Ballistic pendulum
Encyclopedia
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 chronograph
s, which allow direct measurement of the projectile velocity.
Although the ballistic pendulum is considered obsolete, it remained in use for a significant length of time and led to great advances in the science of ballistics
. The ballistic pendulum is still found in physics
classrooms today, because of its simplicity and usefulness in demonstrating properties of momentum and energy. Unlike other methods of measuring the speed of a bullet, the basic calculations for a ballistic pendulum do not require any measurement of time, but rely only on measures of mass
and distance.
In addition its primary uses of measuring the velocity of a projectile or the recoil of a gun, the ballistic pendulum can be used to measure any transfer of momentum. For example, a ballistic pendulum was used by physicist C. V. Boys to measure the elasticity
of golf ball
s, and by physicist Peter Guthrie Tait
to measure the effect that spin had on the distance a golf ball traveled.
(1707–1751), and published in his book New Principles of Gunnery, which revolutionized the science of ballistics, as it provided the first way to accurately measure the velocity of a bullet.
Robins used the ballistic pendulum to measure projectile velocity in two ways. The first was to attach the gun to the pendulum, and measure the recoil
. Since the momentum of the gun is equal to the momentum of the ejecta, and since the projectile was (in those experiments) the large majority of the mass of the ejecta, the velocity of the bullet could be approximated. The second, and more accurate method, was to directly measure the bullet momentum by firing it into the pendulum. Robins experimented with musket
balls of around one ounce in mass (30 g), while other contemporaries used his methods with cannon
shot of one to three pounds (0.5 to 1.4 kg).
Robins' original work used a heavy iron
pendulum, faced with wood, to catch the bullet. Modern reproductions, used as demonstrations in physics classes, generally use a heavy weight suspended by a very fine, lightweight arm, and ignore the mass of the pendulum's arm. Robins' heavy iron pendulum did not allow this, and Robins' mathematical approach was slightly more complex. He used the period of oscillation
and mass of the pendulum (both measured with the bullet included) to calculate the rotational inertia of the pendulum, which was then used in the calculations. Robins also used a length of ribbon
, loosely gripped in a clamp, to measure the travel of the pendulum. The pendulum would draw out a length of ribbon equal to the chord
of pendulum's travel.
The first system to supplant ballistic pendulums with direct measures of projectile speed was invented in 1808, and used a rapidly rotating shaft of known speed with two paper disks on it; the bullet was fired through the disks, parallel to the shaft, and the angular difference in the points of impact provided an elapsed time over the distance between the disks. A direct electromechanical clockwork measure appeared in 1840, with a spring-driven clock started and stopped by electromagnets, whose current was interrupted by the bullet passing through two meshes of fine wires, again providing the time to traverse the given distance.
the acceleration due to gravity, and 
the height of the pendulum, it is possible to calculate 
the pendulum velocity:
We can then use momentum (
) conservation to get the speed of the bullet, 
, as:
Or:
the following year. German mathematician Leonhard Euler
, unaware of this correction, independently corrected this omission in his annotated
German
translation of the book. The corrected formula, appearing in a 1786 edition of the book, was:
where:
Robins used feet for length and ounces for mass, though other units, such as inches or pounds, may be substituted as long as consistency is maintained.
and published in The Mécanique Physique, for measuring the bullet velocity by using the recoil of the gun:
where:
can be calculated with the equation:
Where
is half the period of oscillation.
, author, and researcher
, described how to construct and use a ballistic pendulum in his 1962 Handbook for Shooters & Reloaders, Volume I. Ackley's pendulum used a parallelogram linkage, with a standardized size that allowed a simplified means of calculating the velocity.
Ackley's pendulum used pendulum arms of exactly 66.25 inches (168.3 cm) in length, from bearing surface to bearing surface, and used turnbuckle
s located in the middle of the arms to provide a means of setting the arm length precisely. Ackley recommends masses for the body of the pendulum for various calibers as well; 50 pounds (22.7 kg) for rimfire up through the .22 Hornet
, 90 pounds (40.9 kg) for .222 Remington
through .35 Whelen
, and 150 pounds (68.2 kg) for magnum rifle calibers. The pendulum is made of heavy metal pipe, welded shut at one end, and packed with paper and sand to stop the bullet. The open end of the pendulum was covered in a sheet of rubber, to allow the bullet to enter and prevent material from leaking out.
