and making ds, St indefinitely small, as v', v" have both the same limit, we get 16. This result includes the case of a constant velocity, because in that case being constant is not altered by taking t 17. The propositions called the parallelogram and parallelepiped of velocities are equally true for varying velocities, as may be clearly seen by considering an indefinitely small parallelogram or parallelepiped similar to those in the figures of Arts. 8, 9, or by comparing the motion of the proposed point with that of another moving with constant velocities equal to those in question. 18. If the velocity of a moving point be continually increased, its motion is said to be accelerated; and if diminished, retarded. The terms acceleration and retardation* are used to express the degree of this change of velocity. The acceleration or retardation is uniform when the velocity is uniformly increased or diminished, i. e. when equal velocities are acquired in equal successive intervals of time, in or opposite to the direction in which the motion is estimated. 19. An uniform acceleration is clearly greater or less in exact proportion as the velocity acquired by the moving point in a given time is greater or less, or as the time requisite for the moving point to acquire a given velocity is less or greater. Therefore if a be the measure of an acceleration, owing to which the moving point acquires a velocity v in time t, a∞ t *N.B. We are here concerned with acceleration and retardation merely as matters of fact, and not considered as resulting from any particular cause whatever. C L and if with the unit of acceleration a velocity v, is acquired in time t1, we have and therefore and v, are at present arbitrary, and the simplest assumption we can make respecting them is that they both = 1, and then 20. These assumptions fix the unit of acceleration to be that owing to which the unit of velocity is acquired by the moving point in the unit of time, and the measure of any acceleration to be the velocity which is acquired in the unit of time. An acceleration then may be represented by a line, both in magnitude and direction. The same remarks apply to retardations. 21. Hence if v be the velocity acquired in time t by means of an uniform acceleration a, v will at, because a velocity a is acquired in each unit of time. = And if v', v be the velocities at the beginning and end of the time t, v=v'+at if the motion is accelerated, and vvat if the motion is retarded. This shews that a negative acceleration is identical with a retardation, as might have been expected, and henceforth the term acceleration will be supposed to include retardation. 22. The motion of a point, however, may not be uniformly accelerated; and we must find a measure of the acceleration in such a case as this. Suppose another point whose motion is uniformly accelerated, and let the accelerations on the motion of this and the proposed point be, at the instant under consideration, equal. Then the acceleration on the motion of this point will be measured by the velocity acquired in an unit of time, and therefore the acceleration on the motion of the proposed point will be measured by the velocity which would be acquired in an unit of time, supposing the acceleration were to remain constant for that unit of time. v' 23. If v be the velocity acquired in time t reckoned from the instant under consideration, by means of an acceleration a, and if v' be the velocity which would be acquired in the time t, supposing the acceleration constant, then a=; and this is always true, and therefore true in the limit when vand t are indefinitely diminished. A. 24. By means of the differential calculus we may obtain the same result. For if dv be the velocity acquired in a small time St immediately subsequent to the instant under consideration, and if a', a" be the greatest and least values of the acceleration during the time St, then dv <a'st and > a"St; and as a', a" have the same limit a when St is indefinitely diminished, we get 25. This result includes the case of a constant acceleration, for then being constant, will = Limit or = v t 26. The Parallelogram of Accelerations. dv dt' "If two adjacent sides of a parallelogram represent in magnitude and direction two accelerations by which the motion of a point is simultaneously affected, the resulting acceleration will be represented both in magnitude and direction by the are acquired in a very small time, AE, AF are proportional to AB, AC, the more nearly as the time in which they are acquired is indefinitely diminished. The resulting velocity will be measured by AG. The resulting acceleration then must be in the direction of AG and be to AB or AC in the same ratio as AG is to AE or AF respectively; i.e. it will be represented in direction and magnitude by the diagonal AD. This may be extended to space of three dimensions as was done for velocities in Art. 9, and all the analytical formulæ and remarks of Arts. 10, 11, 12, will equally hold good for accelerations. 27. As we have shewn (Art. 6) how to transform the measure of a velocity from one set of units of space and time to another, we must now shew how to transform the measure of an acceleration. Let the new units of space and time be a and b times respectively as great as the original units. Then a, being the measure of the acceleration, is equal to the measure of the velocity acquired in an unit of time; therefore the measure of this velocity, when referred to the new unit of time, will be ba. (See Art. 6.) But this velocity is not acquired in the new unit of time, but only in the 6th part of it; therefore the velocity acquired in the new unit of time is 6 times as great as this, i. e. it = b'a. This velocity has now to be referred to the new unit of space, and therefore (see Art. 6) its measure is b3a α This, then, is the measure of the acceleration in terms of the new units. E. g. If the acceleration on the motion of a point be measured by 20 when a foot and a second are the units, what will be its measure when a yard and a minute are the units? The velocity acquired in 1′′ is 20 feet per 1", i. e. 20 × 60 feet per 1'; therefore the velocity acquired in 1' is 60 x (20 × 60) feet per 1', therefore the measure of the acceleration is 24,000. We have purposely chosen the same numbers as those in the transformation of the velocity given in Art. 6, in order that the distinction may be clearly seen. No errour can well arise if it be borne in mind constantly, that the measure of an acceleration is the velocity acquired in an unit of time, estimated per that unit of time. 28. All that has been said respecting constant accelerations will equally apply to varying accelerations, by means of considerations analogous to those employed in Art. 17, and by making the necessary changes in phraseology. |