Ballistics: Measure the velocity of the projectile with the ballistic pendulum.
If you don't want to buy an expensive coronograph, you can use the ballistic pendulum, a somewhat dated method, but which is always accurate and reliable. The ballistic pendulum, whose origins date back to the 1700s, laid the foundations of modern ballistics.
Ballistic experts and lovers of refilling cartridges often use the chronograph, an instrument with which the speed of a bullet is precisely established. Those who do not want to buy the chronograph, on the other hand, can use the old method of the ballistic pendulum, with which more than reliable data is obtained without any expense.
Cassini Junior in 1707 had the idea of using a pendulum to determine the speed of a bullet, but later scholars such as B. Robins, Diddion, Morin and Piobert developed the derivative instrument, called the ballistic pendulum. The latter was the tool that laid the foundations of modern ballistics. The theoretical principle behind the ballistic pendulum is very simple: the weapon must be pointed towards a pendulum mass, composed of material capable of holding the bullet and creating an inelastic collision.
By firing the bullet, it introduces itself into the pendulum mass and transmits an impulse to it. Starting from the conservation of momentum theorem and recalling the laws of pendular motion, we arrive at the momentum of the pendular mass from the amplitude of its oscillation, and therefore at the speed of the projectile.
Given the weight P of the pendulum mass and p the weight of the projectile, by adding P and p the weight of the pendulum with the driven projectile is obtained; given the velocity acquired by the pendulum la V and v the velocity of the projectile before the impact, we will obtain the following formula: p • v = (P + p) • V, from which we arrive at the inverse formula: V = (p • v) / (P + p).
Through the influence of the impact, the pendulum acquires a speed and therefore a living force E, expressed by the formula E = (P + p) xVª / 2 × 9.81, which causes it to rise by a certain space h where it is transformed into potential energy Ep = (P + p) • h.
Since E = Ep, we can put the two expressions together and derive that V is given by the square root of hx2x9.81, that is the formula relating to the fall of the bodies. This value must correspond to the one derived from the impulse and, consequently, the formula is obtained:
The height h cannot be determined directly, but can be expressed trigonometrically as a function of the length l of the pendulum and the amplitude of the angle of the oscillation with the formula h = l • (1-cosã).
Ultimately, the formula to be applied is the following:in which the only unknown is given by the alpha angle, which must be measured from time to time. The length l of the pendulum can be indirectly determined by the physical laws of pendular motion: the duration D in seconds of an oscillation is identified with a chronometer, for example by counting the number of oscillations performed in one minute, and from here the length in meters according to the formula l = 0,248 • D².
Instead of measuring the amplitude of the oscillation angle of the pendulum, it may be easier to measure the amplitude s of the arc crossed by one of its points and from there go back to the angle alpha using the formula:
where L represents the effective distance between the point around which the pendulum swings and the point tracing the arc.
In the next article we will explain how to make the ballistic pendulum.