Calculating ballistics coefficient from two velocities

[ER-01]

Homework Statement: I am trying to figure out how to calculate the BC from two velocity readings with a known distance between the two readings
Relevant Equations: BC = (AirDensity * Distance) / ( SQRT(Velocity0) - SQRT(Velocity1))

In long range rifle shooting, knowing your true ballistics coefficient is very useful. You might get a ballistics coefficient listed on your ammunition, but depending on your range that ballistics coefficient might not be accurate. I have two velocity readings at 0 range and (let’s say) 100m. How can I calculate the ballistics coefficient for that shot?

I have been looking at this thread to solve ballistics coefficient (BC) with two velocity readings:
https://www.physicsforums.com/threa...t-bc-from-two-velocities.958802/#post-7069285
The author gives this formula to calculate BC,

BC = (AirDensity * Distance) / ( SQRT(Velocity0) - SQRT(Velocity100))
but in the variables, he provides a "air resistance" value and not "air density". I am aware these can't be used interchangeably and cannot find a clear answer on the internet.

Values given by author:
Air Resistance = 0.0052834
Distance = 100 meters
Velocity0 = 914.4 mps
Velocity100 = 838 mps @ 100 meters
BC = .451

Can someone help me out?

 

[BvU]

Your new thread leans on a five year old one that at least explains what BC stands for: a Ballistic coefficient.

This is a physics forum, and it would be nice to know what this is about.
Please post a complete problem statement, list and explain known variables (include units),
list all relevant equations you need and post your work.

And convert variables to a consistent set of units, preferably SI.

The arms trader reference no longer explains the relevant equation used in the old thread (${\bf BC} = {\rho d\over \sqrt v_0-\sqrt v_1} \qquad (1)$ ).
($\rho$ is air density, $d$ is distance -- we must guess there are two velocity measurements, $v_0$ and $v_1$, with a distance $d$ between them)

Regrettable, since the dimension of this ballistic coefficient should be [mass]/[distance]² and that's not the dimension on the righthand side of $(1)$.

We can ask @jrmichler if he/she can recall

Translating into metric, and entering into their equation

about translation and equation...


[edit] Ah, you added some context. Thanks

$\$

 

[Filip Larsen]

For what its worth and as I understand it from [1], the BC for a particular bullet varies as a function of speed since the drag coefficient varies with speed with a profile that is somewhat dependent on the shape, as shown below (also from [1]).

I also understand that the BC profile for most long range ammunition match well with a scaled G7 bullet (bottom curve) rather than the G1 (upper curve) and if that is also the case for your bullet you may be able to establish a BC speed profile from that. However, I am not aware of the exact data set for the G7 drag profile and [1] is also written using imperial units making it difficult to compare with your data. If you "just" need the BC data I would suggest that you use a ballistic calculator that is able to base its calculations on the proper bullet shape or (constant) form factor relative to a standard bullet (like G7). If you need theoretical insight I suggest you read [1] or similar.

[1] Applied Ballistics For Long-Range Shooting 2nd Edition, Bryan Litz, 2011.

Later: I missed this problem was listed as homework so the input above may not be very useful in that context.

 

[haruspex]

According to https://en.wikipedia.org/wiki/Balli...h,and hence, different ballistic coefficients and my algebra,
$BC=\frac{\rho s}{2(\ln(v_0)-\ln(v_1))}$

 

[Jim Hahn]

I have been working on the derivation of this formula for several weeks. I can get the form of "L/difference of square roots of velocities." Not there yet on the multiplier. The formula is based on recognizing that the drag vs Mach number curve has several sections. Low velocity is constant. At speed of sound, it's crazy. Above that up to say Mach 2.5 drag is a constant divided by sqrt of Mach number. All we care about is this last section. (There's another one for hypersonic stuff which we can safely ignore.)

There's a really great book on ballistics and it's available for free on the internet. Search for: "Ballistics Theory and design of guns and ammunition" by Carlucci and Jacobson. Some sections are completely unreadable in complexity of equations but the writings of the basics is straightforward.

See the discussion around figure 8.4 re Mach number.

