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question. Without supposing any such definite adjustment of the driving-power to be made, we can evidently obtain an approximation to the velocity at a particular instant, by considering (825) the motion for so short a time, that during that time the actual variation of speed may be small enough to be neglected.
28. In fact, if u be the velocity at either beginning or end, or at any instant, of an interval t, and s the space actually described in that interval; the equation v=;(which expresses the definition of the average yelocity, fe6) is more and more nearly true, as the velocity is more nearly úniform during the interval t; so that if we take the interval small enough the equation may be made as nearly exact as we choose. Thus the set of values
Space described in one second,
hundredth and so on, give nearer and nearer approximations to the velocity an the beginning of the first second.
The whole foundation of Newton's differential calculus is, in fact, contained in the simple question, "What is the rate at which the space described by a moving point increases ?' i.e. What is the velocity of the moving point? Newton's notation for the velocity, i.e. the rate at which s increases, or the fluxion of s, is s. This potation is very convenient, as it saves the introduction of a second letter.
29. The preceding definition of velocity is equally applicable whether the point move in a straight or a curved line; but, since, in the latter case, the direction of motion continually changes, the mere amount of the velocity is not sufficient completely to describe the motion, and we must have in every such case additional data to thoroughly specify the motion.
In such cases as this the method most commonly employed, whether we deal with velocities, or (as we shall do farther on) with accelerations and forces, consists in studying, not the velocity, accele, ration, or force, directly, but its resolved parts parallel to any three assumed directions at right angles to each other. Thus, for a train moving up an incline in a N.E. direction, we may have the whole velocity and the steepness of the incline given; or we may express the same ideas thus the train is moving simultaneously northward,
eastward, and upward and the motion as to amount and direction will be completely known if we know separately the northward, eastward, and upward velocities
these being called the components of the whole velocity in the three mutually perpendicular directions N Eng and up
80. A velocity in any direction may be resolved in, ana perpendicular to, any other direction. The first component is found by multiplying the velocity by the cosine of the angle between the two
directions; the second by using as factor the sine of the same angle. Thus a point moving with velocity V up an Inclined Plane, making an angle a with the horizon, has a vertical velocity V sin a and i horizontal velocity V cosas
Or it may be resolved into components in any three rectangular directions, each component being found by multiplying the whole velocity by the cosine of the angle between its direction and that of the component. The velocity resolved in any direction is the sum of the resolved parts (in that direction) of the three rectangular components of the whole velocity. And'if we consider motion in one plane, this is still true, only we have but two rectangular com. ponents.
31. These propositions are virtually equivalent to the following
OA, for instance, represents in magni-
tude and direction the space which
and equal to OB. Join OC: then OC
is the resultant velocity in magnitude and direction.
OC is evidently the diagonal of the parallelogram two of whose sides are OA, OB.
Hence the resultant of any two velocities as OA, AC, in the figure, is a velocity represented by the third side, OC, of the triangle QAÇ.
Hence if a point have, at the same time, velocities represented by OA, AC, and Co, the sides of a triangle taken in the same order, it is at rest.
Hence the resultant of velocities represented by the sides of any closed polygon whatever, whether in one plane or not, taken all in the same order, is zero.
Hence also the resultant of velocities represented by all the sides of a polygon but one, taken in order, is represented by that one taken in the opposite direction.
When there are two velocities, or three velocities, in two or in three rectangular directions, the resultant is the square root of the sum of their squares, and the cosines of its inclination to the given directions are the ratios of the components to the resultant.
82. The velocity of a point is said to be accelerated or retarded according as it increases or diminishes, but the word acceleration is generally used in either sense, on the understanding that we may regard its quantity as either positive or negative : and (8 34) is farther generalized so as to include change of direction as well as change of speed. Acceleration of velocity may, of course be either
uniform or variable. It is said to be uniform when the point receives equal increments of velocity in equal times, and is then measured by the actual increase of velocity per unit of time. If we choose as the unit of acceleration that which adds a unit of velocity per unit of time to the velocity of a point, an acceleration measured by a will add a units of velocity in unit of time-and, therefore, at units of velocity in t units of time. Hence if v be the change in the velocity during the interval +,
v=at, or a = 33. Acceleration is variable when the point's velocity does not receive equal increments in successive equal periods of time. It is then measured by the increment of velocity, which would have been generated in a unit of time had the acceleration remained throughout that unit the same as at its commencement. The average accelera. tion during any time is the whole velocity gained during that time divided by the time. In Newton's notation v is used to express the acceleration in the direction of motion; and, if v = $ as in $ 28, wehave
a=j= $." E 34. But there is another form in which acceleration may manifest itself. Even if a' point's velocity remain unchanged, yet if its direc tion of motion change, the resolved parts of its velocity in fixed directions will, in general, be accelerated.
Since acceleration is merely a change of the component velocity in a stated direction, it is evident that the laws of composition and resolution of accelerations are the same as those of velocities.
We therefore expand the definition just given, thus:-Acceleration is the rate of change of velocity whether that change take place in the 'direction of motion or not.
