When equal spaces are not described in equal times, we can imagine the rate of motion to change either gradually or discontinuously. Thus it can be imagined that a body which moves for some seconds uniformly at the rate of 10 feet in a second, may at once, without any intermediate state, take a velocity of 20 feet. But such a conception cannot be realised on any material body, though there may be all the appearance of it. [IMPULSE.] When the rate of motion is changing perfectly gradually, there seems to be no direct method of obtaining the rate at any one instant ; for no successive equal spaces are described in equal times. This difficulty will be discussed in the article VELOCITY : for the present, it may be considered sufficient to take a length so small that the change of rate undergone in passing through it is insensible, and to consider the point as moving uniformly through that length. Let the very small portion of a foot represented by as be described in the small fraction of a second, represented by a t. Let r be the number of feet which would be described in one second, at the same rate.
5 s ' Then, :Ss= I v • and as and at being taken as small as at • s wo please, 5 t — may be taken for the velocity at any instant.
E The existence of motion is detected either by a change of the distance of an object, or of its direction, or both ; but it is not necessarily the object which moves. The spectator himself may be in motion uncon sciously, and it is matter of common experiment that every motion of the spectator of which he is not conscious, and every rapid motion, whether he be conscious of it or not, causes surrounding objects to appear in motion. In walking, the effort necessary to maintain motion perpetually reminds us that it is ourselves who move ; in a carriage, at an ordinary pace, we can always destroy the illusion of surrounding motions by a moment's thought. But if the attention drop, and we ook at objects with the mind intent on other things, they soon take the motion of the carriage in a contrary direction. In the smooth motion of a boat, no effort of thought will enable the spectator to realise his own motion, and destroy that of the shore or a neighbouring vessel. We state that which we find to happen to ourselves ; perhaps the experience of other persons may be different.
It may also happen that the object is in motion as well as the spectator, in which case the latter motion will be transferred to the former, in the manner in which we shall describe. The whole motion of the object, compounded of that which it has of its own, and that which it appears to have from the motion of the spectator, is called the apparent or relative motion.
The method of ascertaining the relative motion is as follows :—Since we only determine the positions of bodies by their distances and directions, and since we suppose the motions both of the spectator and the object to be given, let a fixed point he taken to represent the position in which the spectator imagines himself to remain, and, laying down the real distances and directions of the object at the end of successive times, set off those distances from the fixed point in the proper directions. The relative positions of the object being thus secured, the line passing through these positions will be that in which the object appears to move. For instance, let the spectator move through 1 2 3... 8 9 while the object moves through A B C H I, so that when the first is at 1, the second is at A; when the first is at 2, the second is at /3, and so on, the last positions being 9 and T. Take o for a fixed point, at which the spectator fancies himself to be, and having joined 1 and A, 2 and B, &c., draw o a parallel and equal to 1 A, 0 b parallel and equal to 2 B, &c., and o i parallel and equal to 9 T. Hence the spectator, fixed at o, will see the object successively at the same distances and in the same directions as a, b, &c., and whence the line abc...hi will be that of its apparent motion.
When both motions are rectilinear and uniform, the apparent motion may be more simply obtained, as follows :—Let the spectator move uniformly from o to A, while, in the same time, the object moves from LI to c. Take the following method of fixing the spectator :—As he moves forward from o to A, let the paper on which the figure is drawn move backward in the direction contrary to on, so that by the time the spectator has reached A, the point A shall have receded to where u was. He will therefore never have changed his place, his progression on the paper having been always compensated by the retrogression of the paper itself. Take c D parallel and equal to A o, whence the point c will, by the motion of the paper, be at the end of the motion, where D was at its beginning. Consequently, the spectator, who imagines himself at rest, will give to B that motion which is compounded of a =lies along s c, whiles o itself is carried into the position ED ; that is, • will appear to move along the line n D, the spectator imagining himself to remain at o. The dotted lines show the method of proceeding, as first described.