, carrying this extra mass was a great inconvenience to the soldier inside the armour, and, to try to keep the mass down, the thickness of the armour varied according to the part of the body being protected. The thickest armour was up to 4 mm thick, and the thinnest about 1 mm. Experiments (not using live-targets!) suggest that, while arrows would easily penetrate 1 mm of armour, the vital areas of the body would have been very unlikely to be hit. Probably the effect of a massive hail of fast-moving heavy arrows, such as the French encountered at Agincourt, would have been to cause very many disabling injuries, but perhaps only one arrow in a hundred would have killed the man it struck. Naturally, the chance of an unarmoured man surviving a blow from such an arrow would have been very much less.

It is sobering to combine these facts with some historical data. Henry had approximately 5,000 archers at Agincourt, and a stock of about 400,000 arrows. Each archer could shoot about ten arrows a minute, so the army only had enough ammunition for about eight minutes of shooting at maximum fire power. However, this fire power would have been devastating. Fifty thousand arrows a minute - over 800 a second - would have hissed down on the French cavalry, killing hundreds of men a minute and wounding many more. The function of a company of medieval archers seems to have been equivalent to that of a machine-gunner, so in modern terms we can imagine Agincourt as a battle between old-fashioned cavalry, supported by a few snipers (crossbow-men) on the French side, against a much smaller army equipped with machine guns. Perhaps from this point of view the most remarkable fact about the battle is that the French ignored the very great military advantages of the longbow.

Gareth Rees

Gareth is Head of the satellite remote-sensing group at the Scott Polar Research Institute in Cambridge. He became interested in the aerodynamics of archery while studying molecular aerodynamics for his PhD.

Reproduced from Physics Review January 1995 by kind permission of the author and publisher and republished in InSight, the Stortford Archery Club Newsletter, Issues 5 & 6, Summer and Autumn 1995.

Photograph: Biblèotheque Nationale, Paris/The Bridgeman Art Library-The Battle of Najera, 1367, between the forces of Edward the Black Prince of England and Enrique II of Castille.

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