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Pistol Physics: Why Your Pistol Works
By: Jim H.

You don't have to know how guns work to shoot one, but improving your knowledge will help you make purchase decisions, read technical material, decide on modifications and talk about guns.  This essay was written to provide a deeper understanding by examining semi-automatic handgun operation from an energy management point of view.  It begins with an informal survey of the physics of ballistics (classical mechanics) and goes on to apply the principles to a 9 mm CZ 75 P-01 compact pistol.

We often do not know what we think we know, so I encourage you to read the section on Classical Mechanics.  It is not long or difficult and you may learn something.  However, you may prefer to skip it and refer back for definitions.

Classical Mechanics

Velocity is the state of motion of an object.  It specifies speed and direction.  Think of velocity as an arrow pointing in the direction of motion and having a length (magnitude) corresponding to speed.

Quantities that specify both magnitude and direction are called vectors.  It takes three numbers to specify a vector, the amounts of the magnitude (components) along each coordinate axis (x, y, z).  Quantities that only have magnitude are called scalars.  Scalars are represented by a single number.

Acceleration is the rate of change of velocity.  When an object speeds up, slows down and/or changes direction it is undergoing acceleration.  Because it specifies the direction of change, acceleration is also a vector.  "Deceleration" is baby talk for negative acceleration.

Force is an influence that accelerates an object; i.e., changes its speed and/or direction.  Multiple forces may act on an object, in which case the net unbalanced force accelerates it.  Forces are vector quantities; they have magnitude and direction.

Mass describes the resistance of an object to acceleration.  Massive objects are hard to start moving, stop and turn.  Mass is a scalar value.

Newton's Laws:

1. An object's velocity (speed and direction) remains constant unless acted upon by a force.

2. The net force acting on an object equals its mass times its acceleration (F = m A).

3. Forces are mutual.  They act between objects.  Forces are felt equally and oppositely by both objects.

Pressure is force distributed over a surface.  Think of pressure as an infinite number of little force vectors distributed over a surface, with each little vector oriented perpendicular to the surface.  The total force acting on a surface is the sum of all the little force vectors pushing on the surface.

Weight is gravitational force, normally measured near the surface of the earth where the gravitational acceleration (g) is nearly constant at 32.2 ft/s/s (9.8 m/s/s); i.e., every second a freely falling object will accelerate by 32.2 feet per second.

Mass Weight specifies the mass of an object in terms of its weight.  Since weighing objects is convenient, mass is usually expressed as equivalent weight.  From Newton's second law, mass is weight divided by gravitational acceleration (m = weight / g).

Momentum is defined as mass times velocity (M = m V).  Because it includes velocity, momentum is a vector.

Conservation of Momentum: The total momentum of a closed system is conserved; i.e., it does not change.  When objects collide or split apart, the total momentum of the objects will be the same after the event as before.  So, when a billiard ball strikes a stationary billiard ball head-on, the approaching ball stops and the stationary one takes off with the other ball's original velocity (speed and direction).  If the stationary ball were half as massive, it would depart at twice the speed.

Energy is a fundamentally important quantity.  It exists in various forms: mechanical, chemical, electromagnetic, etc.  The kinetic energy of a non-rotating object is half its mass times its velocity squared (ke = 1/2 m V^2).

Mechanical systems may also store potential energy.  An obvious example is a coil spring that exerts a force proportional to its compression (F = k x, where k is the stiffness constant and x is the compression distance).  The potential energy in a coil spring is pe = 1/2 k x^2.

Conservation of Energy: The important thing about energy is that there is a fixed amount of it.  It cannot be created or destroyed.  It only changes forms between say chemical and mechanical form or between kinetic (moving) and potential (stored).

Work is the energy transferred to an object and/or dissipated (spent) when a force moves an object.  It is defined as force times distance (w = F x).  Unless a force moves an object, no work is done, no energy is expended.

Power is the rate at which energy is transferred or transformed.  It is defined as energy per unit time (p = e / t).

Cartridge Dynamics

The powder in a cartridge contains a fixed amount of chemical potential energy that is released when it ignites.  Let's call it the powder energy (pe).  Everything that happens is powered by parceling out pe into everything that moves (the bullet, the air, the target, the gun mechanism and the shooter's body) and wasted as heat, light and noise.  The amount of pe spent on ancillary functions and wasted reduces the amount available to the bullet as muzzle velocity.  An efficient gun would minimize the amount of pe spent ejecting the empty casing, cocking the hammer, chambering the new round and pushing the shooter around.  But, that would not be a practical gun.

When the bullet completes its transit and everything settles down, all of pe will be transformed into heat.  The shell casing and gun will be warmer, the air will be warmer, the bullet and target will be warmer and even the shooter's body will be warmer.

When combustion occurs, the gas pressure in the sealed cartridge builds with the rate of combustion.  The increasing pressure is felt over all of the surfaces inside the cartridge.  The bullet sees a force equal to the pressure times the area of its rear face; the higher the caliber, the greater the force.  At some point, the bullet breaks free from the casing and things begin to happen.  When this occurs, the pressure on the bullet equals the pressure on the rear face of the casing.  Since the two areas are the same, the force on the bullet and on the rear of the casing are equal.  Therefore, the bullet and the slide/barrel assembly accelerate apart under equal and opposite forces.

