External Ballistics Calculations - James Prescott

External Ballistics Calculations

Here is a detailed description of the calculations that I do, in pseudo-code. I actually use Prograph, an interesting true visual programming language that didn't catch on. The calculations include a significant loop (thousands of iterations), which makes them unsuitable for implementation in a regular spreadsheet. These calculations do not include my more recent modifications that take into account angled fletching and projectile rotation.

These calculations are suitable for projectiles with velocities up to about 240 m/s (790 fps), at Standard Temperature and Pressure.

The following calculations are done in SI metric units. To calculate using units of pounds and feet per second (fps), I recommend taking your input units and converting them to metric. Then do the calculation. Then convert back to Imperial units. Here are the conversion factors required.

1 m = 3.28084 ft
1 kg = 2.20462 lb

Initial variables:

A : Angle of elevation, in radians
V : Velocity at release, in m/s
M : Mass of projectile, in kg
C : Coefficient of drag, pure number
D : Diameter of head of projectile, in m
F : Drag factor, in m^(-1)
T : Time increment, in s
H : Height at release, in m
G : Acceleration due to gravity, in m/s^2

F = -0.473595 * C * D^2 / M
T = 0.004 : for most computations
T = 0.001 : to be painfully accurate
G = -9.80665

Loop variables:

v : Velocity, in m/s
x : Horizontal velocity, in m/s
y : Vertical velocity, in m/s
s : Range, in m
h : Height, in m
t : Flight time, in s

Prior to the first iteration:

v = V
x = V * cos ( A )
y = V * sin ( A )
s = 0
h = H
t = 0

Iterate generating new values for the loop variables:

sA : Sine of angle of travel
cA : Cosine of angle of travel
d : Acceleration due to drag
dx : Horizontal acceleration
dy : Vertical acceleration

sA = y / v : avoid doing trig functions
cA = x / v : avoid doing trig functions
d = F * v^2 : drag depends on square of velocity at these speeds

dx = d * cA : drag slows horizontally
dy = G + d * sA : drag slows vertically, plus gravity slows on ascent and accelerates on descent

s = s + ( x * T ) + ( 0.5 * dx * T^2 ) : s = ut + 1/2 at^2
h = h + ( y * T ) + ( 0.5 * dy * T^2 ) : s = ut + 1/2 at^2

x = x + ( dx * T ) : v = u + at
y = y + ( dy * T ) : v = u + at
v = sqrt ( x^2 + y^2 ) : avoid doing trig functions

t = t + T : tick

Exit if h<=0 : we've hit the ground

After the final iteration, correct s, h, and t so that they match the exact moment of impact with ground:

dh : Correction to h
ds : Correction to s
dt : Correction to t

dh = h
h = h - dh
dt = dh / y
t = t - dt
ds = x * dt
s = s - ds

And that's it.

Here is what the interface for the program currently looks like. You'll note that for convenience I take some inputs in Imperial units. They are immediately converted to SI metric units before use. You'll also note that the first three values (elevation, velocity, and drag coefficient) are set up to permit iterations so as to facilitate solving some problems by successive approximation. For example, given a number of measurements of maximum range and flight time one can solve for the coefficient of drag by successive approximation.