Sound speed vs pellet impact.

As a bow hunter I run into this issue all the time and have found that a simple math equation works best for me, P.S this also works very well in black powder shooting due to the fact that for side lock guns there is a pop from the side lock before the gun goes off
Take your FPS and divide it by 10 so if you are shooting a projectile that is traveling at 970 FPS then your effective range is 97 yards. This is not the killing range of the projectile but the maximum range that an animal like a deer would likely hear the pop and not be able to have time to react to the sound before it hits them.
Naturally the type of animal, its speed, and if they are looking at or away from you, and how they normally react to sprint away from danger will play its own part on how much you can fudge those numbers.
For good example if a deer is looking directly at you when you pull the trigger the condensed air vapor that a PCP creates when shot may be enough to startle it into a sprint causing a miss even if you are within that range, Where if it is head down feeding and or distracted by rutting then it might be possible to take a further range shot than the equation comes out to be. Keep in mind also that at 97 yards a moving target can change your POI at 5 inches or more.
Again this has nothing to do about the power of the shot itself only about the ability of the target to hear the shot and possibly react to it and yes it is perhaps slightly conservative as I would like to stay on the inside of the line of caution but it has always worked well for me.

Willie

The sound will get there about 1/10th of a second before the pellet based on average pellet speed of 830fps. An animal is not likely to react at all in that time but probably could start to at 2/10ths or more.

GQ the rest of your questions are way above my math abilities but in laymans terms you do have a pretty good gap between the pop of the air rifle and the projectile hitting the target. Do a Youtube Search for Airforce Texan shooting at over 100 yards look at this one about 5 minutes into the video
https://www.youtube.com/watch?v=4VKWBQLkEbU

It's simple math. 1 divided by 1125 is .00088 seconds per foot. 100 yards (300 feet) times .00088 equals .266 seconds for the sound to reach the target. The pellet at an average speed of 830fps. 1 divided by 830 is .0012 seconds per foot. 100 yards (300 feet) times .0012 equals .36 seconds. That's 1/10th of a seconds difference. Can the animal hear it? That depends on how loud your gun is, what's between you and it, other noises in the area, how well can the animal hear. Many animals can hear far better than we can.

This will depend on the type of air rifle you are shooting the bigbores like the Texan, Bulldog, BushBucks all are pretty darn loud and so it would be enough to startle an animal,
for your smaller calibers or silenced versions perhaps depending on variables like wind and humidity.
The Higher the Humidity the further sound can travel, if your a fisherman have you ever noticed you can hear folks talking in a boat way away from you.
The Speed stays constant however the ability of your atmosphere around you to carry that sound further can change dramatically and surprisingly. I have had times when I would have sworn I could here cows on the property I hunt on yet the closest place to me that has cows is a mile away.

"outdoorman"It's simple math. 1 divided by 1125 is .00088 seconds per foot. 100 yards (300 feet) times .00088 equals .266 seconds for the sound to reach the target. The pellet at an average speed of 830fps. 1 divided by 830 is .0012 seconds per foot. 100 yards (300 feet) times .0012 equals .36 seconds. That's 1/10th of a seconds difference. Can the animal hear it? That depends on how loud your gun is, what's between you and it, other noises in the area, how well can the animal hear. Many animals can hear far better than we can.

Not so sure about simple math, you just made my brain hurt,.,.,.,. LOL ...... Not trying to disprove your math and again this is from my experience in shooting bows at speeds of 400 fps
But I do know that at 75 yards and a 400 fps crossbow I have had deer duck my shot and have had belly shots from hogs when they sprinted away

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That is a very good point thanks for pointing that out it is something that I did not consider
Thanks

Given that pellets lose speed especially at long range, at 100 yards if the pellet leaves your barrel at say 900 fps, by the time it gets to the target it will be down to 680-700ish fps even with an outstanding BC, therefore bringing the time of flight nearer to .39 seconds. Very unlikely anything would react in that time.

Hi GQ. 
When I first got an airrifle good enough to hit anything at that distance I would think the starlings would hear the shot as they appeared to move just before I would hear the impact! 
Happened a couple times before the penny dropped