The answer to whether falling bullets can kill is complicated, and requires more specificity. If the question is “has any falling bullet of any type ever killed anyone”, the answer is yes.
But are falling bullets in general a deadly risk? In what cases? Are there risk differences between different types (caliber, shape, spin) of falling bullets? How do different trajectories (90 deg, 80 deg, 70 deg) affect the risk?
In the early 20th century, Army General Julian S. Hatcher supervised experiments to determine whether a .30 caliber bullet returning by gravity from rifle fired vertically represented a danger.
“One of the shots that hit the platform was a Service .30-'06, 150 grain flat based bullet, which came down base first (as that bullet usually does) and bounced into the water after striking the edge of the lower platform. It left a mark about 1/16 inch deep in the soft pine board.”
“It was concluded from these tests that the return velocity was about 300 feet per second. With the 150 grain bullet, this corresponds to an energy of 30 foot pounds. Previously the Army had decided that on the average, an energy of 60 foot pounds is required to produce a disabling wound. Thus Service bullets returning from extreme heights cannot be considered lethal by this standard.”
However what’s not lethal to an adult soldier may be lethal to an unshielded child.
Also the Hatcher study focused only on bullets fired vertically. Such returning bullets would have energy limited by atmospheric terminal velocity. Much celebratory gunfire is at angles, maybe even 45 degrees. In that case more of the horizontal muzzle energy would be retained, making it more deadly.
All bullets are not equal: we can calculate the effect caliber has on impact energy, simply by knowing the terminal velocity of the bullet and the caliber. By actual observation this site gave the following bullet fall rates:
http://tinyurl.com/5axrf
.22 rimfire, 25 grains, terminal velocity 60-80mph
.223, 55 grains, terminal velocity 80-100 mph
9 mm, 115 grains, terminal velocity 100-110 mph
.45, 230 grains, terminal velocity 120 mph
From those numbers this equation calculates the kinetic energy (I used the highest estimate):
KE = 1/2mv^2, where
KE = kinetic energy in Joules
m = mass in kg
v = velocity in meters/sec
Kinetic Impact Energy:
.22 rimfire - 1 Joule (0.74 ft lbs)
.223 - 3.55 Joules (2.62 ft lbs)
9 mm - 9 Joules (6.63 ft lbs)
.45 - 21.4 Joules (15.78 ft lbs)
The question is almost always phrased as “will a falling bullet kill you”, not could you be injured, is it dangerous, etc.
The answer to that is complicated, since in some cases you clearly will not be killed. E.g. the popular .22 round is unlikely to kill anyone, if fired vertically. Based on the above recent tests, it has a total energy 0.74 ft lbs.
But as bullet size increases, energy goes up rapidly. A .44 bullet is extremely dangerous. A .50 cal is certainly deadly, even if fired at 90 degrees. The celebratory skyward anti aircraft 23mm artillery fire sometimes seen in the middle east is deadly for non-exploded rounds. A high velocity rifle fired at 45 degrees would seem very dangerous.
So firing guns into the air isn’t a good idea. In many cases the returning bullets won’t kill you, but in some cases they can. However we should also place that risk in context of other life risks. A few people have definitely been killed by falling bullets. However 11,000 EVERY YEAR die from gunshots in the US alone. Firing skyward isn’t wise, is always dangerous, and is potentially fatal. However your probability of dying from a gunshot is vastly higher from direct fire, especially in the U.S, Brazil and Mexico.