ballistics physics question

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

 

Any of the usual ballistic programs will give time of flight for the bullet. If you know the speed of sound at the ambient conditions, it is not hard to figure.

 

Thank you. Actually, I know you can use TOF, which some ballistics programs calculate, and figure it out, but I'm one of those guys who likes to figure out square roots by hand, and make ice cream in a hand-crank bucket. I've taken a few courses in physics, but I don't remember how to set up the math to figure this one.

 

I don't know the answer, but it might be never if I understand the question. Are you asking at what range would someone be able to hear the gunshot at the same time or before the bullet arrives? Most rifle bullets start out 4-5X the speed of sound and are still double the speed of sound at 1000 yards. That is quite a head start. Considering roughly 30' of bullet drop at 1000 yards it would be pretty hard for someone to hear a shot fired at them before the bullet hit the ground or was already past.

The only scenario I can thing of is firing a shot up at a very steep angle. The bullet may well travel quite a distance up and back down, but only strike the ground a short distance away.

 

You understand the question correctly, but I don't think many bullets travel at 4 times the speed of sound (4444.52 fps at sea level). It's definitely not true that MOST rifle bullets travel at 4-5 times the speed of sound. Also, I didn't have rifle bullets in mind when I thought of the question. Many pistol-caliber bullets start off at just above or just below the speed of sound. Finally, for me, it doesn't have to be practical to be interesting. Some people do crossword puzzles. Sometimes, just finding the answer is its own reward.

 

I was wondering if, like the bullet, the sound of the rifle shot also hits the ground?

 

Sound is not affected by gravity.

 

Good man! I like your sense of humor.

 

The distance will be the point at which the bullet's average velocity for the trip equals the speed of sound.

A bullet with muzzle velocity less than the speed of sound could never catch up to the sound wave of the muzzle blast. If muzzle velocity is just a bit faster than the sound wave, the sound wave will catch up when the bullet slows down enough that it's average velocity equals the speed of sound. Think muzzleloaders.

If the muzzle velocity is more than twice the speed of sound and the bullet is launched fairly horizontally, then the average velocity for the trip will be greater than the speed of sound and the sound wave can never catch up:

(Assume speed of sound is 1100 feet per second; for the sound wave to overtake the bullet, [muz. vel. + final bullet velocity] / 2 cannot be more than 1100. Or to say it differently, if the muzzle velocity is 2200, the final bullet velocity must be zero for the average velocity to be 1100.)

So jmr40 is right; for any muzzle velocity much greater than twice the speed of sound, you are going to have to make a high angle lob shot to get the AVERAGE velocity below the speed of sound.

Good luck!

 

Or black powder cartridge rifles. My Shiloh Sharps .45-110 launches a 500 grain bullet at about 1,350fps - just a couple of hundred fps faster than the speed of sound in air. Yet that 500 grain bullet can arrive on a distant target a full 2 seconds before the report from the rifle gets there. And I know that's true because I saw it in the movie "Quigley Down Under" - which I own a copy of.
Seriously Eleanor416Rigby, I hope you get an answer, because like you, I too would like to know how to "set up the math" to figure it out.

 

While there was an element of humor in my post, at some distance the sound of a rifle report will fall below the threshold of human hearing. A 6 decibel drop for each doubling of distance from the source may need to be considered. The sound and the bullet both have an expiration date.

 

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

The "final" bullet velocity is always zero. Again, pistol cartridges often take off near the speed of sound. (9mm averages MV right around the speed of sound.) The sound can catch up within actual shooting distances. The original question does not limit the formula to shooting two or more times the speed of sound, nor to horizontal launches.

For example, the sound of a .45 colt bullet weighing 310 grains fired at 1300 fps will catch up to the bullet between 330 and 340 yards downrange. (This is approximate, due to estimating the speed of sound and BC.)

My "formula" takes a lot of trial and error, toggling between the calculator and the ballistics program; it's inefficient and it made me decide to try to figure how to find it with a nice tidy formula and some math. It's not for any one practical purpose. It's just something to cause synapses to fire.

 

Oh, I see, bikemutt. It "falls to the ground" in that nobody could hear it. I didn't make that connection and thought it was just a funny. That's a good point. Hmmmm, 6 decibel drop for each doubling of distance. That's interesting. I guess I've heard rifle and pistol shots that were in the neighborhood of a couple miles off, but it wasn't exactly measured distance.

 

OP now you made my head hurt.

 

In three weeks, there will be a mountain man rendezvous held on the other side of the valley, right at 2.5 miles from our house. We'll be able to hear the reports from the muzzle loaders, and especially the reports from the cannons, on our back porch. Unlike our gun-shy Cocker Spaniel, we look forward to it every year.

 

OK, I understand what it is you're asking. Time for the sound of the shot to catch up to the bullet in flight.

To solve the problem, you need to know what the rate of the drop in bullet velocity is. For relatively short distances, for most rounds, you can say that the drop in bullet velocity is negligible. So we've got to be talking about a fairly significant distance...several hundred yards.

