Useful calculations
Ballistic Coefficient (BC)
A measure of aerodynamic efficiency. Primarily varying with the projectile’s Sectional
Density, the BC value is altered to some extent by its shape (or form).
BC = SD / FF
where:
BC = Ballistic Coefficient (Lb/in²)
FF = Form Factor = Cd.proj / Cd.ref
where:
Cd.proj is the actual Drag Coefficient of the projectile at some given velocity and
Cd.ref is the Drag Coefficient of a reference projectile at the same velocity.
SD = Sectional Density = W / d² (Lb/in²)
where:
W = Projectile weight (Lb) and
d = Projectile diameter (Inch)
Since the Form Factor is a ratio (a dimensionless quantity), the Ballistic Coefficient
has the same units as Sectional Density (Lb/in²) although these are rarely stated
explicitly in common usage.
Note also that the BC value - given the appropriate Drag Law - must be constant
since its constituent parts (FF, W and d) are all constant.
i.e., The BC value is not a function of velocity, temperature, pressure, altitude or
anything else - just the projectile weight, diameter, Form Factor and the applicable drag law.
Density (Air)
Drag Force
Pr = P / 0.00029529983071
Pv = H * 6.1078 * Pow(10.0, ((7.5 * (T))/((T)+237.3)))*100.0
ρ =((Pr - Pv) / (287.058*(T+273.15)) + (Pv/(461.495*(T+273.15))) / 16.02 Lbm/Ft³
If the Ambient Air Pressure is unknown then it can be estimated from:
P = MSLP * exp(-Alt / 7.0) “Hg
where:
P = Ambient Air Pressure “Hg
MSLP = Mean Sea Level Pressure “Hg
Alt = Elevation = Y * 0.3048 / 1000.0
where:
Y = height above sea level Ft
In approximate terms - and ignoring Temperature and Relative Humidity - the Ambient Air Pressure will decrease by approximately 1” Hg for every 800 Ft of elevation.
i.e.,
P = MSLP - Elevation / 800 “Hg
where:
P = Ambient Pressure “Hg
MSLP = Mean Sea Level Pressure “Hg
Elevation = Height above sea level Ft
Air density (as used for the Drag Force calculations below) can be calculated from:
1) the current Ambient Air Pressure P (“Hg) , - most important
2) the current Ambient Air Temperature T (ºC) - not too important, and
3) the Ambient Relative Humidity H. - usually negligible
Classical Fluid Dynamics has it that:
Fd = ½ * ρ * v² * Cd.proj * A
where, in a ballistics context:
Fd = Drag Force (Lbf)
ρ = air mass density (Lbm/Ft³)
v = projectile velocity relative to the surrounding air (Ft/s)
Cd.proj = Drag Coefficient at velocity ‘v’
A = projectile cross sectional ‘projected’ area (Ft²)
... which is not much use in itself but given that ...
BC = Sd / FF = W / (d² * FF) = W * Cd.ref / (d² * Cd.proj)
from Ballistic Coefficient above and, by Newton ...
F = M * A
where:
F = Force (Lbf)
M = Mass (Poundal) and
A = Acceleration (Ft/s²)
... the three formulae can be (and are) combined and manipulated to produce the basis of the time-slice / point-mass method for calculating trajectory parameters against a known reference Drag Law:
Kinetic Energy
R = Cd.ref(v) * v² * ρ * Q / BC
where:
R = retardation (Ft/s²)
v = instantaneous velocity (Ft/s)
Cd.ref(v) = reference Cd value at velocity ‘v’
ρ = air mass density (Lb/Ft³)
Q = cumulative constant derived from the conversion of the units of the
various elements = 𝝅 / (2³ * 12²) = 𝝅 / 1152 = ~0.00272707696
BC = Ballistic Coefficient (Lb/in²)
From general physics: KE = ½ * M * V²
if the projectile weight is in Grains and the velocity in Ft/s then:
Magnification (Calibration / Actual / Indicated)
KE = W * V² / K
where:
KE = Kinetic Energy (Ft·Lb)
W = projectile weight (Grains)
V = velocity (Ft/s) and
K = (2 * 7000 * 32.17405) = 450436.7
The units of Kinetic Energy are Distance (Ft) * Force (Lb), hence Ft·Lb
This is sometimes written as ‘Ft.Lbf’ or ‘FtLb’ or - mostly in North America -
as ‘FPE’ (Foot-Pound-Energy). Never, ever as ‘Ft/Lb’ which is clearly nonsensical.
Calibration Magnification: The design magnification at which one milliradian is subtended by exactly one mil-dot.
Actual Magnification: The real magnification with respect to the calibration magnification.
