The symbolic algebra of the medieval longbow

Today’s post is going to be a bit more abstract and a bit less applied than usual, because today I want to take Python’s Sympy symbolic algebra library for a spin. As a narrative subject, we’ll be looking at the medieval English Longbow. We’ll start with some physics by modelling the deterministic trajectory of the arrow. Then we’ll add some statistical noise to make it a bit more realistic. Lastly, we’ll do some simple Bayesian inference on the range of the longbow. And we’ll do as much of it as we can via symbolic algebra.

These are English longbowmen. The English (or Welsh) Longbow is around 1.8-2m tall, can fire 10-12 arrows per minute in the hands of a skilled archer, and its steel-tipped arrows can penetrate the armour of a medieval knight. The “draw-weight” of a longbow is considerable, and skeletons of longbow archers often have enlarged left arms. Over a lifetime, the archer’s body was actually deformed by their tool of trade. So, how far could it shoot an arrow? Let’s figure that out with physics. According to Longbow Speed Testing (!), the velocity of an arrow leaving the bow is 172-177 feet per second. Say 53m/s in metric.

The trajectory of a projectile can be computed using classical mechanics:

$y = x \tan{\left (\alpha \right )} - \frac{g x^{2}}{2 v^{2} \cos^{2}{\left (\alpha \right )}}$

Where:

• $y$ is the height of the arrow
• $x$ is the horizontal distance that the arrow has travelled
• $\alpha$ is the angle at which the arrow is fired
• $v$ is the velocity of the arrow as it leaves the bow
• $g$ is the acceleration due to gravity (~$9.8 m/s^2$ on the surface of the earth)
• (We’re ignoring wind resistance)