VELOCITY. 



VELOCITY 



G02 



another moving point, and if y=<j>x, we have by the rules of the 

 differential calculus : 



dy rfy dz^ 



dt dx dt 



Here dy : dt and dj; : dt, represent y and x, Newton's FLUXIONS of 

 y and x ; and dy : dx is obviously the same thing as y : x. The term 

 fluxion merely means velocity, and, after all, there can be formed no 

 clearer notion of a differential coefficient than one which is formed 

 from a consideration closely resembling the fluxional one. If y be a 

 function of x, dy : dx is the rate at which y is increasing, as compared 

 with that at which x increases. Thus if y=x*, dy : dx^ix 3 , which 

 when:r = 10 is 4000. What does this mean? We say that nothing 

 can answer more clearly than the following : If a number be imagined 

 to be gradually increased [VARIABLE], by the time it becomes 10 its 

 fourth power will be, at that instant, increasing 4000 times as fast as 

 itself. 



ACCELERATION is the increase of velocity ; and in the article cited 

 uniform acceleration has been considered, and its laws deduced, if not 

 with the forms, yet on the principles, of the differential calculus. 

 Precisely the same difficulties come before us in the development of 

 the measure of acceleration as in that of velocity, and they are to be 

 met in the same manner. In fact, by the acceleration is meant >the 

 rate of increase of the velocity, the velocity of the velocity. Suppose 

 the velocity, first, to increase uniformly : that is to say, let 6 feet be 

 added to it in every second, and in that proportion for all times 

 elapsed ; if then a be the initial velocity, that at the end of the time t 

 isa + lt, and we have 



dt 



if be measured from the point of starting. Here at is the length 

 due to the initial velocity o, and J& 2 the effect of the continual 

 acceleration. Now suppose, returning to the diagram, that the velocity 

 at B i greater by I than that at A, and the fraction h of a second 

 having elapsed between the two positions : that is, suppose that at A 

 the point begins to move as if it meant (continuing our illustration) to 

 describe r feet in the next second ; but that by the time of coming to 

 B it begins (from B), as if it would describe v+l feet in the next 

 second. If this increase of velocity were uniformly given, that is, if 

 in the time J A its velocity had become v + ^l, in 4/t, r+ J/, and so on 

 for every fraction of h, we might then infer that the acceleration at A, 

 that is, the rate at which velocity is thn increasing, measured by the 

 quantity which would be gained in a second at the same rate, is l-^-h : 

 for as A is to one second, so is / gained in the time A to what would be 

 pained in a second at that rate. But if this supposition be not true, 

 that is, if the speed receive unc'jual additions in equal times, we must 

 then begin to reason as before, and to find what (pursuing the same 

 illustration) we may call the intention of the velocity. If I be added 

 to the velocity in the small time A, it will be added nearly uniformly ; 

 if h be still smaller, still more nearly, and so on : in such manner that 

 while, practically speaking, f-=-A is a sufficiently good representative of 

 the current rate of acceleration, when A is small, the (uniform) rate 

 of acceleration which best represents the state of things at A is the 

 limit which is deduced by making A diminish without limit. And 

 here again, copying our own preceding words, we do not undertake to 

 say at what rate the velocity is increasing when the moving point is at 

 A, but what fictitious uniform rate of increase best represents, at the 

 instant, the variable rate of increase which would be detected if 

 the changes of velocity between A and B could be noted. And hence, 

 if ir lie called the acceleration, the student of the differential calculus 

 will eusily deduce, 



rfr <Ps dt 



ill 



dt- 



since t= -r. 



li 



and also re/r = icd>. 



Thus if the motion of the point be such that in t seconds there are 

 described (' + ( 4 feet, we have as follows : 



At the end of two seconds, then, the state of things is this : the 

 point has advanced 8 + 16 or 24 feet, and if allowed to move on 

 without further change of velocity, would describe 12 + 32 or 44 feet 

 in the next second, and hag the velocity 44 (feet) in one second; while 

 at the same time there is an acceleration taking place which would, 

 if allowed to remain uniform for one second, add 12 + 48 or 80 to the 

 velocity, making it 44 + 60 or 104 at the end of the third second. But 

 this rate of acceleration is itself increasing, since at the end of the 

 third second the velocity is 27 + 324 or 351. 



