Free-falling bodies
Projectile motion without air resistance
Projectile motion with air resistance
Resources
Ballistics is the scientific study of the motion of bodies projected through space. A projectile is an object that has been launched, shot, hurled, thrown, or by other means projected, and continues in motion due to its own inertia. Such objects can include projectiles fired from cannons or small arms, bombs dropped from airplanes, or powered rockets. The path of the projectile is determined by its initial velocity (direction and speed) and the forces of gravity and air resistance. For objects projected close to Earth and with negligible air resistance, the flight path is a parabola. When air resistance is significant, however, the shape and rotation of the object are important and determining the flight path is more complicated. Ballistics influences many fields of study ranging from analyzing a curve ball in baseball to developing missile guidance systems in the military.
Free-falling bodies
In order to understand projectile motion it is first necessary to understand the motion of free-falling bodies—objects that are simply dropped from a certain height above Earth. For the simplest case, when air resistance is negligible and when objects are close to the Earth’s surface, Italian astronomer and physicist Galileo Galilei (1564–1642) was able to show that two objects fall the same distance in the same amount of time, regardless of their weights. It is also true that the speed of a falling object will increase by equal increments in equal periods of time. For example, a ball dropped from the top of a building will start from rest and increase to a speed of 32 ft (9.8 m) per second after one second, to a speed of 64 ft (19.6 m) per second after two seconds, to a speed of 96 ft (29.4 m) per second after three seconds, and so on. Thus, the change in speed for each one second time interval is always 32 ft per second. The change in speed per time interval is known as the acceleration and is constant. This acceleration is equal to 1 g (commonly pronounced: one gee), where g stands for the acceleration due to the force of gravity (g). By comparison, a pilot in a supersonic jet pulling out of a nose dive may experience an acceleration as high as 9 g (nine gee). (Of course, a jet is not in free fall but is being accelerated by its engines and gravity.)
The acceleration of gravity, g, becomes smaller as the distance from Earth increases. However, for most earthbound applications, the value of g can be considered constant (it only changes by 0.5% due to a 10 mi [16 km] altitude change). Air resistance, on the other hand, can vary greatly depending on altitude, wind, and the properties and velocity of the projectile itself. It is well know that skydivers may change their altitude relative to other skydivers by simply changing the shape of their body. In addition, it is obvious that a rock will fall more quickly than a feather. Therefore, when treating problems in ballistics, it is necessary to separate the effects due to gravity, which are fairly simple, and the effects due to air resistance, which are more complicated.
Projectile motion without air resistance
The motion of projectiles without air resistance, can be separated into two components. Motion in the vertical direction where the force of gravity is present, and horizontal motion where the force of gravity is zero. As English physicist and mathematician Sir Isaac Newton (1642–1727) proposed, an object in motion will remain in motion unless acted upon by an external force. Therefore, a projectile in motion will remain with the same horizontal velocity throughout its flight, since no force exists in the horizontal direction, but its velocity will change in the vertical direction due to the force of gravity. For example, if a cannon ball is fired in the horizontal direction, the velocity of the ball will remain constant in the horizontal direction but will accelerate toward the Earth in the vertical direction with an acceleration of 1 g. The combination of these two effects produces a path that describes a parabola. Since the vertical motion is determined by the same acceleration that describes the motion of objects in free fall, a second cannon ball that is dropped at precisely the same instant as the first cannon ball is fired, will reach the ground at precisely the same instant. Therefore, the motion in the horizontal direction does not affect the motion in the vertical direction. This fact can be confirmed by knocking one coin off the edge of the desk with a good horizontal whack, while a second coin is simultaneously knocked off the desk with a gentle nudge. Both coins will reach the ground at the same time.
By increasing the amount of gunpowder behind the cannon ball, one could increase the horizontal velocity of the cannon ball as it leaves the cannon and cause the cannon ball to land at a greater distance. If it were possible to increase the horizontal velocity to very high values, there would come a point at which the cannon ball would continue in its path without ever touching the ground, similar to an orbiting satellite. To attain this orbiting situation close to the Earth’s surface, the cannon ball would have to be fired with a speed of 17,700 mph (28,500 km/h). In most instances, projectiles, like cannon balls, are fired at some upward angle to the Earth’s surface. As before, the flight paths are described by parabolas. (The maximum range is achieved by aiming the projectile at a 45° angle above the horizontal.) Angles equally greater or less than 45°will produce flight paths with the same range (for example, 30° and 60°).
Projectile motion with air resistance
If projectiles were only launched from the surface of the moon where there is no atmosphere, then the
KEY TERMS
Acceleration of gravity— The vertical downward acceleration equal to 32 ft (9.8 m) per second per second experienced by objects in flight close to Earth.
Air resistance— The drag force on an object in flight due to the interaction with air molecules.
Free falling body— A falling object in one dimensional motion, influenced by gravity when air resistance is negligible.
Gyroscope— A device similar to a top, which maintains rotation about an axis while maintaining a constant orientation of that axis in space.
Inertia— The tendency of an object in motion to remain in motion in a constant direction and at a constant speed, and the tendency of an object at rest to remain at rest.
Projectile— An object that is projected close to Earth and whose flight path is determined by gravity, air resistance, and inertia.
effects of gravity, as described in the previous section, would be sufficient to determine the flight path. On Earth, however, the atmosphere will influence the motion of projectiles. As opposed to the situation due to purely gravitational effects, projectile motion with air resistance will be dependent on the weight and shape of the object. As one would suspect, lighter objects are more strongly affected by air resistance. In many cases, air resistance will produce a drag force that is proportional to the velocity squared. The effects of increased air drag on an object such as a cannon ball will cause it to fall short of its normal range without air resistance. This effect may be significant. In World War I, for instance, it was realized that cannon balls would travel farther distances if aimed at higher elevations, due to the decreased air density and decreased drag.
More subtle effects of air resistance on projectile motion are related to the shape and rotation of the object. Clearly, the shape of an object can have an effect on its projectile motion, as anyone has experienced by wadding up a piece of paper before tossing it into the waste can. The rotation of an object is important, too. For example, a good quarterback always puts a spin on a football when making a pass. By contrast, to produce an erratic flight, a knuckle ball pitcher in baseball puts little or no spin on the ball. The physical property that tends to keep spinning objects spinning is the conservation of angular momentum. Not only do spinning objects tend to keep spinning but, also, the orientation of the spin axis tends to remain constant. This property is utilized in the design of rifle barrels that have spiral grooves to put a spin on the bullet. The spinning of the bullet around its long axis will keep the bullet from tumbling and will increase the accuracy of the rifle. This property is also utilized in designing guidance systems for missiles. These guidance systems consist of a small spinning device called a gyroscope, which keeps a constant axis orientation and, thus, helps to orient the missile. Small deviations of the missile with respect to the orientation of the gyroscope can be measured and corrections in the flight path can be made.
See also Conservation laws.
Resources
BOOKS
Hewitt, Paul. Conceptual Physics. Englewood Cliffs, NJ: Prentice Hall, 2001.
Munson, Bruce, et al. Fundamentals of Mechanics. 4th ed. New York: John Wiley and Sons, 2002.
Young, Hugh D. Sears and Zemansky’s University Physics. San Francisco, CA: Pearson Addison Wesley, 2004.
Kurt Vandervoort