To use the pendulum, it is set up with a device to measure the horizontal distance of the pendulum swing, such as a light rod that would be pushed backwards by the rear of the pendulum as it moved. The shooter is seated at least 15 feet (5 m) back from the pendulum (reducing the effects of muzzle blast on the pendulum) and a bullet is fired into the pendulum. To calculate the velocity of the bullet given the horizontal swing, the following formula is used:
where:
For more accurate calculations, a number of changes are made, both to the construction and the use of the pendulum. The construction changes involve the addition of a small box on top of the pendulum. Before weighing the pendulum, the box is filled with a number of bullets of the type being measured. For each shot made, a bullet can be removed from the box, thus keeping the mass of the pendulum constant. The measurement change involves measuring the period of the pendulum. The pendulum is swung, and the number of complete oscillations is measured over a long period of time, five to ten minutes. The time is divided by the number of oscillations to obtain the period. Once this is done, the formula
generates a more precise constant to replace the value 0.2018 in the above equation. Just like above, the velocity of the bullet is calculated using the formula:
Bullet
A bullet is a projectile propelled by a firearm, sling, or air gun. Bullets do not normally contain explosives, but damage the intended target by impact and penetration...
's momentum
Momentum
In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...
, from which it is possible to calculate the velocity
Velocity
In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...
and kinetic energy
Kinetic energy
The kinetic energy of an object is the energy which it possesses due to its motion.It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes...
. Ballistic pendulums have been largely rendered obsolete by modern chronograph
Gun chronograph
A gun chronograph is an instrument used to measure the velocity of a projectile fired by a gun.-History:Around 1800, the ballistic pendulum was used to measure the momentum of the projectile fired by a gun; dividing the momentum by the projectile mass gives the velocity.An early chronograph that...
s, which allow direct measurement of the projectile velocity.
Although the ballistic pendulum is considered obsolete, it remained in use for a significant length of time and led to great advances in the science of ballistics
Ballistics
Ballistics is the science of mechanics that deals with the flight, behavior, and effects of projectiles, especially bullets, gravity bombs, rockets, or the like; the science or art of designing and accelerating projectiles so as to achieve a desired performance.A ballistic body is a body which is...
. The ballistic pendulum is still found in physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
classrooms today, because of its simplicity and usefulness in demonstrating properties of momentum and energy. Unlike other methods of measuring the speed of a bullet, the basic calculations for a ballistic pendulum do not require any measurement of time, but rely only on measures of mass
Mass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...
and distance.
In addition its primary uses of measuring the velocity of a projectile or the recoil of a gun, the ballistic pendulum can be used to measure any transfer of momentum. For example, a ballistic pendulum was used by physicist C. V. Boys to measure the elasticity
Elasticity (physics)
In physics, elasticity is the physical property of a material that returns to its original shape after the stress that made it deform or distort is removed. The relative amount of deformation is called the strain....
of golf ball
Golf ball
A golf ball is a ball designed to be used in the game of golf.Under the Rules of Golf, a golf ball weighs no more than 1.620 oz , has a diameter not less than 1.680 in , and performs within specified velocity, distance, and symmetry limits...
s, and by physicist Peter Guthrie Tait
Peter Guthrie Tait
Peter Guthrie Tait FRSE was a Scottish mathematical physicist, best known for the seminal energy physics textbook Treatise on Natural Philosophy, which he co-wrote with Kelvin, and his early investigations into knot theory, which contributed to the eventual formation of topology as a mathematical...
to measure the effect that spin had on the distance a golf ball traveled.
History
The ballistic pendulum was invented in 1742 by English mathematician Benjamin RobinsBenjamin Robins
Benjamin Robins was a pioneering English scientist, Newtonian mathematician, and military engineer.He wrote an influential treatise on gunnery, for the first time introducing Newtonian science to military men, was an early enthusiast for rifled gun barrels, and his work had substantive influence...
(1707–1751), and published in his book New Principles of Gunnery, which revolutionized the science of ballistics, as it provided the first way to accurately measure the velocity of a bullet.
Robins used the ballistic pendulum to measure projectile velocity in two ways. The first was to attach the gun to the pendulum, and measure the recoil
Recoil
Recoil is the backward momentum of a gun when it is discharged. In technical terms, the recoil caused by the gun exactly balances the forward momentum of the projectile and exhaust gasses, according to Newton's third law...
. Since the momentum of the gun is equal to the momentum of the ejecta, and since the projectile was (in those experiments) the large majority of the mass of the ejecta, the velocity of the bullet could be approximated. The second, and more accurate method, was to directly measure the bullet momentum by firing it into the pendulum. Robins experimented with musket
Musket
A musket is a muzzle-loaded, smooth bore long gun, fired from the shoulder. Muskets were designed for use by infantry. A soldier armed with a musket had the designation musketman or musketeer....
balls of around one ounce in mass (30 g), while other contemporaries used his methods with cannon
Cannon
A cannon is any piece of artillery that uses gunpowder or other usually explosive-based propellents to launch a projectile. Cannon vary in caliber, range, mobility, rate of fire, angle of fire, and firepower; different forms of cannon combine and balance these attributes in varying degrees,...
shot of one to three pounds (0.5 to 1.4 kg).
