Testing Firearms: Measuring Bullet Velocity - III

In our last post, we studied a few methods that measure the time taken by the bullet to travel a known distance and thereby calculate its velocity. These methods fall under the class of chronograph methods. All these methods were generally inferior to the ballistic pendulum method for a variety of reasons, chiefly the inability to measure small amounts of time accurately and also the inability to ensure that the targets were moving at uniform rates.

While the ballistic pendulum method was superior for a long time, one of the issues it had was the weight of the apparatus. For testing the velocity of ordinary rifles or shotguns, the ballistic pendulum alone needs to weigh around 25 kg. (55 lbs.), without considering the weight of the supporting frame. When people try to scale this method for larger cannon balls, the weight of the pendulum apparatus increases exponentially. For example, in 1781, one Mr. Hutton tried to measure the velocity of cannon balls weighing just 3 pounds and his pendulum weighed about 315 kg. (approx. 700 lbs.). During the period of 1842 to 1847, one Major Alfred Mordecai from the United States Army tried to determine the muzzle velocity of larger guns and built a ballistic pendulum weighing over 4215 kg. (approx. 9300 pounds) and was mounted between two large brick towers. This could only measure velocities for 32 pounders at most. It was estimated that to build a ballistic pendulum to measure velocities for even larger weapons, one would need to build a massive pendulum suspended by the two Brooklyn bridge sized towers!

Meanwhile, the discovery of electricity made chronograph methods much more accurate. It was now possible to measure the beginning and end of a period of time using some sort of electrically operated mechanism. It also became possible to measure very small intervals of time accurately, making chronograph methods much more accurate than was achievable previously.

One of the early chronoscopes was invented by Charles Wheatstone, a noted scientist of the Victorian era. Among his other inventions were a stereoscope, an encryption system, several developments in telegraphy and the wheatstone bridge. His chronoscope consisted of a wooden ring fixed to the muzzle of a gun, with a thin wire running through the middle of it. The target was placed at a known distance and consisted of two plates which were arranged so that the least impact would result in a permanent contact between the two plates. The wires were hooked to an electromagnet mechanism and a small battery. Initially, a continuous circuit would be maintained. Then when the firearm would be discharged, the bullet would leave the muzzle and pass through the wooden ring and cut the thin wire running through the center. This would deactivate the electromagnet, which would then start a special clock driven by a falling weight. When the bullet would hit the target, the second circuit would be completed, which would re-energize the electromagnet and stop the clock. This mechanism was capable of measuring time with a resolution accurate to approx. 1/7300 of a second, which allowed it to calculate bullet velocities very accurately.

A more modernized version used a tuning fork to measure small increments of time. The tuning fork would have a thin stylus attached to one of the arms and a roll of paper would be gradually moved over the stylus. The tuning fork would be activated and deactivated by electricity and the tester would count the number of vibrations of the tuning fork, inscribed on the paper roll, to determine the elapsed time.

Using electricity to measure time became much more popular because the same setup could be used on just about any caliber firearm.

These days, modern chronographs use optical detection or to determine the passage of a bullet through a known distance. For instance, photo-transistors (such as those that work with infrared frequencies) could be used to start and stop a highly accurate stopwatch. The passing bullet shadow causes the circuit to be activated and deactivated at the two ends of a known distance and very high time resolutions can be obtained. Such devices are pretty cheap as well and available for around $100-$200 or so.

In the above images, we have a modern chronograph that can measure bullet velocities between 30 - 7000 feet/sec with 99.5% accuracy. The V arms indicate the area through which the bullet should be shot for the sensors to detect it. The two white strips on the top are light diffusers, so that this device can be used even in bright sunlight. They also help to provide a uniform background so the photo-transistor sensors can easily detect the contrast a passing bullet. In less than bright daylight conditions, the two diffusers can be optionally removed. The digital display automatically calculates the bullet velocity, so all the user needs to do is position this device on a flat surface, such as a table, turn it on and then shoot through the two V areas. High velocity rifles should be shot from at least 3 meters (10 feet) away and lower velocity weapons may be shot from around half that distance to get accurate readings.

Modern chronographs such as the one above make it a breeze to calculate bullet velocities. Due to their lightness and low cost, these are overwhelmingly the method of choice to measure bullet velocities these days.