The total momentum of the two masses was zero before ignition and the conservation of momentum principle demands that it remain zero as they split apart.  Therefore, mV for the bullet and mV for the slide/barrel assembly must have equal magnitudes.  So, the velocity of the bullet must be proportionately higher than that of the heavier slide/barrel assembly.  That is, the ratio of the velocities must be the inverse of the ratio of the masses.  For the example pistol, the slide/barrel assembly weighs 13 oz, or 5688 gr, and a 9 mm Luger bullet might weigh 124 gr.  The ratio of the two is 46:1.  This is an important ratio because it tells us (1) that at every moment the bullet has traveled 46 times farther up the barrel than the slide/barrel assembly has traveled backward, (2) that the bullet speed will be 46 times the slide/barrel assembly speed and (3) that the bullet carries 46 times the kinetic energy as the slide/barrel assembly does.

Strictly speaking, this analysis is only approximate since the bullet and the slide/barrel assembly are working against different forces as they move and the work they do must come out of their kinetic energy; i.e., out of their velocities.  So, their velocities depreciate at different rates.  But this is not a significant consideration while the bullet is in the barrel.

Bullet Transit

When the bullet breaks free of the casing, it accelerates down the barrel, forced by combustion pressure.  The ideal acceleration rate is reduced by barrel friction and rifling.  The friction imposes drag force on the bullet, transforming some of its kinetic energy into heat at a rate that increases with bullet speed.  Since rotation is a form of kinetic energy, the spin also slows the bullet, but only slightly.

The pressure builds to a maximum and falls off as the burn rate increases and declines.  The remaining pressure plummets when the bullet exits the muzzle.  Exactly how the pressure varies with bullet position depends on barrel length, caliber, bullet weight, powder composition, powder charge and slide/barrel assembly weight.  This is a critical issue in designing guns and ammunition to make best use of pe.

The bullet and slide/barrel assembly continue to separate in the 46:1 ratio until the bullet exits the muzzle.  Where is the slide/barrel assembly when that happens?  The answer is 1/46 of the distance the bullet traveled.  The 3.8" barrel of the example pistol gives the bullet 3" of travel in the barrel, so the slide/barrel assembly quits accelerating at 3"/46 (.065").  The design of the gun must ensure that the rear end of the barrel does not begin pulling downward until after this point for the maximum bullet and powder loads.

At that point, the slide/barrel assembly will have its maximum velocity, about 1/46 of the muzzle velocity.  So, if the muzzle velocity is say 1200 f/s, then the slide/barrel assembly will be moving back at 26 f/s.  As we shall see, this is well before the first mechanical event of the semi-automatic cycle.  At this point, the slide/barrel assembly has all the energy it is going to get.  It must have enough to extract and eject the spent casing, cock the hammer (charging the main spring with enough energy to initiate the next cycle), store enough energy in the recoil spring for the return trip and push the shooter around.

Since the slide/barrel assembly's kinetic energy is 1/46 of the muzzle energy.  A typical muzzle energy of say 385 ft-lb gives the slide/barrel assembly 8.4 ft-lb for all the things it needs to do.

Slide Back Travel

Soon after the bullet exits the muzzle, the rear of the barrel is forced downward so that it may disengage from the slide and stop moving.  This permits the extractor on the slide to pull the spent casing from the chamber.  The barrel stops abruptly when it collides with the frame after .25" of travel.  Together, they impart their energy to the shooter as the first of two crisp recoil impulses.  Since the respective weights of the barrel and slide are 2.6 oz and 10.4 oz, only 20% of the slide/barrel energy (1.7 ft-lb) goes into this impulse.

The slide, now free of the barrel, with 80% of the energy (6.7 ft-lb), continues moving back into the opposing force of the recoil spring.  The rear end of the recoil spring is held by the frame that in turn is backed up by the shooter who feels approximately 17 lb from the spring while the slide is moving.  This is because the factory spring is essentially linear throughout the slide's travel.

During back travel, the slide slows down as it loses energy compressing the recoil spring.  At some point, the slide takes on the additional job of cocking the hammer.  This slows the slide faster as it charges the main spring for the next cycle.  When the hammer cocks, the main spring no longer opposes the slide.

Slide Reversal

Now the slide must reverse direction and begin traveling forward under force from the relaxing recoil spring.  But, how does that occur?  The answer depends on how much kinetic energy remains in the slide, the stiffness of the recoil spring and the distance to the slide stop.  With the factory's linear recoil spring, the slide has enough energy to strike the slide stop quite hard.  At that point, the slide picks up the mass of the frame and barrel, slowing down as together they impact the shooter with a second recoil impulse.

Slide Forward Travel

Now the slide may then begin accelerating forward under force from the recoil spring.  As the slide accelerates, potential energy in the compressed recoil spring is transformed into kinetic energy in the accelerating slide.  When the slide strikes the top round in the magazine, it works against friction and the inertia of the round to push it out of the magazine and into the chamber.

Next, the slide picks up the rear end of the barrel, engaging it.  When that happens, the moving weight increases from 10.4 oz to 13 oz, reducing the speed of the combined assembly.  The shooter does not feel this event because it does not involve a collision with the frame.

Finally, the slide/barrel assembly does collide with the frame, ending the forward motion.  The remaining kinetic energy of the slide/barrel assembly is shared with the frame and passed on to the shooter as a forward impulse.  With the slide, barrel and frame now joined, there remains no unbalanced force on the recoil spring and the shooter no longer feels its force.

Summary

We have reviewed the physical principles governing ballistics and studied the operation of a semi-automatic handgun from an energy point of view according to those principles. I hope this has helped you understand better not only how but why semi-automatic pistols work.

Figure 1 below is a simplified hypothetical representation of the recoil forces produced by a semiautomatic handgun that is clamped in a vice. Numeric values in Fig. 1 are from a northeastshooters.com forum article: Time Evolution of Recoil – A Calculation (http://www.northeastshooters.com/vbulletin/firearms/18066-time-evolution-recoil-%96-calculation.html).﻿
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