Robins' original work used a heavy iron
Iron
Iron is a chemical element with the symbol Fe and atomic number 26. It is a metal in the first transition series. It is the most common element forming the planet Earth as a whole, forming much of Earth's outer and inner core. It is the fourth most common element in the Earth's crust...
pendulum, faced with wood, to catch the bullet. Modern reproductions, used as demonstrations in physics classes, generally use a heavy weight suspended by a very fine, lightweight arm, and ignore the mass of the pendulum's arm. Robins' heavy iron pendulum did not allow this, and Robins' mathematical approach was slightly more complex. He used the period of oscillation
Oscillation
Oscillation is the repetitive variation, typically in time, of some measure about a central value or between two or more different states. Familiar examples include a swinging pendulum and AC power. The term vibration is sometimes used more narrowly to mean a mechanical oscillation but sometimes...
and mass of the pendulum (both measured with the bullet included) to calculate the rotational inertia of the pendulum, which was then used in the calculations. Robins also used a length of ribbon
Ribbon
A ribbon or riband is a thin band of material, typically cloth but also plastic or sometimes metal, used primarily for binding and tying. Cloth ribbons, most commonly silk, are often used in connection with clothing, but are also applied for innumerable useful, ornamental and symbolic purposes...
, loosely gripped in a clamp, to measure the travel of the pendulum. The pendulum would draw out a length of ribbon equal to the chord
Chord (geometry)
A chord of a circle is a geometric line segment whose endpoints both lie on the circumference of the circle.A secant or a secant line is the line extension of a chord. More generally, a chord is a line segment joining two points on any curve, such as but not limited to an ellipse...
of pendulum's travel.
The first system to supplant ballistic pendulums with direct measures of projectile speed was invented in 1808, and used a rapidly rotating shaft of known speed with two paper disks on it; the bullet was fired through the disks, parallel to the shaft, and the angular difference in the points of impact provided an elapsed time over the distance between the disks. A direct electromechanical clockwork measure appeared in 1840, with a spring-driven clock started and stopped by electromagnets, whose current was interrupted by the bullet passing through two meshes of fine wires, again providing the time to traverse the given distance.
Mathematical derivations
Most physics textbooks provide a simplified method of calculation that uses the mass of the bullet and pendulum and the height of the pendulum's travel to calculate the amount of energy in the pendulum, and thus the amount of momentum in the system. This ignores the small amount of momentum involved in rotating the pendulum. Robins' calculations were significantly more involved, and used a measure of the period of oscillation to determine the rotational inertia of the system.Simple derivation
Given the acceleration due to gravity, and the height of the pendulum, it is possible to calculate the pendulum velocity:We can then use momentum (
) conservation to get the speed of the bullet, , as:Or:
Robins' formula
Robins' original book had some omitted assumptions in the formula; for example, it did not include a correction to account for a bullet impact that did not match the center of mass of the pendulum. An updated formula, with this omission corrected, was published in the Philosophical Transactions of the Royal SocietyPhilosophical Transactions of the Royal Society
The Philosophical Transactions of the Royal Society is a scientific journal published by the Royal Society of London. It was established in 1665, making it the first journal in the world exclusively devoted to science, and it has remained in continuous publication ever since, making it the world's...
the following year. German mathematician Leonhard Euler
Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...
, unaware of this correction, independently corrected this omission in his annotated
Annotation
An annotation is a note that is made while reading any form of text. This may be as simple as underlining or highlighting passages.Annotated bibliographies give descriptions about how each source is useful to an author in constructing a paper or argument...
German
German language
German is a West Germanic language, related to and classified alongside English and Dutch. With an estimated 90 – 98 million native speakers, German is one of the world's major languages and is the most widely-spoken first language in the European Union....
translation of the book. The corrected formula, appearing in a 1786 edition of the book, was:
where:
-
is the velocity of the ball in units per second -
is the mass of the ball -
is the mass of the pendulum -
is the distance from pivot to the center of gravity -
is the distance from pivot to the point of the ball's impact -
is the chord, as measured by the ribbon described in Robins' apparatus -
is the radius, or distance from the pivot the attachment of the ribbon -
is the number of oscillations made by the pendulum in one minute
Robins used feet for length and ounces for mass, though other units, such as inches or pounds, may be substituted as long as consistency is maintained.
