## Monday, January 10, 2011

### 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.