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I heard this anecdote from someone who watched this QI episode:

The reason people fall over when they have been shot is because they have seen people do it in films. You never fall from being shot unless you know you have been shot, according to the FBI Academy Firearms Training Unit. (Forfeit: Because they've been shot; The impact)

Searching around, I saw this more detailed and nuanced look:

It's likely that naiveté about the damaging effects of guns, coupled with battle-fueled adrenaline, rendered the Moros and the Boxers disconcertingly impervious to bullets.

[...]

But many experts don't subscribe to the psychological explanation of why people fall when shot, instead preferring one that's neurologically based. Roach spoke with neurologist Dennis Tobin, who theorizes that an area of the brain stem called the reticular activating system (RAS) can become overloaded by impulses precipitated by intense pain. Thus, the agony of being shot triggers the RAS to send out a signal weakening the muscles of the leg and causing the person to collapse.

So, what is the truth?

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    @PlasmaHH I think there's an inverse law between reaction and gun caliber in Hollywood: isn't it always the guy who just took the cannonball through the chest that's still standing so he can silently contemplate the cavity before keeling over? – Lilienthal Sep 30 '15 at 13:47
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    @Lilienthal, I hear that most people hit by a cannonball stay standing because that's what they saw in a movie. – user11522 Sep 30 '15 at 14:08
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    It obviously depends on where you've been shot. It might very well be that people surviving non-lethal gun shots fall because they think they are supposed to do so. A gunshot can however also damage many areas of the body, where it may cause (near) instantaneous paralysis or unconsciousness with the result that the target falls due to lack of muscle control. It is not uncommon to see that shot animals fall as well and hunters are usually trained to ensure immobilisation with a single shot to prevent a hurt animal from escaping. I doubt that the animals have seen this in movies. – Tor-Einar Jarnbjo Sep 30 '15 at 21:27
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    The bullet can't knock you over. Newton's Third Law--the gun pushes back on the shooter harder than the bullet hits the target (some energy goes into gas.) Guns don't knock over the shooter, the bullet shouldn't knock you over. – Loren Pechtel Oct 1 '15 at 0:02
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    @LorenPechtel The Newton's law argument is unfortunately too simplified. Using the same argument, you could also claim that it's impossible for one person to hit another person so it falls, since the hitter is pushed back with the same force applied to the person hit. Untrained shooters may also very well fall because of the unexpected recoil force from larger weapons and if a shot person is not expecting the force applied from the bullet, it is not impossible that it will cause him to fall, even if the shooter stays standing. – Tor-Einar Jarnbjo Oct 1 '15 at 8:27
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+100

Per Anthony J. Pinizzotto, Harry A kern and Edward F. Davis in FBI law enforcement bulletin October 2004, with the exceptions of hits to the brain or upper spinal cord, the concept of reliable and reproducible immediate incapacitation of the human target by gunshot wounds to the torso a.k.a one shot drop is a myth. However, many shooting scenes in majority of films and television programs show unrealistic reactions and expectations regarding ballistic effects.

Physiologically, a determined adversary can be stopped reliably and immediately only by a shot that disrupts the brain or upper spinal cord. Failing to hit the center nervous system, massive bleeding from holes in the heart, or major blood vessels of the torso causing circulatory collapse is the only way to force incapacitation upon an adversary, and this takes time. For example, there is sufficient oxygen within the brain to support full, voluntary action for 10 to 15 seconds after the heart has been destroyed.

Realistic and regular law enforcement training must counterbalance and mentally and emotionally override the fallacy of the one-shot drop still promoted by some media. Short of disrupting the brain or severing the upper spinal column, immediate incapacitation does not occur. Therefore, the threat remains to the officer.

Yet, implicit in the media presentations of law enforcement encounters is the belief that with the “proper handgun” and the “proper ammunition,” officers will inflict immediate incapacitation if they shoot offenders anywhere in the torso. Varied and multiple real-life law enforcement experiences contradict this false and dangerous belief. Social science discloses that if people expect to see something, they well may see it.

Per J. Scott Denton et.al. in 2006,

Generally, 3 factors determine the severity of a gunshot injury: the amount of kinetic energy transferred by the bullet to the surrounding tissues, the internal organs and structures damaged by the bullet, and the bullet’s final disposition. Obviously, the anatomic location of the wound is critical—a gunshot wound to the central nervous system, even one of low velocity, can be more life threatening than a high-velocity wound through the arm. Involvement of vital structures such as the heart, aorta, lung, liver, spleen, and kidneys can quickly lead to hemorrhage, hypoxia, and death.

Per Special Agent Urey W. Patrick who belongs to the Firearms training Unit, FBI Academy in 1989,

A bullet simply cannot knock a man down. If it had the energy to do so, then equal energy would be applied against the shooter and he too would be knocked down. This is simple physics, and has been known for hundreds of years. The amount of energy deposited in the body by a bullet is approximately equivalent to being hit with a baseball. Tissue damage is the only physical link to incapacitation within the desired time frame, i.e., instantaneously.

The human target can be reliably incapacitated only by disrupting or destroying the brain or upper spinal cord. Absent that, incapacitation is subject to a host of variables, the most important of which are beyond the control of the shooter. Incapacitation becomes an eventual event, not necessarily an immediate one. If the psychological factors which can contribute to incapacitation are present, even a minor wound can be immediately incapacitating. If they are not present, incapacitation can be significantly delayed even with major, unsurvivable wounds.

Further, it appears that many people are predisposed to fall down when shot. This phenomenon is independent of caliber, bullet, or hit location, and is beyond the control of the shooter. It can only be proven in the act, not predicted. It requires only two factors to be effected: a shot and cognition of being shot by the target. Lacking either one, people are not at all predisposed to fall down and don't. Given this predisposition, the choice of caliber and bullet is essentially irrelevant. People largely fall down when shot, and the apparent predisposition to do so exists with equal force among the good guys as among the bad. The causative factors are most likely psychological in origin.

Physiologically, no caliber or bullet is certain to incapacitate any individual unless the brain is hit. Psychologically, some individuals can be incapacitated by minor or small caliber wounds. Those individuals who are stimulated by fear, adrenaline, drugs, alcohol, and/or sheer will and survival determination may not be incapacitated even if mortally wounded.

The will to survive and to fight despite horrific damage to the body is commonplace on the battlefield, and on the street. Barring a hit to the brain, the only way to force incapacitation is to cause sufficient blood loss that the subject can no longer function, and that takes time. Even if the heart is instantly destroyed, there is sufficient oxygen in the brain to support full and complete voluntary action for 10-15 seconds.

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