Physics of firearms

Last updated February 07, 2021

From the viewpoint of physics (dynamics, to be exact), a firearm, as for most weapons, is a system for delivering maximum destructive energy to the target with minimum delivery of energy on the shooter.^{[ citation needed ]} The momentum delivered to the target, however, cannot be any more than that (due to recoil) on the shooter. This is due to conservation of momentum, which dictates that the momentum imparted to the bullet is equal and opposite to that imparted to the gun-shooter system.^{[ failed verification ]}

Firearm energy efficiency

From a thermodynamic point of view, a firearm is a special type of piston engine, or in general heat engine where the bullet has a function of a piston. The energy conversion efficiency of a firearm strongly depends on its construction, especially on its caliber and barrel length. However, for illustration, here is the energy balance of a typical small firearm for .300 Hawk ammunition:^[1]

Barrel friction 2%
Projectile motion 32%
Hot gases 34%
Barrel heat 30%
Unburned propellant 1%.

which is comparable with a typical piston engine.

Higher efficiency can be achieved in longer barrel firearms because they have better volume ratio. However, the efficiency gain is less than corresponding to the volume ratio, because the expansion is not truly adiabatic and burnt gas becomes cold faster because of exchange of heat with the barrel. Large firearms (such as cannons) achieve smaller barrel-heating loss because they have better volume-to-surface ratio. High barrel diameter is also helpful because lower barrel friction is induced by sealing compared to the accelerating force. The force is proportional to the square of the barrel diameter while sealing needs are proportional to the perimeter by the same pressure.

Force

Assuming the gun and shooter are at rest, the force on the bullet is equal to that on the gun-shooter. This is due to Newton's third law of motion (For every action, there is an equal and opposite reaction). Consider a system where the gun and shooter have a combined mass $M$ and the bullet has a mass $m$ . When the gun is fired, the two systems move away from one another with new velocities $V$ and $v$ respectively. But the law of conservation of momentum states that the magnitudes of their momenta must be equal:

MV+mv=0\qquad (1)

Since force equals the rate of change in momentum and the initial momenta are zero, the force on the bullet must therefore be the same as the force on the gun/shooter.

Gunshot victims frequently fall or collapse when shot; this is less a result of the momentum of the bullet pushing them over, but is primarily caused by physical damage or psychological effects, perhaps combined with being off balance. This is not the case if the victim is hit by heavier projectiles such as 20 mm cannon shell, where the momentum effects can be enormous; this is why very few such weapons can be fired without being mounted on a weapons platform or involve a recoilless system (e.g. a recoilless rifle).

Example: A .44 Remington Magnum with a 240-grain (0.016 kg) jacketed bullet is fired at 1,180 feet per second (360 m/s)^[2] at a 170-pound (77 kg) target. What velocity is imparted to the target (assume the bullet remains embedded in the target and thus practically loses all its velocity)?

Let $m$ _$b$ and $v$ _$b$ stand for the mass and velocity of the bullet, the latter just before hitting the target, and let $m$ _$t$ and $v$ _$t$ stand for the mass and velocity of the target after being hit. Conservation of momentum requires

m

_$b$

v

_$b$ =

m

_$t$

v

_$t$.

Solving for the target's velocity gives

v

_$t$ =

m

_$b$

v

_$b$ /

m

_$t$ = 0.016 kg × 360 m/s / 77 kg = 0.07 m/s = 0.17 mph.

This example shows the target barely moves at all. That's not to say one couldn't stop a train by firing bullets at it, it's just completely impractical.^[3]

Velocity

From Eq. 1 we can write for the velocity of the gun/shooter: V = mv/M. This shows that despite the high velocity of the bullet, the small bullet-mass to shooter-mass ratio results in a low recoil velocity (V) although the force and momentum are equal.

