VELOCITY. 



VKLOCITY. 



portion of Urn* bears to one second. But this is a presumption only. 

 It doe* n..t follow, because ten feet are described in one Kcond, that 

 fire feet are denribed in each half of second, and one foot in each 

 tenth of a aecond. If the Mcond could be divided into a million of 

 part*, and it could be shown that the millionth part of ten feet ia 

 described in each and all of thews part*, it would be no doubt a very 

 strong presumption that the motion is really uniform, but still not 

 amounting to certainty ; for it is possible that in each of those part* 

 .I time there may be a variation of speed : for instance, the moving 

 1 ..int may do all its work in the first half of the small interval, and 

 rat during the remainder. Something of this kind takes place in the 

 motion of the minute-hand of a clock, which ia propelled once in each 

 second during a portion of the second, and rests during the remainder. 

 But so rapidly do the small propulsions follow one another, and so 

 small are their individual effects, that, even when the hand has been 

 watched until its motion is certain, there is no irregularity discoverable 

 by ordinary eyes. And, speaking with reference to common purposes, 

 there is no occasion to deny uniformity of motion so long as the lengths 

 described in those times which are convenient to be mentioned are 

 equal or nearly equal. It would be useless, in speaking of tbe pace 

 with which a man walks four miles an hour, to remind the hearer that 

 no person walks uniformly , and that in every step the centre of gravity 

 of the body moves up and down, advancing most rapidly when it is at 

 the highest, and most slowly when it is lowest. But for mathematical 

 purposes a correct measure of speed must be obtained, and the pre- 

 ceding account would at first seem to lead to the inference that it is 

 impossible to have such a measure. Nor indeed has velocity yet 

 received its definition in this article, at least not its measure : we have 

 spoken of velocity and of its changing, but without alluding to any 

 mode of estimating the quantity of change. But there is that about 

 the word which needs no definition : when we Bay that the railroad 

 carriage mores " faster " than the old stage-coach, or that two bodies 

 which set out together and keep together are always moving " at the 

 same rate," there is no need of explanation of the words which are in 

 marks of quotation. And we must now refer to the considerations in 

 UNIFORM, as a constituent part of this article, showing that we may 

 have a perfect idea, both of velocity that it is a magnitude, and that 

 there is such a thing as uniform velocity, previous to any definite ideas 

 of the most proper mode of measuring even that uniform velocity, to 

 go no farther. 



If a body move uniformly, it is customary at once to lay down as 

 the measure of the velocity the space described in a given time, 

 usually the unit of time, a second, a minute, an hour, as the case 

 may be. So far as the great object of calculation is concerned, this 

 definition is perfect : by instituting measures of velocity, we can but 

 want to answer one or other of these questions : Where will the moving 

 point be at the end of a given time ? or, In what time will the moving 

 point pass over a given length ? The body moves at the rate of v feet 

 per second, it moves over r feet in t seconds, and moves over the 

 length j feet in -=-* seconds. Let us now take a point moving with 

 a variable motion, that is, not describing equal lengths in equal times, 

 say a particle descending by its own weight in a vacuum. In the first 

 second it falls 16 feet; but in the first half of this second it fulls 

 only 4 feet, and the remaining 12 feet in the second half-second. 

 The space described in one second is therefore no measure of the rate 

 of motion during that second, and it is now to be asked, What is the 

 way of obtaining a measure of the speed after any interval has elapsed ? 

 What is velocity itself, when it cannot for want of uniformity be 

 ascertained by the space described in any given time ? If the action 

 of gravity were removed at the end of that time, so that the point 

 would go on uniformly with its last acquired velocity, how much 

 would it then describe in one second ? All these questions are the 

 same, and the answer cannot be given without the introduction of 

 the notion of a limit, whether with or without the forms of the 

 differential calculus. At the end of the time t seconds, let the moving 

 point be at A, distant by < feet from the fixed point o. During 



o ' " 



the ensuing fraction A of a second, let it describe the further space 

 AB(-i). The length k is then moved over in the time A, and, 

 if the velocity were uniform, that velocity would be k/i feet in one 

 second ; for as k is to 1 (second), so (on the supposition of uniform 

 Telocity) is i- to the space which would be described in one second. If 

 AH were very small, we might reason (with tolerable exactness) as 

 follows : In a very small time the change of speed will be slight, and the 

 motion of the point nearly uniform, though not absolutely so ; whence 

 we may say, without material error, that AD is described as with a 

 uniform velocity at the rate of t~H feet per second. The process 

 which the mathematician adds u the following : The error of the 

 preceding process, small when A is small, becomes smaller when A is still 

 smallcr,and may be diminished to any extent : that is, little as may be the 

 departure from uniform motion in moving through a small length, it is 

 leas in moving through a smaller. If, then, instead of making A simply 

 small, and then finding jt-~:-A,wo diminish A without limit, and find the 

 limit towards which k-:-h approaches, we find that uniform velocity 

 which may be naid to represent tbe speed of the point in pasting 



through A, so far as any uniform velocity can be said to do so. Using 

 such language as supposes the point to have volition, we have, in the 

 limit of X-t-A, the length per aecond with which tbe point shows an 

 intention of proceeding at the instant when it passes through A, though 

 it does not preserve that intention wholly unaltered for any portion of 

 time, however small. 