Poisson's formula
A rotational inertia based formula similar to Robins' was derived by French mathematician Siméon Denis PoissonSiméon Denis Poisson
Siméon Denis Poisson , was a French mathematician, geometer, and physicist. He however, was the final leading opponent of the wave theory of light as a member of the elite l'Académie française, but was proven wrong by Augustin-Jean Fresnel.-Biography:...
and published in The Mécanique Physique, for measuring the bullet velocity by using the recoil of the gun:
where:
-
is the mass of the bullet -
is the velocity of the bullet -
is the distance from pivot to the ribbon -
is the distance from bore axis to pivot point -
is the combined mass of gun and pendulum -
is the chord measured by the ribbon -
is the radius from pivot to the center of mass of gun and pendulum (measured by oscillation, as per Robins) -
is gravitational acceleration -
is the distance from the center of mass of the pendulum to the pivot
can be calculated with the equation:Where
is half the period of oscillation.Ackley's ballistic pendulum
P. O. Ackley, a gunsmithGunsmith
A gunsmith is a person who repairs, modifies, designs, or builds firearms. This occupation is different from an armorer. The armorer primarily maintains weapons and limited repairs involving parts replacement and possibly work involving accurization...
, author, and researcher
Researcher
A researcher is somebody who performs research, the search for knowledge or in general any systematic investigation to establish facts. Researchers can work in academic, industrial, government, or private institutions.-Examples of research institutions:...
, described how to construct and use a ballistic pendulum in his 1962 Handbook for Shooters & Reloaders, Volume I. Ackley's pendulum used a parallelogram linkage, with a standardized size that allowed a simplified means of calculating the velocity.
Ackley's pendulum used pendulum arms of exactly 66.25 inches (168.3 cm) in length, from bearing surface to bearing surface, and used turnbuckle
Turnbuckle
A turnbuckle, stretching screw or bottlescrew is a device for adjusting the tension or length of ropes, cables, tie rods, and other tensioning systems. It normally consists of two threaded eyelets, one screwed into each end of a small metal frame, one with a left-hand thread and the other with a...
s located in the middle of the arms to provide a means of setting the arm length precisely. Ackley recommends masses for the body of the pendulum for various calibers as well; 50 pounds (22.7 kg) for rimfire up through the .22 Hornet
.22 Hornet
The .22 Hornet is a low-end vermin, small-game and predator centerfire rifle cartridge. It is considerably more powerful than the .22 WMR and the .17 HMR, achieving higher velocity with a bullet twice the weight. The Hornet also differs very significantly from these in that it is not a rimfire...
, 90 pounds (40.9 kg) for .222 Remington
.222 Remington
The .222 Remington aka the Triple Deuce/Triple Two/Treble Two is a centerfire rifle cartridge introduced in 1950, and was the first commercial rimless .22 cartridge made in the United States...
through .35 Whelen
.35 Whelen
The .35 Whelen is a powerful medium-bore rifle cartridge that does not require a magnum action or a magnum bolt-face. The parent of this cartridge is the .30-06 Springfield, which is necked-up to accept a bullet diameter of...
, and 150 pounds (68.2 kg) for magnum rifle calibers. The pendulum is made of heavy metal pipe, welded shut at one end, and packed with paper and sand to stop the bullet. The open end of the pendulum was covered in a sheet of rubber, to allow the bullet to enter and prevent material from leaking out.
To use the pendulum, it is set up with a device to measure the horizontal distance of the pendulum swing, such as a light rod that would be pushed backwards by the rear of the pendulum as it moved. The shooter is seated at least 15 feet (5 m) back from the pendulum (reducing the effects of muzzle blast on the pendulum) and a bullet is fired into the pendulum. To calculate the velocity of the bullet given the horizontal swing, the following formula is used:
where:
-
is the velocity of the bullet, in feet per second -
is the mass of the pendulum, in grains -
is the mass of the bullet, in grains -
is the horizontal travel of the pendulum, in inches
For more accurate calculations, a number of changes are made, both to the construction and the use of the pendulum. The construction changes involve the addition of a small box on top of the pendulum. Before weighing the pendulum, the box is filled with a number of bullets of the type being measured. For each shot made, a bullet can be removed from the box, thus keeping the mass of the pendulum constant. The measurement change involves measuring the period of the pendulum. The pendulum is swung, and the number of complete oscillations is measured over a long period of time, five to ten minutes. The time is divided by the number of oscillations to obtain the period. Once this is done, the formula
generates a more precise constant to replace the value 0.2018 in the above equation. Just like above, the velocity of the bullet is calculated using the formula: