Mathematics for the Nonmathematician by Morris Kline

Author:Morris Kline
Language: eng
Format: epub, pdf
Publisher: Dover Publications
Published: 1967-04-07T16:00:00+00:00

14–3 THE MOTION OF A PROJECTILE DROPPED FROM AN AIRPLANE

Let us see now how parametric formulas arise in the study of physical phenomena and how they can be useful in deducing new information about the phenomena. Suppose a bomb is released from an airplane which is flying horizontally at 60 miles per hour (an unrealistic figure used for computational convenience). If there were no gravity, the bomb would continue to move forward alongside the airplane at the rate of 60 miles per hour. This fact seems surprising, but it is a consequence of the first law of motion, which states that if an object is in motion and no force is applied to alter that motion, then the object will continue to move indefinitely at the speed it already has. Since the bomb has been moving with the airplane, it already possesses a horizontal speed of 60 miles per hour. We have assumed that no forces are acting on the bomb and hence it will continue to move forward at that speed. There are more familiar analogous situations which may make the truth of what was just said a little more acceptable. Suppose that a person rides in an automobile which is moving at the rate of 60 miles per hour and the driver suddenly applies the brakes. The automobile’s motion is then checked, but the passenger’s motion is not, and he continues to move forward at 60 miles per hour, at least until he hits the windshield.

Let us return to the motion of the bomb released from the plane. We had assumed that gravity was not acting. But it does act and it pulls the bomb downward at the same time as the bomb moves forward so that the bomb follows a curved path. Here Galileo made a discovery applying to projectile motion, namely, that one could study its horizontal and vertical motions as though they were occurring separately, and that the position of the bomb at any time could be determined by finding how far it had traveled horizontally and vertically. This idea was new and radical in Galileo’s time. Aristotle had argued that one motion would interfere with the other, and that only one could operate at any given time. Thus he would have said that the violent motion imparted to the bomb by the airplane would prevail until the acting force was used up, and then the natural motion downward would take over and cause the bomb to fall straight down.

Let us apply Galileo’s way of analyzing the motion. The bomb moves horizontally at the constant speed of 60 miles per hour, or 88 feet per second. If we measure time from the instant the bomb is released from the plane, and if we measure horizontal distance from the point at which it is released, then the horizontal distance x covered by the bomb in t seconds is given by the formula

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