Now, some might say you need calculus for this. I don't think so, if you make the right assumptions. Let's assume, for example, that the rate of drop in bullet velocity is linear over time and distance. If you know the rate of drop in bullet velocity over distance, you can calculate how much it drops per foot traveled. For simplicity, let's make the following assumptions...you can change the numbers to more realistic values later:

Assume the speed of sound is 1125 fps.
Assume initial bullet velocity is 1325 fps.
Assume the bullet loses 100 fps in 100 yards and the loss in velocity is linear.

If the bullet loses 100 fps in 100 yards, with a linear loss, then the bullet looses 1 fps per yard.

So...how much velocity does the bullet need to lose to equal the speed of sound? That's the initial bullet velocity minus the speed of sound...which is 200 fps.

How far does the bullet have to travel to lose 200 fps? At a loss rate of 1 fps per yard, the bullet must travel 200 yards.

At this point, bullet velocity matches the speed of sound. Still assuming a linear drop in velocity over distance, the bullet must now travel an additional 200 yards for sound to catch up to it. (An equal distance with the same linear velocity drop.)

So, the bullet must travel 400 yards for the sound of the shot to catch up to the bullet in flight.

How much time is that? It's the distance traveled (400 yards, which is 1,200 feet) divided by the speed of sound (1,125 fps).

In this instance, that's 1.07 seconds.

 

The drag on the bullet surface declines (not linearly) as the speed declines (not linearly); therefore the drag force is not constant. The decrease in velocity, likewise, decreases with the decreasing drag that results from the decreasing speed. In other words, we need calculus.

 

I understand that.

However, you can still make valid working assumptions which will get you some excellent results with respect to practical accuracy.

For example, I assumed the loss of velocity was linear over time and distance. I should have clarified WHY I made that assumption, which was because the non-linear effects of drag would be insignificant for all practical purposes for the ranges of velocity and the times (distances) we're working with.

What are the effects of drag at 1325 fps, 1125 fps, and 925 fps? Over a distance of a few hundred yards (and resulting fractions of seconds travel time involved), how much difference would this really make? (I'm not asking for answers...just questions for perspective.)

The effects of drag increases greater than linearly as velocity increases. Which translates to a higher rate of velocity drop initially, decreasing as distance downrange increases.

How would this affect the 1.07 second answer I arrived at previously? Probably not much. It might make a bit more difference if you started off with a bullet velocity of 3750 fps. Or if the distances involved resulted in a few thousand yards.

The question is really "what are my assumptions and are they reasonable under the circumstances?"

If you're looking for practical...it's probably a fair assumption for the range of velocities discussed. Especially so, since I believe you were talking about pistol ammo velocities not much above the speed of sound to start with (which means minimizing the change in velocity required for sound to catch up...which minimizes the effects of drag.)

If you're looking for exact...then there's a LOT of other information also required, not the least of which are specific bullet ballistics actually obtained experimentally over distance as well as actual speed of sound based on atmosoheric pressure, temperature, and humidity.


An interesting question, though! I had fun thinking about it!

 

A bullet with a BC of .5 started at 3000 fps will be going about 1400 fps at 1000 yards, depending.

A bullet with a BC of .1 started at 2000 fps will travel about 1200 feet in 1.1 second.

A bullet with a BC of .1 started at 2500 fps will take about 4 seconds to go 1000 yards. The sound would get there more than a second ahead of the bullet.

 

My claim about the fastest bullet in the books was based on the 220 swift, which does not have a BC anywhere near .5. Most bullets don't have BC of .5, but if you can get one going 3000 fps, it will retain speed a lot better.

 

Analytical solution is possible. Back of the keyboard approximation and guessing is not required. If you want to work it out by hand, there are tables in the back of Hatcher's Notebook for the purpose. Sharpe instructed on the use of the slide rule to carry out such computations.

 

It's pretty easy to look this stuff up on a ballistics calculator.

 

It's really pretty easy to get a close answer.

You know the speed of sound. Distance = rate x time. So, for any time interval, you know how far the sound has traveled.

Your ballistics program will give you the time to any range. So all you have to do is choose your range, note the time of flight, and see whether the sound has traveled that far yet.

With any muzzle velocity above about 2500 FPS or so, I'm pretty sure the bullet will hit the ground before the sound is heard, if you're shooting near level.

 

This might be true (likely is) for something like the 220 Swift you mentioned, but is absolutely not true for many other calibers used for long range shooting. 50 BMG's starting out at 2750 fps can stay supersonic...depending on the projectile....well past 1500 yds and the 338 Lapua over 1800 which is why they're often chosen for extreme long range shooting. The newer VLD bullets fly really, really well....so long range isn't bothered by that pesky sound barrier thing getting in the way on the way back down.

 

You can't do it by hand. Ballistics programs compute trajectories by numerically integrating the differential equations that make up the point mass model or the modified point mass model. The equations need to be integrated numerically (cannot be done by hand) for several reasons, the biggest of which is because the drag coefficient of bullets is not constant, but changes with velocity.

Having a reasonably accurate ballistic coefficient, accurate ambient conditions, and a good muzzle velocity will get you close, but having an accurate custom drag model (drag coefficient vs. velocity) and a ballistics program that knows how to use it will get you even closer, because lots of bullets differ a lot from the standard ballistic coefficient drag curves when making the transition from supersonic to subsonic motion, which always happens if the sound is going to catch up with the bullet.