Kinetic energy

However, the smaller mass of the bullet, compared to that of the gun-shooter system, allows significantly more kinetic energy to be imparted to the bullet than to the shooter. The kinetic energy for the two systems are ${\begin{matrix}{\frac {1}{2}}\end{matrix}}MV^{2}$ for the gun-shooter system and ${\begin{matrix}{\frac {1}{2}}\end{matrix}}mv^{2}$ for the bullet. The energy imparted to the shooter can then be written as:

{\frac {1}{2}}MV^{2}={\frac {1}{2}}M\left({\frac {mv}{M}}\right)^{2}={\frac {m}{M}}{\frac {1}{2}}mv^{2}

If we now write for the ratio of these energies we have:

{\frac {{\frac {1}{2}}MV^{2}}{{\frac {1}{2}}mv^{2}}}={\frac {m}{M}}\qquad (2)

The ratio of the kinetic energies is the same as the ratio of the masses (and is independent of velocity). Since the mass of the bullet is much less than that of the shooter there is more kinetic energy transferred to the bullet than to the shooter. Once discharged from the weapon, the bullet's energy decays throughout its flight, until the remainder is dissipated by colliding with a target (e.g. deforming the bullet and target).

Transfer of energy

When the bullet strikes, its high velocity and small frontal cross-section means that it will exert highly focused stresses in any object it hits. This usually results in it penetrating any softer material, such as flesh. The energy is then dissipated along the wound channel formed by the passage of the bullet. See terminal ballistics for a fuller discussion of these effects.

Bulletproof vests work by dissipating the bullet's energy in another way; the vest's material, usually Aramid (Kevlar or Twaron), works by presenting a series of material layers which catch the bullet and spread its imparted force over a larger area, hopefully bringing it to a stop before it can penetrate into the body behind the vest. While the vest can prevent a bullet from penetrating, the wearer will still be affected by the momentum of the bullet, which can produce contusions.

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Momentum Conserved physical quantity related to the motion of a body

In Newtonian mechanics, linear momentum, translational momentum, or simply momentum is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If $m$ is an object's mass and $v$ is its velocity, then the object's momentum is:
$In SI units, momentum is measured in kilogram meters per second (kg\cdotm/s).$

A kinetic energy penetrator is a type of ammunition designed to penetrate vehicle armour using a flechette-like, high-sectional density projectile. Like a bullet, this type of ammunition does not contain explosive payloads and uses purely kinetic energy to penetrate the target. Modern KEP munitions are typically of the armour-piercing fin-stabilized discarding sabot (APFSDS) type.

In firearms, rifling is the helical groovings that are machined into the internal (bore) surface of a gun's barrel, for the purpose of exerting torque and thus imparting a spin to a projectile around its longitudinal axis during shooting. This spin serves to gyroscopically stabilize the projectile by conservation of angular momentum, improving its aerodynamic stability and accuracy over smoothbore designs.

Recoil is the rearward thrust generated when a gun is being discharged. In technical terms, the recoil is a result of conservation of momentum, as according to Newton's third law the force required to accelerate something will evoke an equal but opposite reactional force, which means the forward momentum gained by the projectile and exhaust gases (ejectae) will be mathematically balanced out by an equal and opposite momentum exerted back upon the gun. In hand-held small arms, the recoil momentum will be eventually transferred to the ground, but will do so through the body of the shooter hence resulting in a noticeable impulse commonly referred to as a "kick".

Muzzle velocity is the speed of a projectile with respect to the muzzle at the moment it leaves the end of a gun's barrel. Firearm muzzle velocities range from approximately 120 m/s (390 ft/s) to 370 m/s (1,200 ft/s) in black powder muskets, to more than 1,200 m/s (3,900 ft/s) in modern rifles with high-velocity cartridges such as the .220 Swift and .204 Ruger, all the way to 1,700 m/s (5,600 ft/s) for tank guns firing kinetic energy penetrator ammunition. To simulate orbital debris impacts on spacecraft, NASA launches projectiles through light-gas guns at speeds up to 8,500 m/s (28,000 ft/s).

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A sabot is a supportive device used in firearm/artillery ammunitions to fit/patch around a projectile, such as a bullet/slug or a flechette-like projectile, and keep it aligned in the center of the barrel when fired. It allows a narrower projectile with high sectional density to be fired through a barrel of much larger bore diameter with maximal accelerative transfer of kinetic energy. After leaving the muzzle, the sabot typically separates from the projectile in flight, diverting only a very small portion of the overall kinetic energy.

Internal ballistics, a subfield of ballistics, is the study of the propulsion of a projectile.

