Variable relations, that in motions uniformly accelerated from a Motions. state of rest, the acquired velocities are proportional to the times from the beginning of the motion. For a e, bf, c.g, dh, represent the velocities gained during the times va, v b, vc, vd, and are in the same proportion with those lines.

37.-1. Also, the momentary increments of velocity, are as the moments in which they are acquired.

2. Also, the spaces described from the beginning of the motion, are as the squares of the times.

3. Also, the increments of the spaces are as the increments of the squares of the times; reckoning from the beginning of the motion.

4. Also, the spaces described from the beginning of the motion, are as the squares of the acquired velo

cities.

5. Also, the momentary increments of the spaces are as the momentary increments of the squares of the velo

eities.

6. Also, the space described during any portion of time by a motion uniformly accelerated from rest, is one half of the space uniformly described in the same time with the final velocity of the accelerated motion.

7. And the space described during any portion of the time of the accelerated motion, is equal to that which would be described in the same time with the mean be tween the velocities at the beginning and end of this portion of time.

In the investigation of all other varied motions, the properties of uniformly accelerated motion stated above, will be found extremely useful, and especially in cases where approximation only can be easily obtained. But for the fuller illustration of these properties the reader is referred to Robison's Elements of Mechanical Philosophy, p. 38.

38. Supposing the acceleration to be always the same, we conceive of this constancy, that in equal times there are equal increments of velocity; and therefore that the augmentations of velocity are proportional to the times in which they are required. That acceleration then, according to this supposition, must be accounted double, or triple, &c. where the velocity acquired is double or triple. And, acceleration being considered as a measurable quantity, the augmentation of velocity uniformly acquired in any given time is its

measure.

COROLLARY.

39. Therefore accelerations are proportional to the spaces described in equal times, with motions uniformly accelerated from a state of rest. For in this case the spaces are the halves of what would be uniformly described in the same time with the acquired final velocities, and are therefore proportional to these velocities, or to the accelerations, since the velocities were acquired in equal times.

40. It is then said that accelerations are proportional to the increments of velocity uniformly acquired, directly, and to the times in which they are acquired, inversely.

V v

A: a=t This relation between acceleration, velocity, and time, is also true, in uniformly accelerated motion, with respect to all momentary changes of velocity, as well as

to those cases of motion passing through all degrees of Variable velocity from nothing to the final magnitude v. For Mutions, the velocity increasing at the same rate with the time, we have v v't: t'; and v′ and t'express the simultaneous increments of velocity and time.

41. But if the augmentation of velocity be the measure of the acceleration, and therefore proportional to it, and if in uniformly accelerated motions, the velocity increases at the same rate with the times, the increments of velocity are as the accelerations and as the times jointly. Hence the proportional equation

vat,

and va t'.

42. It appears from (39.), that when the velocity has uniformly increased from nothing, the spaces described in equal times are proper measures of acceleration. And in (37-3.) uniformly accelerated motions, the spaces are as the squares of the times. Therefore, when the acceleration continues the same, the fraction

品 must also remain of the same value, and a

บ

vv

vt

43. And since a, we have a; but vts, therefore a Therefore we have another measure of acceleration, viz. Accelerations are directly as the squares of the velocities, and inversely as the spaces along which the velocities are uniformly aug. mented.

44. But when the spaces are equal, we have av3, and in uniformly accelerated motions, that is, when a remains constant, the space being increased in any proportion, increases in the same proportion; it follows that increases in the proportion both of the aeceleration and of the space. And therefore, in general, we have vas. And, as in 41, 42, we shall have a S, and V-va Sas, or S-s, which may be thus expressed, vva s', that is, in a motion uniformly accelerated, the momentary change of the square of the velocity is proportional to the accelera tion and to the space jointly. Thus it appears, that the acceleration continued during a given time t, or t, produces a certain augmentation of the simple velocity; but the acceleration continued along a given space or S', produces a certain augmentation of the square of the velocity.

45. But accelerations which are constant and uniform, and such as have been considered, are very rare in the phenomena of nature. They are as variable as velocities, and therefore it is not less difficult to discover their actual measure. By changes of velocity_only we obtain any knowledge of the changing cause. From the continual acceleration of a falling body we learn, that the same power which makes it press on the hand, presses it downward, as it falls through the air; and whatever be the rapidity of its descent, it is from observing that it acquires equal increments of velocity in equal times, that we know the downward pressure to be the same.

Variable In the same way that we obtain measures of a veloMotions. city which is continually varying, we may obtain accurate measures of a similarly varying acceleration. A line may be conceived to increase along with the velocity, and at the same rate; and this rate of increase of velocity is what is called acceleration, in the same way as the rate at which the line increases, is what is called velocity. If, then, we consider the areas (fig. 5.) or the line AD, as representing a velocity; the ordinates to the line eg h, which were demonstrated to be proportional to the rate of variation of the area, will be proportional to the variation of the velocity, that is, to the acceleration.