Testing Firearms: Measuring Bullet Velocity - II

In our previous post, we studied the first accurate method of identifying bullet velocities: the ballistic pendulum method. While the ballistic pendulum method was accurate and remained in use for many decades, it wasn't initially adopted all over Europe perhaps because the inventor was English. Instead there were some other methods developed in some other countries of Europe, all of which were developed after the ballistic pendulum method. Some of these were not as accurate, but were still preferred over the ballistic pendulum method, perhaps because of nationalistic reasons. The interesting thing about these methods is that while they were more inaccurate than the ballistic pendulum method at the time they were invented, they laid the foundations for more modern and accurate methods of determining bullet velocities.

In 1767, an Italian named Mattei came up with a method to measure bullet velocities. His method consisted of a vertical paper cylinder which was mounted on a wooden frame. The frame was made to rotate by using a cord and a weight. Once a uniform known speed was attained by it, the bullet was fired through it, perpendicular to the axis of the cylinder. The bullet passed through the paper cylinder and left two holes on the surface. The two holes gave the arc through which the cylinder had rotated as the bullet passed through it. By computing the length of the arc and knowing the diameter and the rotational speed of the cylinder, one could compute the bullet velocity. However, this method's precision was dependent on three factors: (a) the diameter of the cylinder, (b) knowing the rotational speed accurately and (c) the uniformity of the rotation. This machine was also not very effective against faster moving bullets, because it could not measure times less than 1/30th of a second, during which time a modern bullet, such as that fired by an M-16 rifle, could easily cover 100 feet (30 meters), which means you'd have to use a cylinder at least 100 feet in diameter to measure the speed of an M-16 bullet reasonably. It was also very hard to make sure the cylinder was rotating uniformly and so the device was not very accurate either.

In 1804, a French officer named Colonel Grobert invented another method to determine bullet velocity based on similar principles as Mattei's method. In Grobert's method, two large disks about 6.5 feet in diameter, made of cardboard, were attached to the same horizontal axle. In Grobert's original design, the two disks were placed 13 feet apart. The axle was rapidly rotated by means of an endless chain in combination with a flywheel and a windlass. When the rotation of the axle was judged to be at a constant speed, the tester then aimed at the disks and fired a shot through them. The bullet pierced through both disks, but since they were rotating rapidly, the exit hole through the second disk was not in the same line as the exit hole through the first disk. By looking at the positions of the two holes, one could determine the angle of revolution. Since the distance between the two disks was known and the speed of rotation was also known, the tester could calculate the bullet velocity. However, this method had the same weaknesses as the Mattei method described above.

The next step to correct these above problems was invented by another French officer, one Colonel Dabooz, in 1818. His method involved a gravity apparatus to measure bullet velocities.

His apparatus consisted of two screens, two pulleys, a cord and a counterweight. In his method, a fixed screen was placed precisely 50 yards from the muzzle of the firearm to be tested. Directly in front of the fixed screen was another screen, which was suspended by the cord. The cord passed through two pulleys and the other end was right in front of the muzzle of the gun and was tied to a counterweight, which held the movable screen in place. The firearm was aimed at the screens. When the firearm was discharged, the bullet would cut the cord as it left the barrel. This would release the movable screen, which would start falling. The bullet would travel 50 yards and pass through both screens. Since the movable screen was falling during this time, the hole in the movable screen would not be at the same height as the hole in the fixed screen. Knowing the acceleration due to gravity and measuring the distance between the holes in the two screens, the tester could calculate how much time the bullet took to travel 50 yards and from this, he could calculate the bullet velocity. This method had the advantage that a constant acceleration was imposed on the falling screen (since acceleration due to gravity remains constant), which made the time measurement more reliable. To measure the velocities of faster moving bullets, the distance between the two pulleys could be increased and reasonably accurate measurements could be made. The one error in this method is that it assumed that the movable screen would begin to fall as soon as the bullet passed through the cord.

The reader might note that while these three method use different ideas, they have one thing in common: they all measure the time taken for a bullet to travel through a known distance and from this, they determine the bullet velocity. This is unlike the ballistic pendulum method, which uses the principle of conservation of momentum to determine the bullet velocity. While these methods were not as accurate as the ballistic pendulum method when they were invented, they laid the foundation for the concept of cronographs. With the invention of electricity, the cronograph concept became more practical and more accurate and it eventually replaced the ballistic pendulum method. We will study practical cronographs in the next post.

