THE
PHYSICS OF
MEDIEVAL ARCHERY
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.
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.
A SIMPLE MODEL OF A BOW
AND ARROW
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.
ENERGY STORAGE IN THE
BOW
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
).
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 SPEED OF THE ARROW
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
or
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
WHAT SHOULD THE BOW BE
MADE OF?
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.'
HOW POWERFUL WERE THE
MEDIEVAL LONGBOWS?
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.
MAXIMUM RANGE OF THE ARROWS
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.
THE EFFECTIVENESS OF MEDIEVAL
ARROWS
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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