Suppose for example that the point moves in such a way as to 

 describe I + C feet in ( seconds, for all values of I, whole or frac- 

 tional. We have then i-*t + (*, + i- (< + A) + (< + A), whence we 

 obtain 



At the end of three seconds, what is the velocity f Judging from tbe 

 length described during the succeeding fraction A (and making t = 3), 

 we should say that, Ir+h being 7 + A, the limit of this, or 7, obtained 

 by diminishing A without limit, is the velocity required ; that is, the 

 point is then moving 7 feet per second. If we suppose 7 feet per 

 second, the length described in the fraction A of a second is the 

 fraction 7 A of a foot; take any other uniform velocity p feet per 

 second, and pA is the length described in the same time. Now what 

 is really described is 7A + A 3 ; so that the errors are A 9 and (7-p) 

 A + A a , which are in the ratio of A to 7 /> + A. Now if p differ (as 

 we have supposed) from 7, the first error diminishes without limit 

 as compared with the second, when A diminishes without limit : 

 so that, of all uniform relodtia, 7 feet per second is the one which 

 best represents the motion of the point in any small time following 

 the end of the third second ; and the better the smaller the time. 



It appears then that we do not, properly speaking, undertake to say 

 at what rate the point is moving at the end of three seconds, but what 

 fictitious uniform rate best represents, at the instant, the rarioMe rate 

 at which it is moving. This will, for a moment, seem rather unsatis- 

 factory to the student who imagines that he has got an absolute idea 

 of velocity, and here he should compare his notion on this subject with 

 that of the direction of a point moving in a curve. [DIRECTION ; 

 TANGENT.] What do we mean by saying that a point which moves in 

 a curve has, at every instant, the direction of motion which is repre- 

 sented by the tangent of that curve? Answer, in nearly th. 

 words as before, We do not, properly speaking, undertake to say in 

 what direction the point is moving at any period of ita motion, but 

 what fictitious line of uniform direction (straight line) best represents, 

 at that instant, the lint of variable direction (curve) on which it in 

 moving. The study of these two things together, velocity and direc- 

 tion, is useful, as each throws illustration upon the difficulties of the 

 other. In both cases the laws of matter agree in preferring that which 

 is indicated as most simple by the laws of mind ; for if a point moving 

 along a curve be suddenly relieved of the forces which keep it in a 

 perpetual change of speed and direction, it will proceed with th;\t \ny 

 velocity which we have said it shows its intention to proceed with, 

 uniformly; and will quit the curve for that straight line which wi- 

 might equally well have said it showed a disposition to prefer to 

 any other while moving on the curve, namely, the tangent of the 

 curve. 



If it should be said that we are reduced, in treating of variable 

 velocities, to a necessity which does not occur in describing those 

 which are uniform, namely, the use of limits, we altogether deny the 

 fact ; that is, we say that we are as much compelled to the use of 

 limits in defining a uniform velocity as a variable one. For what does 

 unifonn velocity mean? A point has uniform velocity when equal 

 spaces, any equal spaces whatsoever, are described in equal times ; or 

 when, it being described in the time A, i-j-A is always the same. That 

 is, I--T-A must retain its value, however small A may be ; or the limit of 

 t-r-h must also have that value. And we have seen that it would be 

 impossible to declare, experimentally, the existence of uniform velocity, 

 even if our senses had no imperfections, upon the experience of com- 

 parisons of any finite equal spaces, however small; nothing but 

 assurance of the limit o/i-=-A being the same thing wherever the point 

 A might be placed, would give mathematical evidence of the velocity 

 being uniform. 



In all cases, then, by the velocity of a point in motion, at any 

 particular period of its motion, is to be understood the limit of the 

 ratio which the increment of the length described bears to tin- 

 ment of the time expended in the description of that increment of 

 length. That is, if the length be measured in feet, and the time in 



. . . ___!. -i i 1 1. _ f ___ A; ___ _ * _ 



.. continually t 



of feet per second which, we may say, expresses the rate of motion at 

 the period in question. The student of the differential calculus will 

 now have no difficulty in altering the preceding into the following 

 form : if the length > be described in the time t, the velocity (r) at 

 the end of the time t is thus expressed : 



.-*. 



dt 



If V be any function of x, and if ? represent the number of units of 

 length de*cril>c>l l>y a moving point in the time ', and y the same for 