35. What is meant by change of velocity is evident from $ 31. For if a velocity OA become OC, its change is AC, or OB.
Hence, just as the direction of motion of a point is the tangent to its path, so the direction of acceleration of a moving point is to be found by the following construction:
From any point 0, draw lines OP, OQ, etc., repre. senting in magnitude and direction the velocity of the moving point at every instant. (Compare $ 49.) The points, P, Q, etc., must form a continuous curve, for 18 7) OP cannot change abruptly in direction. Now if ( be a point near to P, OP and OQ represent two successive values of the velocity. Hence PQ is the whole change of velocity during the interval. As the 0 interval becomes smaller, the direction PQ more and more nearly becomes the tangent at P. Hence the direction of acceleration is that of the tangent to the curve thus described.
The magnitude of the acceleration is the rate of change of velocity, and is therefore measured by the velocity of P in the curve PQ.
> 86. Let a point describe a circle, ABD, radius R, with uniform velocity V. Then, to determine the direction of acceleration, we must draw, as below, from a fixed point. O, lines OP, OQ, etc, representing the velocity at A, B, etc., in direction and magnitude. Since the velocity in ABD is constant, all the lines OP, OC, etc.,
will be equal (to V), and there fore POS is a circle whose centre is O. The direction of
acceleration at A is parallel to S
the tangent at P, that is, is perpendicular to OP, i.e. to AQ, and is therefore that of the radius AC.
Now P describes the circle D
POS, while A describes ABD.
Hence the velocity of P is to that of A as OP to CA, i.e. as V to R; and is therefore equal to
Ř R' and this ($ 35) is the amount of the acceleration in the circular path ABD. 37. The whole acceleration in any direction is the
sum of the components in that direction) of the accelerations parallel to any three rectangular axes-each component acceleration being found by the same rule as component velocities (8 34), that is, by multiplying by the cosine of the angle between the direction of the accelera. tion and the line along which it is to be resolved.
38. When a point moves in a curve the whole acceleration may be resolved into two parts, one in the direction of the motion and equal to the acceleration of the velocity; the other towards the centre of curvature (perpendicular therefore to the direction of mo. tion), whose magnitude is proportional to the square of the velocity
and also to the curvature of the path. The former of these changes speed the velocity, the other affects only the form of the path, or the
direction of motion. Hence if a moving point be subject to an acceleration, constant or not, whose direction is continually perpen. dicular to the direction of motion, the velocity will not be altered and the only effect of the acceleration will be to make the point move in a curve whose curvature is proportional to the acceleration at each instant, and inversely as the square of the velocity.
89. In other words, if a point move in a curve, whether with a ad 야 ñ uniform or a varying velocity, its change of direction is to be re
garded as constituting an acceleration towards the centre of curva. ture, equal in amount to the square of the velocity divided by the radius of curvature. The whole acceleration will, in every case, be the resultant of the acceleration thus measuring change of direction, and the acceleration of actual velocity along the curve.
40. If for any case of motion of a point we have given the whole velocity and its direction, or simply the components of the velocity in three rectangular directions, at any time, or, as is most commonly the case, for any position, the determination of the form of the path described, and of other circumstances of the motion, is a question of pure mathematics, and in all cases is capable (if not of an exact solution, at all events) of a solution to any degree of approximation that may
be desired. This is true also if the total acceleration and its direction at every instant, or simply its rectangular components, be given, provided the velocity and its direction, as well as the position of the point, at any one instant be given. But these are, in general, questions requiring for their solution a knowledge of the integral calculus.
41. From the principles already laid down, a great many interest. ing results may be deduced, of which we enunciate a few of the simpler and more important.
(a) If the velocity of a moving point be uniform, and if its direction revolve uniformly in a plane, the path described is a circle. 7 (6) If a point moves in a plane, and its component velocity parallel to each of two rectangular axes is proportional to its distance from that axis, the path is an ellipse or hyperbola whose principal diameters coincide with those axes; and the acceleration is directed to or from the centre of the curve at every
instant (S$ 66, 78).
() If the components of the velocity parallel to each axis be equimultiples of the distances from the other axis, the path is a straight line passing through the origin.
(d) When the velocity is uniform, but in a direction revolving uniformly in a right circular cone, the motion of the point is in a circular helix whose axis is parallel to that of the cone.
42. When a point moves uniformly in a circle of radius R, with velocity V, the whole acceleration is directed towards the centre, and has the constant value See $ 36.
R → 43. With uniform acceleration in the direction of motion, a point describes spaces proportional to the squares of the times elapsed
since the commencement of the motion. This is the case of a body falling vertically in vacuo under the action of gravity.
Is this case the space described in any interval is that which would be described in the same time by a point moving uniformly with a velocity equal to that at the middle of the interval. In other words, the average velocity (when the acceleration is uniform) is, during any interval, the arithmetical mean of the initial and final velocities. For, since the velocity increases uniformly, its value at any time before the middle of the interval .is as much less than this mean as its value at the same time after the middle of the interval is greater than the