Ballistic pendulum

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.

The ballistic pendulum is still found in physics classrooms today, because of its simplicity and usefulness in demonstrating properties of momentum and energy.

For example, a ballistic pendulum was used by physicist C. V. Boys to measure the elasticity of golf balls,[2] and by physicist Peter Guthrie Tait to measure the effect that spin had on the distance a golf ball traveled.

[3][4] The ballistic pendulum was invented in 1742 by English mathematician Benjamin Robins (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.

[2][5] Robins used the ballistic pendulum to measure projectile velocity in two ways.

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 (28 g), while other contemporaries used his methods with cannon shot of one to three pounds (0.5 to 1.4 kg).

[6] Robins' original work used a heavy iron pendulum, faced with wood, to catch the bullet.

Robins also used a length of ribbon, loosely gripped in a clamp, to measure the travel of the pendulum.

[7] The first system to supplant ballistic pendulums with direct measures of projectile speed was invented in 1808, during the Napoleonic Wars 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 1848, 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.

Robins' calculations were significantly more involved, and used a measure of the period of oscillation to determine the rotational inertia of the system.

We begin with the motion of the bullet-pendulum system from the instant the pendulum is struck by the bullet.

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 Society the following year.

[6] 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.

[7] A rotational inertia based formula similar to Robins' was derived by French mathematician Siméon Denis Poisson and published in The Mécanique Physique, for measuring the bullet velocity by using the recoil of the gun: where:

Ackley's pendulum used a parallelogram linkage, with a standardized size that allowed a simplified means of calculating the velocity.

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.

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 pendulum is swung, and the number of complete oscillations is measured over a long period of time, five to ten minutes.