Indicated Magnification:
SFP scopes: The magnification indicated on the scope’s ‘zoom’ ring.
Since - by definition - one mil-dot subtends 3.6” at 100 Yards (or 10cm at 100m) at the calibration magnification, the basic equation can be written as:
Ma = R * 0.036 * D * Mc / H;
where:
Ma = Actual Magnification
R = Range (Yards)
Mc = Calibration Magnification
H = Target Height and
D = the number of Mil-dots spanned by H at R
and this can be rearranged as:
R = H * Ma / (D * Mc * 0.036);
or
Mc = Ma * H / (R * 0.036 * D )
or
H = R * 0.036 * D * Mc/ Ma;
or
D = H * Ma / (R * 0.036* Mc)
... depending of the parameter to be calculated.
Speed of Sound
Temperature conversion
For ambient temperature in Fahrenheit:
SoS = 49.0223 * ✓( °F + 459.67 ) Ft/s
where:
SoS = Speed of Sound in Ft/s
°F = ambient temperature in Fahrenheit
For ambient temperature in Celsius:
SoS = 65.7703 * ✓(°C + 273.15 ) Ft/s
where:
SoS = Speed of Sound in Ft/s
°C = ambient temperature in Celsius
Parallax Error (Maximum)
°C = (°F - 32) / 1.8
or
°F = (°C * 1.8) + 32
where:
°C = degrees Celsius and
°F = degrees Fahrenheit
PE = Abs( D * (F - T) / (2.0 * F) )
where:
PE = maximum parallax error (units any of Inch/cm/mm)
D = objective diameter (units same as PE)
F = Focus distance (units any of Ft/Yard/Metre)
T = Target distance (units same as F)
Clearly, Parallax error can be eliminated by accurately focusing on the target
(i.e., if F = T then F - T = 0 so PE = 0) or mitigated by using a scope with a small objective lens (since PE is proportional to D)
Horizontal Wind Drift
Vertical Wind Drift (aka. Aerodynamic Jump)
Dz = W * (T - R / MV) Ft
where:
Dz = lateral deflection at target (Foot)
W = crosswind vector (Ft/s)
T = time of flight between muzzle and target (sec)
R = range between muzzle and target (Ft)
MV = muzzle velocity (Ft/s)
The general solution for Aerodynamic Jump is:
Dv = ((Ix * CLa) / ( m * d² . CMo)) * (2𝜋 / n) * (W / Vo) * R
or Dv = Cg * Kb * Kv * R
where:
Dv = vertical displacement
Projectile-related item - Cg :
Cg = (Ix * CLa) / ( m * d² * CMo)
where:
Ix = projectile axial moment of inertia
CLa = projectile lift force coefficient
m = projectile mass
d = projectile reference diameter
CMo = projectile overturning moment coefficient
Barrel-related item - Kb :
Kb = 2π /n
where n = the number of calibres per turn = p / d
where:
d = projectile reference diameter
p = rifling pitch
so: Kb = 2π / (p / d) = 2π * d / p
Velocity-related item - Kv :
Kv = (W / MV)
where
W = Crosswind vector magnitude
MV = Muzzle velocity
Range variable - R :
R = Range
Since Dv has the same dimension as R and the Kb and Kv terms are ratios (and therefore dimensionless quantities) then Cg must also be dimensionless.
Re-arranging and expanding for a specific solution:
Cg = Dv / (Kb * Kv * R)
In the specific solution, we can supply all of the outstanding values from measurement, experimentation and calculation.
e.g., A 0.177 diabolo airgun pellet (d = 0.177 Inch,BC[GA₂] = 0.02), shot from a barrel having a right-hand 15” (p = 15.0 Inch) rifling pitch at 780 Ft/s (MV = 780 Ft/s) is zeroed (under still conditions) and the impact point at 60 Yards (R = 60 * 3 = 180 Ft.) is noted. Under windy conditions the pellet is seen to impact 1.0” high (Dv = 0.083 Ft.) and 6” (Dh = 0.5 Ft.) to the right when aimed at the same point as per the previous still conditions.
Time of flight (from MERO) = T = 0.2741 sec. so
the windspeed W = Dh / (T - R / MV) = 0.5 / (0.2741 - 180 / 780) ≈ 11.5 Ft/s
Dv = 1.0 Inch = 0.083 Ft,
Kv = W / MV = 11.5 / 780 = 0.015,
Kb = 2π * d / p = 6.28318 * 0.177 / 15.0 = 0.074,
and R = 180 Ft
so Cg = Dv / (Kb * Kv * R) = 0.083 / (0.074 * 0.015 * 180.0) = 0.4154