Fig. 8.

PROP. VII.

46. If the abscissa a d of a curve line e gh represent the time of a motion, and if the areas abfe, acge, adhe, &c. are proportioned to the velocities at the instants b, c, d, &c. then the ordinates a e, bf, cg, dh, &c. are proportional ta the acceleration at the instants a, b, c, d, &c.

By substituting the word acceleration for the word velocity, the same demonstration may be applied here as in Prop. 6. (28.). From this proposition may be deduced soine corollaries of practical use in mechanical

discussions.

47. The momentary increments of velocity are as the accelerations, and as the moments jointly.

For the increment of velocity in the moment c d is accurately represented by the area cd hg, or by the rectangle c d n k; and c d accurately represents the moment. Also, the ultimate ratio of c k to such another ordinate bi, is the ratio of cg to bf; that is, the ratio of the acceleration in the instant a to that in the instant b. And therefore the increment of velocity during the moment pa is to that during the moment c d as paxa e to cdxdg. Or it may be expressed by the proportional equation vat.

48. Conversely. The acceleration a is proportional

celerated (40.).

And as the area of this figure is analogous to the sum of all the inscribed rectangles, when the circumstances of the case admit of its being measured, it may be expressed by fai; and thus is obtained the whole velocity acquired during the time AC, and we say v fai.

The intensities (or at least their proportions) of the accelerating power of nature in the differents points of the path being frequently known, we wish to discover the velocities in those points. This may be done by the following proposition.

PROP. VIII.

49. If the abscissa AE (fig. 8.) of a line a ce be the space along which a body moves with a motion continually varied, and if the ordinates A a, Bb, Cc, &c. are proportional to the accelerations in the points A, B, C, &c. then the areas AB ba, AD da, AE ea, &c. are proportional to the augmentations of the square of the velocity in A at the points B, D, E, &c.

Take BC, CD, as two very small portions of the line Compound AE, and draw bf, cg, parallel to AE. Then, sup- Motions. posing the acceleration B b, to continue through the space BC, the rectangle BbfC will express the augmentation made on the square of the velocity in B. În the same way Ccg D will express the augmentation of the square of the velocity in C; and, in like manner, the rectangles inscribed in the remainder of the figure will express the increments of the squares of the velocity acquired, while the body moves over the corresponding portions of the abscissa. And, therefore, the whole augmentation of the square of the velocity in A (should there be any velocity in that point) during the time of moving from A to B, will constitute the aggregate of these partial increments. The same thing must be affirmed of the motion from B to E. And, when the subdivision of AE is carried on without end, it is plain that the ultimate ratio of the area AE e a to the aggregate of inscribed rectangles, is that of equality; that is, when the acceleration varies continually, the area AB ba will express the increment made on the square of the initial velocity in A, while the body moves along AB; and the same must be affirmed with respect to the motion along BE. And, therefore, the intercepted areas AB ba, BD db, DE e d, are proportional to the changes made on the squares of the velocities in the points A, B, and D.

COROLLARIES.

50. COR. 1. If the body had no velocity in A, the areas AB ba, AD da, &c. are proportional to the squares of the velocity acquired in the points B, D, &c.

Cor. 2. The momentary change on the square of the velocity, is as the acceleration and increment of the space jointly; or we have v vas.

Cor. 3. vv being equal to half the increment of the square of the velocity, it follows that the area AE ea, or the fluent fa's is only equal to

v and V as the velocities in A and E.

2

taking

51. What has now been said of the acceleration of motion, is equally applicable to motions that are retarded, whether these motions be uniform or unequable. The momentary variations in this case are to be taken as decrements of velocity instead of increments. A moving body, subject to uniform retardation till it come to rest, will continue in motion during a time proportional to the initial velocity; and describe a space proportional to the square of this velocity; and the space which is so described, is one half what it would have been if the initial velocity had continued undiminished.

Compound former. What is taken for the beginning of motion in Motions every observation of surrounding bodies, is nothing more than a change induced on a motion already produced.

How to as

56. The actual composition of motion being so general in the phenomena of the universe, it obtains in all motions and changes of motion produced or observed, and the characteristic which has been assumed of a change of motion being the same, whatever may have been the previous motion, and this being equally applicable to simple motions, it is evident that a knowledge of the general results of this composition of motion will be of essential service in acquiring a knowledge of mechanical nature.

57. The following is the general theorem to which all others may be reduced.

PROP. IX.

