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Started 27-Jun by hobbes154; 607 views.

27-Jun

Hi all,

This is just a quick step-by-step summary I have put together (including from posters here and Bryan Litz) of how ballistic coefficients are derived and what they mean. Trying to go a bit beyond Wiki or the section in Tony's BASIC BALLISTICS. Would appreciate

i. Any corrections.

ii. Link to a better explanation than this, if it exists?

Not looking for all the practical details, just making sure I have the concepts straight.

- Measure bullet velocity vs. distance or time (ideally a continuous measurement - Doppler radar? - otherwise at discrete points with a fitted curve to interpolate).

- Take the derivative (slope) with respect to time to calculate acceleration (deceleration).

- Combine this with the known mass of the bullet to calculate the drag force F = ma.

- Convert the drag force to a dimensionless drag coefficent Cd= F/PA, where P is pressure (itself a function of air density and bullet velocity) and A is bullet frontal area. A Cd of 1 represents a completely unaerodynamic "wadcutter" shape, lower means less drag.

*The curve of drag coefficient vs velocity (typically measured as Mach numbers to allow for variable air density) is the real aerodynamic "fingerprint" of the bullet, for which ballistic coefficients (and form factors) are only a scalar approximation.*

- The
~~ballistic coefficient~~form factor is a single number that scales the reference Cd-Mach curve for a standard projectile such as G1 or G7 to be the best fit with the empirical curve for the bullet.

*If I understand correctly, the BC can be interpreted as the rate of deceleration of the reference projectile relative to the test bullet (yes, that way round, so bigger is better). E.g. a G1 BC of 0.5 means that the bullet decelerates twice as fast as the G1 reference projectile.*

- The form factor is defined as FF = SD/BC where SD is sectional density (mass divided by frontal area). Unlike BC it stays constant as the bullet is scaled up or down.

*As SD scales linearly with caliber, so does BC. Incidentally, because BCs were first developed for artillery, the reference projectiles are 1-pounders of 1 inch caliber. Therefore a .30 inch G1 or G7 bullet will have a corresponding BC of 0.3.*

Strictly, we can only convert between G1 and G7 BCs for a given velocity/Mach number. However, for velocities between Mach 1.5-3, where the G1 and G7 drag curves are not wildly dissimilar in shape, G1 BCs will be about double G7 BCs for the same bullet.

- Edited 30 June 2020 21:50 by hobbes154

27-Jun

https://www.thefirearmblog.com/blog/2016/05/13/ballistics-101-ballistic-coefficient/

https://www.thefirearmblog.com/blog/2016/05/16/more-on-ballistic-coefficients/

https://www.thefirearmblog.com/blog/2016/06/09/ballistics-101-form-factor/

Coefficient of Drag:

Cd = (2Fd)/(A(rho)(mu^2))

Fd - Normal Force in N (kg*m/s^2)

rho - 1.225 kg/m^3 for air

mu - Velocity in m/s

A - pi(r^2) in meters

i7 FF from Cd:

i7 = Cd(test)/Cd(G7)

Where Cd(G7) = the Cd for the G7 model projectile at the given speed (mu).

G7 Reference values:

Mach Cd

0.00 0.1198

0.05 0.1197

0.10 0.1196

0.15 0.1194

0.20 0.1193

0.25 0.1194

0.30 0.1194

0.35 0.1194

0.40 0.1193

0.45 0.1193

0.50 0.1194

0.55 0.1193

0.60 0.1194

0.65 0.1197

0.70 0.1202

0.725 0.1207

0.75 0.1215

0.775 0.1226

0.80 0.1242

0.825 0.1266

0.85 0.1306

0.875 0.1368

0.90 0.1464

0.925 0.1660

0.95 0.2054

0.975 0.2993

1.0 0.3803

1.025 0.4015

1.05 0.4043

1.075 0.4034

1.10 0.4014

1.125 0.3987

1.15 0.3955

1.20 0.3884

1.25 0.3810

1.30 0.3732

1.35 0.3657

1.40 0.3580

1.50 0.3440

1.55 0.3376

1.60 0.3315

1.65 0.3260

1.70 0.3209

1.75 0.3160

1.80 0.3117

1.85 0.3078

1.90 0.3042

1.95 0.3010

2.00 0.2980

2.05 0.2951

2.10 0.2922

2.15 0.2892

2.20 0.2864

2.25 0.2835

2.30 0.2807

2.35 0.2779

2.40 0.2752

2.45 0.2725

2.50 0.2697

2.55 0.2670

2.60 0.2643

2.65 0.2615

2.70 0.2588

2.75 0.2561

2.80 0.2533

2.85 0.2506

2.90 0.2479

2.95 0.2451

3.00 0.2424

3.10 0.2368

3.20 0.2313

3.30 0.2258

3.40 0.2205

3.50 0.2154

3.60 0.2106

3.70 0.2060

3.80 0.2017

3.90 0.1975

4.00 0.1935

4.20 0.1861

4.40 0.1793

4.60 0.1730

4.80 0.1672

5.00 0.1618

27-Jun

Ignoring some minor details, as you requested, your description is OK in my view.

A Cd of 1 is associated with a flat plate that is infinite in dimension. The Cd of a cylinder (wad cutter) is usually higher, the British proof slug around 1.3, for example.

I think your comparison G1 versus G7 is too optimistic. Lets take a bullet with Cd 0.38 at Mach 2. If you modify G1 (i = 0.63) and G7 (i = 1.27) in a manner so that they agree at about Mach 2, the G1 starts to look much better from about Mach 1.6 downwards. At Mach 1.5, computations with G1 will use Cd 0.42 versus 0.49 for G7. At Mach 1, G1 will use Cd 0.3 versus Cd 0.5 for G7. (rounded figures from a graph)

G1 simply does not fit modern slender bullets. It is still used by industry, in my opinion, because on paper it results in much better bullet velocities. Even Lapua, which has published radar data for the S538 bullet does it. For 955 m/s muzzle velocity, the G1 based catalog figure for V300 (meters) is 605 m/s. In reality (Lapua radar data) v300 is 581 m/s.

- Edited 27 June 2020 15:09 by JPeelen

27-Jun

Ta, I had the first and third of those but not the second - that JBM BC converter looks very cool!

27-Jun

Yeah, 2x rule not something to build a ballistic table for target shooting but just to get a first impression. E.g. assuming .303 has a G1 BC of 0.467, it is better than .30-06 M2 but worse than M1.

30-Jun

Just edited the original post to say it is the form factor, not the ballistic coefficient, that scales the drag model to match the empirical data. E.g. if the G7 Cd is the same as the empirical Cd, then the G7 form factor is 1 (for a particular Mach number, or on average over a range of Mach numbers).

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