Finally, let us consider the formula for an ellipse. This example is important also since it is the figure that describes planetary orbits. This formula is d1 b2

This ellipse has its center at the origin and semi-axes a and b. One could map out this figure in the same way as for the circle, that is, by solving the equation for y and then substituting in values for jc. If one has a = 4 and b = 3, one obtains the ellipse shown in Fig. 3.O.2. Note that if a = b, the ellipse becomes a circle with R = a.

An ellipse can be drawn in the following simple fashion. Drive two nails into a board a few inches apart. Cut a string somewhat longer than

the distance between the nails and tie the ends to the nails. Place a pencil against the string until it is taut and then make a figure by moving the pencil along the string always keeping it taut. The mark will form an ellipse.

In our discussion of types of motion and Galileo's studies of falling objects, we sometimes found it necessary to describe the motion on a speed versus time graph, and sometimes on a distance versus time graph. We indicated how some of these graphs were related, for example, how uniform motion would appear on both kinds of graphs. In general, the mathematical techniques for relating curves on these two kinds of graphs are part of the subject known as calculus. For example, we discussed how the area under a speed versus time curve corresponds to the total distance traveled.

The subject of integral calculus deals with mathematical techniques for finding the area under any curve for which a mathematical formula exists (analytical geometry). Similarly, the slope of a distance versus time graph indicates the speed of an object, and the slope of a speed versus time graph gives the acceleration. The mathematical techniques for determining slopes of curves described by mathematical formulas are known as differential calculus.

Newton and the German mathematician-philosopher Gottfried Leibnitz (16461716) are credited as independent inventors of the calculus. They performed their work separately, but arrived at the same basic results on differential calculus. That Newton developed the ideas and methods of calculus certainly enabled him to arrive at some of his conclusions regarding mechanics and the universal law of gravity. He used calculus not only to translate from speed versus time graphs to distance versus time graphs, and vice versa, but also to find areas and volumes of figures and solids.

Finally, it is necessary to discuss the fact that some quantities have direction, as well as magnitude (i.e., size). For example, if we indicate that an automobile is traveling at 30 mph, we have not fully described its motion. We need to indicate also in what direction the car is traveling. Quantities that logically require both a direction and a magnitude are called vector quantities. The vector quantity associated with motion is called velocity. If we indicate the magnitude of the motion only, we refer to its speed. Thus if we say that a car is traveling 30 mph in a northerly direction, we have specified its velocity.

Velocity is not the only quantity that is a vector; force is another example. Whenever a force is impressed on an object, it not only has a certain strength, but also a specific direction, which must be indicated in order to describe it completely. Similarly, acceleration (positive or negative) is always at a certain rate and in a specific direction, so that it also is a vector quantity. Thus circular motion, even with constant speed, is accelerated motion.

Some quantities do not require a direction, but only a magnitude. Such quantities are called scalar quantities. The mass of an object (in kilograms, for example) needs no direction and is only a scalar quantity. The length of an object is also a scalar quantity. Other examples include time, electric charge, temperature, and voltage. Speed is a scalar quantity, but velocity is a vector. Because a proper discussion of motion, including the actions of impressed forces, must include directions, it is appropriate to use vector quantities. As we will see shortly, Newton was well aware of the need to use vector quantities in his descriptions of motion, and made a special point to introduce them carefully.

Because vector quantities have both magnitude and direction, they can be represented graphically by arrows, as shown in Fig. 3.14. The length of the arrow is proportional to the magnitude of the vector, and the orientation of the arrow and position of the arrowhead represent the direction. The simple mathematical operations carried out with ordinary numbers are only somewhat more complicated when carried out with vector quantities—including addition, subtraction, and multiplication. For the purposes of this and subsequent chapters, it is sufficient to describe graphically how vectors are added. To add two or more vectors,

Figure 3.14. Vectors, (a) Examples of vectors represented by arrows, (b) Examples of vector addition in graphical representation.

Figure 3.14. Vectors, (a) Examples of vectors represented by arrows, (b) Examples of vector addition in graphical representation.

they are simply placed head to tail, maintaining their orientation and direction; and a final arrow or resultant vector is drawn from the tail of the first arrow to the head of the last arrow. This final arrow represents the vector sum of the individual vectors, as shown in Fig. 3.14b. (One interesting result of vector addition is that a vector sum can have a smaller magnitude than each of the magnitudes of the vectors contributing to the sum.)

As an example of vector addition, let us consider an airplane flying in a diagonal cross wind. Let us suppose that the airplane is flying at 100 mi/h in a due east direction and that there is a cross wind of 50 mi/h moving due north. The resultant motion will be somewhat north of east and can be determined quantitatively from Fig. 3.O.3.

As shown, triangle ABC is a right triangle, hence by the Pythagorean theorem,

The direction is given by 0. We see that tan 0 = y/x = 50/100 = 0.5, or 0 = 27°. (If one is not familiar with the trigonometric functions, the angle can be found graphically, by plotting the triangle to scale and measuring the angle with a protractor.) The resultant motion of the airplane is that it travels at an angle of 27° north of due east at 112 mi/h.

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