The multiplier usually quoted (0.00528) has some crazy units but I believe they work if x is in feet and v is in ft/sec. It's definitely not air density. My derived value is off by a factor of four and hopefully I'll find my error.

This whole thing is ascribed to Boris Karpov, 1944. Many of the papers from the US Army Research Lab (aka the ballistics lab) are photocopied and available in pdf. Alas, the key paper does not seem to be available.
--jim

 

[pbuk]

I have been looking at this thread to solve ballistics coefficient (BC) with two velocity readings:
https://www.physicsforums.com/threa...t-bc-from-two-velocities.958802/#post-7069285
The author gives this formula to calculate BC,

BC = (AirDensity * Distance) / ( SQRT(Velocity0) - SQRT(Velocity100))
but in the variables, he provides a "air resistance" value and not "air density". I am aware these can't be used interchangeably and cannot find a clear answer on the internet.

The author of that thread was mistaken: the constant equal to 0.0052834 is not 'air density' and 'air resistance' doesn't have a specific meaning. I suggest you forget about that thread.

I have been working on the derivation of this formula for several weeks. I can get the form of "L/difference of square roots of velocities." Not there yet on the multiplier.

Calculation of the constant is not trivial and I doubt that you have access to all the information required. Note that the value is anything but 'constant' and depends on barometric pressure/altitude, temperature, humidity, the ballistic model used and the down-range distances over which it is applicable, what you ate for breakfast and a number of other things.

A good introduction to this subject can be found at http://www.frfrogspad.com/extbal.htm.

 

[Jim Hahn]

Several places on the Internet reference the following equation for calculating BC:



$$G1 = 0.0052834 * \frac{L}{\sqrt{V_0}-\sqrt{V_L}}$$



Unfortunately, the equation seems to have raised many questions and confusion on how to use the equation. I derived the more useful, and less confusing to use, equation below.



Measuring two velocities separated by a known distance is particularly easy now that inexpensive Doppler radar chronographs have come on the scene. (LabRadar, Garmin) One clever YouTube video extracts the LabRadar data to produce BC measurements.



The problem with the “0.0052834” multiplier equation is there are no units given anywhere. Meters, inches, feet? Where you want to end up is pounds per square inch for BC. It’s also set up for a G1 based BC but modern bullets want G7.



Another problem with the equation is that the constant “0.0052834” is based on physical parameters involving air pressure and temperature. Unless velocity measurements are made at exactly the same conditions which produced the “0.0052834”, you’re going to be off in your final BC. The difference between air density at sea level and low temperature versus 10,000 feet is rather immense. (0.08 versus 0.052 lb/ft^3) The utility of five significant digits is kinda debatable.



My derivation’s final equation:



$$BC=\frac{1}{4} * r h o * K_3 * \sqrt{a} * \frac{L}{\sqrt{V_M}-\sqrt{V_L}}$$



a = Speed of sound



rho = density of air



V0 = The first measured velocity.

VL = Velocity at distance L. The second measured velocity.

L = distance between measuring points



K3 = a dimensionless number extracted from the drag coefficient curve.



I calculate K3 for the G7 curve to be 0.55. For G1, it’s 0.834. $K_3 = drag * \sqrt{Mach}$

 

[Jim Hahn]

Derivation's available. 10 pages, hand written, scanned.

 

[erobz]

Derivation's available. 10 pages, hand written, scanned.

I’m not seeing your notes?

 

[Jim Hahn]

Did not scan it yet. Wasn't sure anybody was interested. Will scan and post, probably today.

 

[Jim Hahn]

Ran into attachment size limit. Very small.

Send me an email and I'll forward to you.

"hahn" underscore "02493" atSign "yahoo.com"

 

[erobz]

What form of Newtons 2nd do you start with?

 

[Jim Hahn]

Split into several sets of pages. To allow me to post as attachments. Several 'chunks'. Should learn how to use Dropbox...

 

[Jim Hahn]

Second chunk

 

[Jim Hahn]

Two more chunks. Scanner acting up...

 

[Jim Hahn]

last chunk

 

[Jim Hahn]

ERROR. I made a mistake in my formula. I wrote that BC = 1/4 times ...

The 1/4 should not be there. The coefficient is ONE. Apologies....