Arrow Ballistics Calculation

This post was inspired by a couple of videos[1] on the excellent YouTube channel Tod's Workshop involving testing the maximum range of crossbow bolts. Comments discussed whether 45 degrees was really the optimal angle for firing crossbow bolts for maximum range, and most agreed that an angle between 30 and 40 degrees was ideal. However, physical testing revealed that maximum range was achieved at close to 45 degrees.

This seemed like an interesting physics problem that could reasonably be written in JavaScript or TypeScript, so I decided to take it on. The partial differential equations are as follows:

$$m \frac{d v}{dt} = -D(\psi) - m g \sin{\theta}$$

$$m v \frac{d \theta}{dt} = L(\psi) - m g \cos{\theta}$$

$$I \frac{d^2 \psi}{dt^2} = M(\psi)$$

$$D = C_D(\psi) A_D(\psi) v^2 \rho$$

$$L = C_L(\psi) A_L(\psi) v^2 \rho \sin{\psi}$$

with the following definitions:


$m$	Mass
$v$	Speed
$\psi$	Angle of attack
$\theta$	Angle of motion
$g$	Gravitational acceleration
$I$	Moment of inertia
$D$	Drag force
$C_D$	Coefficient of drag
$A_D$	Drag area
$L$	Lift force
$C_L$	Coefficient of lift
$A_L$	Lift area
$\rho$	Air density
$M$	Pitching moment

The lift and drag forces depend on the angle of attack. This relationship is approximated by finding the projected area of the body in the direction of motion. Motion is calculated with a 4-th order Runge Kutta integrator.

Lift was added for completeness and doesn't have much of an effect at close to zero angle of attack. However, setting a high lift area and a low mass leads to interesting results.

Total Distance
Time of Flight
Energy
Momentum

I liked learning TypeScript for this project. It was easy to learn coming from a python background and npm made it easy to manage the build environment. However, I found a lot of documentation and community around JavaScript and TypeScript to be a bit industrial. The focus is not on helping people make little calculators for their personal blogs. Finding how to get a slider to interact with a JavaScript object was a chore, because almost all of the results were something to the effect of use the [] framework on your webserver running node.

The source code for this is available on github.

[1]: Distance - Which angle is best? 850lbs Crossbow DISTANCE TEST


\(m\)	Mass
\(v\)	Speed
\(\psi\)	Angle of attack
\(\theta\)	Angle of motion
\(g\)	Gravitational acceleration
\(I\)	Moment of inertia
\(D\)	Drag force
\(C_D\)	Coefficient of drag
\(A_D\)	Drag area
\(L\)	Lift force
\(C_L\)	Coefficient of lift
\(A_L\)	Lift area
\(\rho\)	Air density
\(M\)	Pitching moment