ballistics physics question

I saw an Army video taken by a soldier of a buddy exiting their large sleeping bunker. A ping is heard as a rifle bullet hits his helmet and is further deflected. With the various sounds in the enclosed bunker, the rifle report is unheard. Might have been a 9x39.

FWIW, here's an interesting video on the problems of estimating range using the crack-thump delay. For example, 5.56 and 7.62x39 timings are pretty different.

I would go for a graphical solution to avoid the trial and error.
Plot the time of flight vs distance for the bullet and the sound wave vs distance on the same scale and see where the lines intersect.

If you can get Phil Sharpe's 'Complete Guide to Handloading', it has a method for determining bullet velocity by ear. Weren't any consumer chronographs in 1951.

Eleanor416Rigby said:

Maybe you can't do it by hand, and I currently can't, but the people who program the ballistics calculators can.

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I've written ballistic calculators and other code that numerically integrates differential equations. The reason it is done by computer rather than by hand is the large number of calculations involved. Most low drag rifle bullets will be in flight for several seconds for the sound to catch up. Typical time steps are less than a millisecond in these numerical integrations. The formula for each time step is fairly complicated and involves 10-50 individual calculations, depending on how it is organized. Do the math on the total number of calculations and you see pretty quickly why one uses a computer to do them rather than doing over 10,000 calculations by hand to obtain a single result.

At its best, my calculus was not equal to the task, today I'm better able to do ten pull-ups than solve it, but the principle is pretty easy.
The distance sound travels will be velocity times time. Since the velocity is constant, a graph of it would look like a rectangle with velocity up the y-axis and time across the x-axis, a nice little rectangle. In calculus terms, the integral.
Now the bullet's distance will be the area under a curve with velocity up the y-axis and time along the x-axis. The shape of this area will be rather like the flight of the bullet itself, starting out horizontal and inflecting down at an ever steepening angle.
The answer to your question is when these two areas are equal.

Eleanor416Rigby said:

Even the fastest bullet in the record books will be well below the speed of sound at 1000 yards.

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shhhhhhhhhhhhh
don't tell the Palma boys

Eleanor416Rigby said:

Even the fastest bullet in the record books will be well below the speed of sound at 1000 yards.

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Sorry, no.

M118 is still above sonic at 1000 yards. And that's not a really fast bullet.

Thanks to people who have written ballistic calculators, it is possible to look this stuff up.
Under one specific set of conditions, a Sierra 30 caliber 150 grain HPBT started at 2600 fps will travel 1723 yards in the same time it will take sound to travel 1723 yards.

Berger.Fan222 said:

I've written ballistic calculators and other code that numerically integrates differential equations. The reason it is done by computer rather than by hand is the large number of calculations involved. Most low drag rifle bullets will be in flight for several seconds for the sound to catch up. Typical time steps are less than a millisecond in these numerical integrations. The formula for each time step is fairly complicated and involves 10-50 individual calculations, depending on how it is organized. Do the math on the total number of calculations and you see pretty quickly why one uses a computer to do them rather than doing over 10,000 calculations by hand to obtain a single result.

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Your answer brought a smile. You're absolutely correct of course. External ballistics involves some pretty snappy math. An exact, analytic answer to the original question would keep you busy for quite a while.

Eleanor416Rigby wrote:
Do you have a formula to calculate how far downrange the sound of a gunshot will catch up with the bullet?

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No.

Each bullet will have a different response to air resistance (this is approximated by the Ballistic Coefficient but not exactly) so it's time to target is not linear. Thus, the equation would be a non-linear differential equation using terms whose coefficients have probably never been directly measured and could only be estimated or determined empirically.

As posters like Berger.Fan222 and JimWatson have already posted, you can approximate this with ballistic computations, but a deterministic formula is would be unique to each bullet and muzzle velocity.

What is the reasoning behind this question?

All this talk about ballistics calculators and computer crunching...I'd be interested in seeing someone actually crunch some numbers for some pistol rounds as they see fit, then compare the results using my methodology and assumptions.

I'm curious to see how the results compare and whether they validate or refute my assumptions.

Sierra 9mm 125 gr. 1325 fps, speed of sound 1125 fps.
Bullet slows to 1125 fps at 64 yards. Sound catches bullet at 140 yards.

Haxby said:

Sierra 9mm 125 gr. 1325 fps, speed of sound 1125 fps.
Bullet slows to 1125 fps at 64 yards. Sound catches bullet at 140 yards.

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So...

0.37 seconds?