Testing Firearms: Measuring Bullet Velocity - I

In our previous posts, we've seen how to measure chamber pressures and trigger pull force. In this post, we will study how to measure bullet velocity. This will be a study in two posts since there is much to discuss on this topic.

First, why does someone need to know the bullet velocity. For one, it is useful to compute the kinetic energy carried by the bullet. We can obtain the bullet mass by weighing it and if we can obtain its velocity, then its kinetic energy is computed by simple mathematics: kinetic energy is calculated as (1/2 * m * v ²), where m = mass of bullet and v = velocity of the bullet. Also, it is useful to know how much velocity a bullet loses over distance to determine effectiveness over various distances.

The first really good method to determine the bullet velocity appeared in a book published in 1742 called New Principles of Gunnery written by Benjamin Robins, an English mathematician with an interest in ballistics. This was a very influential book, as it introduced military men to the teachings of Newtonian physics. This book also contributed to the development of artillery towards the end of the 18th century and was responsible for introducing calculus to the syllabus of many military academies. In fact, Benjamin Robins is considered one of the founders of modern aerodynamics and the father of modern gunnery. Before this book appeared, gunnery was simply a matter of guesswork. After this book was published, it became an exact science. The work was so influential that the famous Swiss mathematician and physicist, Leonhard Euler, himself translated this book into German.

In this book, Robins introduced the concept of a ballistic pendulum. In his original book, this is a heavy iron weight with a wooden board covering its face. The bullet is fired into the pendulum weight and gets embedded into the wooden board. The act of the bullet hitting the pendulum transmits the bullet's momentum into the pendulum, causing it to swing, as shown in the image below:

Public domain image

The pendulum also has a ribbon attached to the arm and gripped loosely by a clamp. As the pendulum swings, it pulls a length of ribbon out with it. By measuring how much of the ribbon was pulled out, we can determine the length of the pendulum's arc. We can also measure how many times the pendulum swings in one minute (i.e. its oscillation period).

An image of the original apparatus as published in Benjamin Robins' book, New Principles of Gunnery

Click on image to enlarge.

Robins' original formula used the oscillation period and mass of the pendulum and the pendulum arm to calculate its rotational moment of inertia and from there, the bullet's velocity. In his original work, he ignored the effect of the bullet not hitting the center of mass of the pendulum weight. The very next year, an updated formula to correct for this omission appeared in a paper published by the Royal Society of England. Meanwhile, Leonhard Euler, who was unaware about the corrected formula, independently determined the same corrected calculation and published the corrected version when he translated Robins' book into German. The formula is computed as:

v = 614.58 * g * c * (p + b) / (b * i * r * n)

where:

• v = Velocity of bullet in meters/sec
• g = Distance from the pivot to center of gravity of the pendulum in meters
• c = The chord (i.e.) length of swing of the pendulum determined by the ribbon in meters
• p = Mass of the pendulum in kg.
• b = Mass of the bullet in kg.
• i = Impact point (i.e.) distance from the pivot to the point of impact of the bullet in meters
• r = Radius (i.e) distance of the pivot to the point of attachment of the ribbon in meters
• n = Number of oscillations made by the pendulum in one minute

The same formula can be switched to get velocity in feet/sec and if one uses feet instead of meters and pounds instead of kg.

If one were to ignore the effects of rotational inertia (whose effect is somewhat small to begin with) and ignore the mass of the pendulum arm (modern day materials technology can make the pendulum arm very lightweight compared to the weight of the pendulum), this formula can be simplified even further as:

v = ( 1 + p / b) * sqrt(2 * G * h)

where G = acceleration due to gravity and h = height of the pendulum's travel and the other terms are the same as the previous formula. If the simplified formula is used, one doesn't even need to calculate the pendulum's period of oscillation and it is sufficient to only weigh the bullet and the pendulum and measure the height of the pendulum's travel. Of course, this simplified formula doesn't provide as accurate an answer as the first formula, but is good enough for many calculations.

This simple experiment was the first really scientific method to determine bullet velocity and the book that it was published in revolutionized military science. This method remained in use for quite a while into the mid 1800s before becoming obsolete. However, it is still seen in high-school physics labs, to teach the concepts of momentum and velocity.