Ballistics Made Really Simple

Today, I would like to give my version of the science of ballistics. That is, I want to explain it here on my own terms.

First, let's define ballistics. An object is said to be in ballistic flight when it is flying as a result of an impact or other input of energy, but does not have a source of power of it's own. An aircraft or bird is not in ballistic flight because both of these have sources of power to enable them to fly.

A rocket is not in ballistic flight if it's engines are on, but it is if the engine thrust has been turned off and it is flying by the resulting momentum. When in the atmosphere, a flying object must be heavier than air to be in ballistic flight. A lighter-than-air balloon is not ballistic.

Ballistics usually refers to the motion of bullets and artillery shells. But it is close to the beginning of baseball season in North America and so, in an effort to make baseball more interesting, I will use the motion of a ball to describe ballistics. But the same principles will apply to any object in ballistic flight. Readers in Commonwealth countries can simply substitute cricket for baseball.

In these examples, I will presume that the ball is caught at the same level that it was thrown or hit, that the playing field is level and that there is no wind. I will neglect any air resistance on the ball.

Consider a ball that is hit or thrown upward at a given angle to the horizontal ground. Just like the hands on a clock, this initial angle determines how long the ball will remain in the air. If the force with which the ball is hit or thrown remains constant, the ball will cover the greatest horizontal distance when it is launched at a 45 degree angle. This is the point at which the horizontal and vertical components of the thrust are equal.

Because the ball is always falling vertically while it is progressing horizontally, a vertical distance is necessary to extend the horizontal distance that the ball travels. This is why the maximum horizontal distance is obtained at 45 degrees and not 0 degrees, because the ball stops when it meets the ground. Angles at which the ball are hit that are equidistant from 45 degrees, such as 30 and 60 degrees, will produce equal horizontal distances covered by the ballistic flight of the ball.

At a launch angle of 45 degrees, the maximum vertical height reached by the ball will be half that of the horizontal distance travelled by the ball. But the total vertical motion will be equal to the total horizontal motion. This is because the vertical motion in the course of the ball's flight consists of both ascension and descension, and the two must be added together.

The impact or throw which launched the ball into ballistic flight is instantaneous, but the gravity which acts on the ball is constant and continuous. This means simply that gravity drops the vertical velocity of the ball at a constant rate down to zero and then reverses it until it is at a negative value of the original velocity at the end of it's flight. The horizontal velocity of the ball, in contrast, is constant throughout the flight.

At launch angles above 45 degrees to the horizontal, the tangent of the angle gives the relationship of total vertical motion (ascension plus descension) to the horizontal distance covered by the ball. This means that at an angle with a tangent of 2, or 63.4 degrees, the maximum height that the ball reaches in flight is equal to the horizontal distance that it covers.

By the way, the way to measure how high a ball has climbed is to time it. Take the total time that the ball was in the air. Then divide that by two because the ball was ascending for only half of that time. Square the number of seconds and then multiply that by 16 feet or 4.9 meters. This presumes, once again, that the ball is caught at the same height above the horizontal that it was thrown or hit.

The total two-dimensional area of the air and the surface of the ground where the ball is overhead at some point during the flight will be shaped like a bell, as long as the ball was hit at an upward angle. This two-dimensional covered area increases as the launch angle of the ball increases, up to 45 degrees and then remains constant. The size of the two-dimensional covered area is in direct proportion to the time that the ball spends in the air, but not to the horizontal distance that it covers.

From the time that the ball is hit by the bat, the angle relative to the horizontal at which it flies is continuously and steadily decreasing. This is why the maximum altitude of a ball launched at 45 degrees is only half of the horizontal distance covered.

At the point of maximum altitude, this angle is zero and the course of the ball is briefly parallel to the horizontal ground. The angle of the ball's course then becomes negative as it heads back toward the ground. Finally, the ball is caught at the end of it's flight with a negative value of it's launch angle and velocity. The total span of angles that the course of the ball goes through is thus twice that of the initial angle at launch.

There, hopefully baseball will be a little bit more interesting this year.