So far the subject, and all notions connected with it, fall within the 

 province of pure mathematics : if there were no such thing as either 

 matter or force, but only motion and a mind to conceive it, all that has 

 been said might be intelligible. It is very much to be regretted that 

 the connection between the mathematical doctrine of motion and the 

 laws of matter is unduly made, and at too early a stage, by the appli- 

 cation of the term accelerating force, instead of simple acceleration, to 

 the result rfr : dt. Acceleration would be what we have described it 



to be, if matter were not inert, if it moved by its own volition, or on 

 any supposition whatever, provided only that it moved. Why then 

 should a theory be made to supply the name of a result antecedent to 

 that theory, and which would be perfectly true even if that theory 

 were false ? The consequence of this is, that when the laws of matter 

 come to be applied to the mathematical expressions of motion, things 

 are taken for granted which ought to be learnt. 



The connection between these two subjects is made in the manner 

 described in the article FORCE, according to which it appears that if 

 the weight of a particle be w, and if (this weight being taken away, as 

 by laying the particle on a table without friction) a pressure be con- 

 stantly exerted upon it such as would be produced by a weight p, in 

 any direction in which it can move freely, the amount added to the 

 velocity will be uniform, at the rate of 32'19 p-j-w feet in every 

 second. Hence the following equation : 



32-19 p dv w dv 



w = dt ' or r = 32-19 dt ' 



For example, what pressure must act uniformly for one second on a 

 particle of 7 ounces weight, to add 13 feet per second to its velocity, or 

 that the rate of motion at the end of that second may be 13 feet per 

 second greater than at the commencement? Here dv : dt = 13, w = 7, 

 and the answer is 



7x 13 

 r = "jjJjTYg = 2-83 ounces. 



The numerical divisor 32-19, the uniform acceleration of bodies 

 falling free in vacuo at the earth's surface, is usually denoted by g, and 

 the factor w-=-$r usually stands for the MASS of the body, or the measure 

 of its quantity of matter. Hence the following equation : 



dv 



and this remains true, whatever unit of mass be employed, provided 

 only that the pressure which is called unity shall be that which, 

 exerted for one second upon the unit of mass, shall add a unit to the 

 velocity. [VARIATION.] And now comes another consequence of the 

 application of the term force to the simple consequence of force, acce- 

 leration. The word is wanted again to signify this pressure which 

 produces acceleration, and for distinction the pressure is called moeimj 

 force. [MOMENTUM ; MOVING FORCE.] So, then, the name of the 

 pressure which acts and produces continual accessions to the speed is 

 moving force, while the name of the rate of acceleration is accelerating 

 force. To mend this confused use of terms, some writers endeavour 

 to create a notion of moving force independent of the pressure ; but as 

 they always end in saying that the moving force varies as the pressure 

 and never tell us more of its definition than that it is the product of 

 the mass and acceleration, they might save themselves trouble, and 

 their readers also, if they would simply establish the above equation, 

 where P means the pressure which produces acceleration, and that 

 pressure is the unit of its kind when it is of that magnitude which 

 creates in the unit of mass a unit of velocity per second. 



As we are now differing from men of deservedly good authority, 

 both at home and abroad, and intending to make our assertion in a 

 more positive manner than is usual with us, we may be excused for 

 dwelling a little more upon the subject. If we consider the natural 

 meaning of moving force and accelerating force, it is obviously as 

 follows : Moving force is force which makes motion ; accelerating 

 force is force which makes acceleration, or increased motion. Were the 

 distinction ever so necessary, these words would be very bad ones, and 

 would always obstruct the learner. Nor does this origin of the word 

 moving force namely, that which produces MOMENTUM give any 

 help ; for the synonyme for momentum namely, quantity of motion, 

 meaning really quantity of matter moved multiplied by the velocity 

 is a perversion of words of the same kind. To momentum we have no 

 objection ; it is a Latin word to which an English ear may easily be 

 familiarised in any sense. If geometers had chosen to call an equd- 

 lateral triangle a momentum, the etymological student might have been 

 startled, but the shock would soon have been got over ; but if they 

 had called the same figure a quantity of motion, every beginner would 

 have been puzzled, and the impression would have been lasting. But 

 returning to the two species of forces, so called, we find a double 

 inconsistency : the idea of motion is introduced into the word which 

 only means pressure (for moving force is but pressure), while the idea 

 of pressure is introduced into a word which has only reference to 

 motion (for accelerating force is but acceleration). There are two 

 distinct and leading ideas in mechanics, pressure and motion : on 

 keeping them perfectly distinct till the time comes for joining them 

 experimentally it must depend whether a student sees mechanics to be 

 a science or not. If any one should say that pressure producing 

 motion ought to be distinguished from pressure which is in a state of 

 equilibrium with other pressures, we could not of course raise any 

 objection : let, then, moviny force and restiny force be used in these 

 two senses, with a clearly expressed distinction. Here force would be 

 synonymous with PRESSURE, in the derived or secondary sense of the 

 article cited. But let acceleration be then acceleration, not accelerating 

 force. 



The COMPOSITION of velocities and accelerations is so easily proved, 



