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How to modify the bullet trajectory based on the ballistic coefficient?

I am new to the physics surrounding bullet trajectory and how it is calculated. I am a software developer and I am working on a ballistics calculator. I am using wiki for the trajectory calculation.

I am currently using the equation under the "Angle $\theta$ required to hit coordinate $(x,y)$" section. This is all well and good, but it doesn't take into account the drag of the bullet(ballistic coefficient).

I have searched all over trying to figure out how to apply the coefficient to this equation. I am really at a loss and and would be very thankful for any direction in this matter. Maybe I have a gap in my understanding, but I have found plenty of other calculators and other documentation on trajectory and the coefficient but nothing that marries to the two together.

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migrated from firearms.stackexchange.com Dec 8 '11 at 21:38

This question came from our site for firearm experts and enthusiasts.

marked as duplicate by Qmechanic Jan 30 '13 at 15:55


My apologies, I posted both places because I wasnt 100% sure where the question would be best served. –  Etch Nov 30 '11 at 14:06
looks like the answer on physics.SE is pretty good :) –  warren Nov 30 '11 at 14:42
Started a meta topic on this: meta.firearms.stackexchange.com/questions/97/… –  kbyrd Dec 8 '11 at 20:32
Possible duplicate cased by cross-posting: physics.stackexchange.com/q/17650/2451 –  Qmechanic Jan 30 '13 at 15:55

3 Answers 3

The problem you're having with not accounting for bullet BC is that it's a relative number. G1 and G7 BC are both based on a prototypical bullet design with a BC of 1.0.

The thing that further complicates your research is that BC is relative to velocity - it changes as the velocity changes. Some bullet manufacturers, such as Sierra, give relative BC's based on velocity bands: http://www.sierrabullets.com/bullets/BallisticCoefficient-rifle.pdf

In other words, ballistics coefficient is more a function of velocity, and therefore elevation, temperature, humidity, etc., than a hard number like c or pi.

Probably the best reference you'll find for this is Bryan Litz's "Applied Ballistics for Long Range Shooting" http://www.appliedballisticsllc.com/Book.htm

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up vote 0 down vote accepted

I was able to use the feedback from the physics forum Ultimately it was that response that gave me the most detail and enabled me to complete my own app.

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The differential equations of motion with drag (K * velocity squared) have never been solved (integrated with respect to time). However, some can be integrated with respect to the flight path angle. I have a paper on this that is being published. For slow speeds the drag is proportional to velocity to the first power. All these equations are easily solvable with conventional math.

However, when you start talking of speeds close to and greater than the speed of sound, all bets are off. The math on anything close to mach 1 is really off the charts. It is already nearly (if not ) impossible.

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