Equating the momentum of the bullet before it impacts the pendulum to that of the bullet-pendulum system as soon as the bullet strikes the pendulum (and using from above), we get: Solving for : 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 Society the following year. Swiss 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: 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 Poisson 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 described how to construct and use a ballistic pendulum in 1962. Ackley's pendulum used a parallelogram linkage, with a standardized size that allowed a simplified means of calculating the velocity. Ackley's pendulum used