A ballistic pendulum is a device that measures the speed of a projectile by catching it in a heavy swinging block and observing how high the block rises. It works by combining two fundamental physics principles in sequence: conservation of momentum during the collision, then conservation of energy during the swing. First developed in the 1700s to measure bullet speeds, it remains one of the most widely used demonstrations in introductory physics courses.
How a Ballistic Pendulum Works
The setup is straightforward. A large block (historically wooden, now often made of foam or fitted with a ball catcher) hangs from a rod or string so it can swing freely. A projectile, usually a small steel ball in classroom versions or an actual bullet in the original design, is fired into the block. The projectile embeds itself in the block, and the combined mass swings upward like a playground swing. The higher it swings, the faster the projectile was traveling.
The measurement happens in two distinct stages, each governed by a different law of physics. During the collision itself, momentum is conserved. The projectile’s momentum before impact equals the total momentum of the block-plus-projectile combination immediately after. Once the projectile is embedded and the block begins to swing, energy conservation takes over. The kinetic energy of the swinging block converts into gravitational potential energy as it rises. By measuring how high the block swings, you can work backward through both stages to calculate the original speed of the projectile.
Why Energy Isn’t Conserved in the Collision
The collision between the projectile and the block is what physicists call a perfectly inelastic collision, meaning the two objects stick together. This is the messiest type of collision in terms of energy loss. A large fraction of the projectile’s kinetic energy gets converted into heat, sound, and deformation of the materials rather than into motion of the block.
How much energy is lost? In a typical classroom setup, only about 3.3% of the original kinetic energy survives the collision, according to calculations from Boston University. That means roughly 96.7% of the energy transforms into thermal energy and other non-mechanical forms. This is exactly why you can’t use energy conservation alone to figure out the bullet’s speed. You need momentum conservation for the collision step, because momentum is conserved even when energy is not. The energy conservation principle only applies to the clean, friction-free swing that happens afterward.
The Math Behind It
The calculation uses two equations chained together. In the first stage, momentum conservation says the mass of the projectile times its velocity equals the combined mass times the velocity right after impact. In the second stage, energy conservation says the kinetic energy of the swinging block converts entirely to potential energy at its highest point. That highest point is measured either as an angle or as a vertical height.
Combining these two relationships gives you a single formula: the projectile’s speed equals the combined mass divided by the projectile’s mass, multiplied by the square root of twice the gravitational acceleration times the height of the swing. The only measurements you need are the mass of the projectile, the mass of the block, and how high the block rises. This simplicity is part of what made the device so revolutionary when it was introduced.
Who Invented It
Benjamin Robins, an English engineer and mathematician, published the design in his 1742 book “New Principles of Gunnery.” His version used a large wooden block suspended in front of a gun. When a bullet struck the block, the momentum transferred to it, and the amplitude of the swing revealed the bullet’s velocity. Before Robins, there was no reliable way to measure how fast a bullet traveled. The French scientist Jacques Cassini may have used a similar concept as early as 1707, but Robins provided the detailed, systematic account that made the device practical.
Robins’ work was significant enough that he received the Copley Medal from the Royal Society in 1747, one of the most prestigious scientific honors of his era. His ballistic pendulum essentially created the field of experimental ballistics, giving military engineers and scientists a quantitative tool for comparing firearms, powder charges, and projectile designs for the first time.
What a Classroom Setup Looks Like
A modern laboratory ballistic pendulum is a compact tabletop apparatus. It typically includes a spring-loaded launcher that fires a small steel ball (about 16 mm in diameter), a pendulum arm with a catcher at the end (often lined with foam or fitted with a cup that traps the ball), and some mechanism for recording the swing angle or height. Many setups use a ratchet system that locks the pendulum at its highest point so you can read the angle after the fact. At Virginia Tech, for instance, the apparatus is calibrated so the ball launches level with the catcher, and a successful shot sends the pendulum above 50 degrees.
Students typically repeat the measurement several times, firing the ball into the pendulum and recording the swing height each time. The main sources of error come from measuring the height precisely (usually accurate to about a millimeter) and from small energy losses during the swing itself, such as friction at the pivot point and air resistance. These losses are generally small enough to ignore in an introductory lab, but they do cause the calculated projectile speed to come out slightly lower than the true value.
Ballistic Pendulums vs. Modern Chronographs
For actually measuring bullet speeds today, ballistic pendulums have been almost entirely replaced by electronic chronographs. These devices use sensors placed a known distance apart to time a projectile as it passes through, giving a direct velocity reading without any collision or energy conversion involved. They’re faster, more precise, and don’t require catching the projectile.
The ballistic pendulum survives, however, as a teaching tool. Its value in physics education comes from the fact that it forces students to apply two conservation laws in sequence, think carefully about when each one applies, and grapple with the reality that most of the energy in a collision can vanish into heat while momentum stays perfectly intact. Few other lab exercises illustrate these concepts as cleanly. That combination of historical significance and pedagogical usefulness keeps the ballistic pendulum a staple of introductory physics courses at universities worldwide.