Muzzle energy is the kinetic energy of a bullet as it is expelled from the muzzle of a firearm. Without consideration of factors such as aerodynamics and gravity for the sake of comparison, muzzle energy is used as a rough indication of the destructive potential of a given firearm or cartridge. The heavier the bullet and especially the faster it moves, the higher its muzzle energy and the more damage it will do.

Stopping power is the ability of a weapon – typically a ranged weapon such as a firearm – to cause a target to be incapacitated or immobilized. Stopping power contrasts with lethality in that it pertains only to a weapon's ability to make the target cease action, regardless of whether or not death ultimately occurs. Which ammunition cartridges have the greatest stopping power is a much debated topic.

Rotation around a fixed axis Type of motion

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A ballistic pendulum is a device for measuring a bullet's momentum, from which it is possible to calculate the velocity and kinetic energy. Ballistic pendulums have been largely rendered obsolete by modern chronographs, which allow direct measurement of the projectile velocity.

Airsoft pellets are spherical projectiles used by airsoft guns. Typically made of plastic, they usually measure around 6 mm (0.24 in) in diameter, and weigh 0.12–0.40 g (1.9–6.2 gr), with the most common weights being 0.12 g and 0.20 g, while 0.25 g, 0.28 g, 0.30 g and 0.40 g pellets are also commonplace. Though frequently referred to as "BBs" among airsoft users, these pellets are not the same as either of the 4.5 mm metal projectiles that BB guns fire, or the 4.6 mm (0.180 in)-sized birdshot from which the term "BB" originated.

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In fluid dynamics, disk loading or disc loading is the average pressure change across an actuator disk, such as an airscrew. Airscrews with a relatively low disk loading are typically called rotors, including helicopter main rotors and tail rotors; propellers typically have a higher disk loading. The V-22 Osprey tiltrotor aircraft has a high disk loading relative to a helicopter in the hover mode, but a relatively low disk loading in fixed-wing mode compared to a turboprop aircraft.

Free recoil is a vernacular term or jargon for recoil energy of a firearm not supported from behind. Free recoil denotes the translational kinetic energy (E_t) imparted to the shooter of a small arm when discharged and is expressed in joules (J), or foot-pound force (ft·lb_f) for non-SI units of measure. More generally, the term refers to the recoil of a free-standing firearm, in contrast to a firearm securely bolted to or braced by a massive mount or wall. Free recoil should not be confused with recoil:

A muzzle blast is an explosive shockwave created at the muzzle of a firearm during shooting. Before a projectile leaves the gun barrel, it obturates the bore and "plugs up" the pressurized gaseous products of the propellant combustion behind it, essentially containing the gases within a closed system as a neutral element in the overall momentum of the system's physics. However, when the projectile exits the barrel, this functional seal is removed and the highly energetic bore gases are suddenly free to exit the muzzle and rapidly expand in the form of a supersonic shockwave, thus creating the muzzle blast.

In astronautics, a powered flyby, or Oberth maneuver, is a maneuver in which a spacecraft falls into a gravitational well and then uses its engines to further accelerate as it is falling, thereby achieving additional speed. The resulting maneuver is a more efficient way to gain kinetic energy than applying the same impulse outside of a gravitational well. The gain in efficiency is explained by the Oberth effect, wherein the use of a reaction engine at higher speeds generates a greater change in mechanical energy than its use at lower speeds. In practical terms, this means that the most energy-efficient method for a spacecraft to burn its fuel is at the lowest possible orbital periapsis, when its orbital velocity is greatest. In some cases, it is even worth spending fuel on slowing the spacecraft into a gravity well to take advantage of the efficiencies of the Oberth effect. The maneuver and effect are named after the person who first described them in 1927, Hermann Oberth, an Austro-Hungarian-born German physicist and a founder of modern rocketry.

References

↑ Thermodynamic Efficiency of the .300 Hawk Cartridge, http://www.z-hat.com/Efficiency%20of%20the%20300%20Hawk.htm Archived 2009-02-28 at the Wayback Machine
↑ "Chuck Hawks".
↑ "XKCD".

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[1] Thermodynamic Efficiency of the .300 Hawk Cartridge, http://www.z-hat.com/Efficiency%20of%20the%20300%20Hawk.htm Archived 2009-02-28 at the Wayback Machine

[2] "Chuck Hawks".

[3] "XKCD".

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