Re: Force of a Bullet vs. Force of an Arrow

Question:  A modern bullet vs. a medieval arrow which would have more 
strength behind it? Wouldn’t it be the bullet since it's made of metal 
while the arrow is simply made of wood?

Dear Krystina,  thank you for your question.  It is not an easy one since 
there are so many things to consider in both the case of bullets and of 
arrows.  I’m no expert in armaments but I’ll point out some of the things 
you need to think about.  

You used the words “force” and “strength” and although we know what you 
mean, we should take some care to use measures of the behaviour of arrows 
and bullets which would let us compare the two on a fair basis.  Let’s be 
honest, both bullets and arrows are used to cause damage, usually to 
animals including humans so we need a measure of this capability to cause 
damage.  In order to cause damage the projectile must use its kinetic 
energy, the energy it need to travel at speed, to split crush and deform 
the material being hit.  Often we use the ability of the bullet or arrow 
to penetrate a solid material.  At least that is a good starting point.

Let’s look at bullets first.  They are propelled by an explosion within 
the gun barrel and the expanding gases force the bullet down the barrel 
and out towards the target.  Guns often need to be carried easily so the 
size of the cartridges and bullets is a compromise between causing the 
maximum damage and being easy to handle.  The kinetic energy of the bullet 
( and arrow ) is  ½ x M x V ^2 where M is the mass, and V the velocity.  
The mass depends on the size and density of the bullet and the velocity 
depends on things like the amount of explosive, the mass of the bullet and 
the design of the gun.

Arrows on the other hand are projected by a bow which has been drawn by 
the archer’s muscles.  This web-site has some interesting information 
about the history of the long-bow:-  
http://www.history-magazine.com.  

We can define energy as the force applied times the distance over which 
the force is applied.  In the case of a bow, the archer pulls back the 
string using at first a small force but as the bow bends the force needed 
to pull further builds up.  Some say that a medieval bow could require 
160lbs “draw strength” which is the force needed for the last part of the 
draw where most of the energy is stored up. On release of the string, most 
of the stored energy is transferred to the arrow. Arrows could be up to 3 
feet long so even though they were made largely of wood this weight of 
wood plus the weight of the normally metal arrow-head could be much 
greater than a rifle bullet.  This is a factor which results in the arrows 
traveling more slowly than bullets which usually travel faster than sound.

Once the projectile is on its way, it can only lose energy through air 
resistance.  This energy loss is important because it is the energy on 
impact not the firing energy that counts. The bullet and arrow are again 
very different.  Because the arrow travels more slowly, the air resistance 
due to pushing air out of the way is lower.  On the other hand the shaft 
and particularly the flights will cause frictional drag.  ( The fact that 
the drag due to the flight is greater than the tip is how flights keep the 
arrow straight ! ) 

The final amount of energy depends on these losses and therefore of course 
how far the projectile travels before impact.  I should think arrows lose 
a smaller fraction of their energy.  This may help to explain why arrows 
can penetrate solid materials so well.

Another factor is their shape.  The arrow-head is very sharp, that is it 
has a very small cross sectional area at the tip.  The size builds up 
slowly and this is good for air resistance and particularly good for 
penetrating solids.  Arrows shot by the famous Welsh bowmen at around the 
time of Henry the Fifth in Britain could penetrate solid oak, four inches 
thick and poke through by over six inches.  

Hope that helps even though there is no clear winner – at least not as 
clear as Henry was against the French thanks to the Welsh ( how did you 
guess I was Welsh ? ).

[note added by MadSci Admin:  If you can find some information,
maybe on the Web, about typical bullet masses and velocities, and
typical arrow masses and velocities, the kinetic energy could be
compared.]