Mechanics

It follows that any special importance which these assumptions may have, in relation to the actual universe, arises solely from the fact that they constitute the simplest possible description. To describe the motions of the bodies which form our solar system, we may either, as in the Ptolemaic system, specify the motions of those bodies relative to the earth, or, as in the Copernican system, specify their motions relative to the sun. Either descrip tion (provided that its details are correct) is equally valid, and the superior merit of the Copernican system consists solely in the fact that its description is simpler. It remained for Newton to dis cover a still more simple description, by inventing a compre hensive theory of dynamics which follows logically from three fundamental assumptions, or "laws." However his theory is still description, it does not explain ; for whilst, like the axioms of geometry, his laws are incapable of proof, they cannot by any stretch of imagination be regarded as self-evident and therefore needing no proof.

Newton's assumptions are incapable of proof, i.e., of direct veri fication by experiment, for reasons which have been indicated already; but, from the standpoint now considered, such verifi cation is in no way essential to the development of his theoretical system, and the appeal to experiment, by which that system is related with the actual universe, can equally well be made when the system is complete. Whatever view be taken of the philo sophical question, it must be admitted that the real evidence for his "laws," as an expression of the facts of nature, is to be found, not in laboratory experiments aimed at direct verification, but in the close accord with experience of every conclusion which has been based upon them.

Speed.

i. The notion of speed, e.g., of a car or railway train, is familiar in every-day experience. When we say that a train is travelling with a speed of 6o miles per hour, we mean that it is moving at a rate which, maintained constant, would take it 6o miles along its route in one hour, one mile in every minute, or 88 feet in every second. We do not mean that this rate is in fact constant : a train which is travelling at 6o miles per hour may, during the next few minutes, increase its speed, stop, or even retrace its path. What we mean is that the actual distance trav elled will be 88 ft., very nearly, in the next second, 8.8 ft., almost exactly, in the next tenth of a second, and so on : given the in stantaneous value of the speed, we can foretell the distance more closely, the shorter the interval of time.

Kinematics gives precision to this idea of speed by fixing at tention on an interval which is indefinitely short. Let s be the distance of the train from its starting point, t the time which has elapsed since the start ; then, as the train moves, s will increase with t. If the speed is constant from the start (say S miles per hour), and if s is measured in miles and t in hours, then evidently s and t will be connected by the relation so the speed S may be found by differentiating the distance s with respect to the time t.

2.

Almost all cars, and occasionally trains, carry an instrument which records the distance travelled. If we take simultaneous readings of this instrument and of a clock, we can plot, in accord ance with the principles of graphical representation, points which relate corresponding values of s and t; and if a large number of such points are plotted and connected by a continuous curve, this curve may be taken to represent the relation between s and t over the whole period of the observations. A curve of this nature is called a distance-time or s-t diagram: if P (fig. i) is a point on the curve, we know that the car or train was, at a time repre- sented (on the time-scale) by the length OL, at a distance from the starting-point, which is repre- N sented (on the distance scale) by 1111 the length OH.

L Now let Q be another point on 0 M the curve. Then we know that the FIG. Idistance from the starting-point, at a time represented by the length OM, is the distance represented by the length OK; and it follows that a distance represented by the length HK was travelled in an interval of time represented by the length LM. This means that the average speed during that interval of time is given by i.e., it is measured by the slope of the line PQ.

Speed Represented by Slope of Tangent.—If the speed were in fact constant, this slope would be constant for all inter vals of time; so we see that the s-t diagram for motion at con stant speed is a straight line. In general the speed will vary during the interval considered; but its variation may be neglected if the interval be sufficiently small ; i.e., if Q be sufficiently near to P. Ultimately, when the interval is infinitesimal, PQ is the tangent at P. We thus obtain a graphical interpretation of the formula (3) : in a distance-time diagram, the speed at a time correspond ing to P is measured (on an appropriate scale) by the slope of the tangent at P. Even when the formula (3) cannot be used, because s is not known as a mathematical function of t, the speed can be estimated by graphical methods.