Two uniform motions, having the directions and velocities represented by the sides AB, AC, of a parallelogram, compose a uniform motion in the diagonal. The demonstration of this has been already given. The motion of the point of intersection of these two lines, each moving uniformly in all its points, in the direction of the other, is, in every instant, composed of the two motions. It is the same as if a point described AB uniformly, while AB is carried uniformly in the direction AC. This motion is along the diagonal A b, and it has been already shewn to be uniform. And, because AB and A b are described in the same time, the velocities of the motions along AB, AC, and A b are proportional to those lines.

COROLLARIES.

Cor. 1. The motion A b, which is compounded of the two simple motions AB and AC, is in the same plane with these motions. For a parallelogram lies all in the same plane.

Cor. 2. The motion A b may be produced by the composition of any twe uniform motions having the direction and velocities which are represented by the sides of any parallelogram, AF b G, or AC b B, which has Ab for its diagonal.

58. Cases are not unfrequent in which the directions certain the of two simple motions composing an observed motion proportion may be discovered; but the proportion of the velocithe ve ties is unknown. This velocity may be ascertained by locity in' means of this last proposition. For the direction of the compound motions. three motions, namely the two simple and the compound motions, determines not only the species of parallelogram, but also the ratio of the sides. Again, in those cases in which the direction and the velocity of one of the simple motions are known, and therefore its proportion to that of the observed compound motion, the direction and velocity of the other may be also found by means of the same proposition; because from these data the parallelogram may be determined.

59. This motion in the diagonal is called the equivalent motion, or the resulting motion; for it is equivalent to the combined motions in the sides. Thus, if the moving body first describe AB, and then Bb or AC, it will be in the same point, as if it had deseribed A b, namely, in the point b.

60. It is often highly useful in investigations of this kind to substitute such motions for an observed motion,

as will produce it by composition. This has been de- Compound nominated the resolution of motions. By this manner Motions. of proceeding, a ship's change of situation at the end of a day, having sailed in different courses, is computed. Thus the distance sailed to the eastward or the westward, as well as that to the northward or southward, on each course, is observed and marked. The whole of the eastings, and the whole of the southings, are added together; and then it is supposed that the ship has sailed for the whole day on that course, which would be produced by combining the same easting and southing.

61. It is also useful to consider how much the body has been advanced in a certain direction by means of the observed motion; let us suppose in the direction AB (fig. 10.). The motion CD is first considered as Fig. 10. composed of a motion CE parallel to the given line AB, and another motion CF perpendicular to AB. CD is the diagonal of a parallelogram CEDF, one of whose sides CE is parallel to AB, and the other CF is perpendicular to AB. It is evident, that the body has advanced in the direction of AB as much as if it had moved from G to H, instead of moving from C to D, so that the motion CF has no effect either in obstructing or promoting the progress in AB. This is called estimating a motion in a given direction, or reducing it to that direction.

62. A motion is also said to be estimated in a given plane, when it is considered as composed of a motion perpendicular to the plane, and of another parallel to it. In a given plane ABCD (fig. 11.), let EF be a Fig. 11. motion compounded of a motion GE perpendicular to the plane, and EH parallel to it. For if the lines GE, FH are drawn perpendicular to the plane, they cut it in two points e and f, and EH is parallel to e f.

63. In the same way a compound motion may be formed of any number of motions. Let AB, AC, AD, AE, &c. (fig. 12.) be any number of motions, of which Fig. 12. the motion AF is compounded. The motion which is the result of this composition is thus ascertained. The motion AG is compounded of AB and AC; and the motion AG compounded with AD, gives the motion AH; which latter being compounded with AE, produces the motion AF. And the same place, or final situation F, will be found by supposing the different motions AB, AC, AD, AE, to be performed successively. The moving body first describes AB; then BG, equal and parallel to AC; then GII, equal and parallel to AD; and lastly, HF, equal and parallel to AE. In this case it is not requisite that all the motions lie in the same plane.

64. Three motions which have the direction and proportions of the sides of a parallelopiped, compose a motion having the direction of its diagonal. Let AB, AC, AD (fig. 13.), be these motions, the compound-Fig. 13. ed motion is in the diagonal AF of the parallelopiped ; because AB and AC compose the motion AE; and AE and AD compose the motion AF.

It is in this way that the mine-surveyor proceeds. He sets down a gallery of a mine, not directly by its real position, but marks the easting and westing, the northing and southing, as well as its dip and rise. All these measures are referred to three lines, of which one runs east and west, one north and south, and a third is perpendicular. These three lines are obviously analo

gous

Compound gous to the angular boundaries of a rectangular box, as Motions. AC, AB, AD.

Other compound motions.

Fig. 14.

Condition

pound mo

tions.

65. The composition of uniform motions only has yet been considered. But it is easy to conceive that any motions may be compounded. It is a case of this kind when a man is supposed to walk on a field of ice along a crooked path, while the ice floats down a crooked stream. Suppose a uniform motion in the direction AB (fig. 14.), to be compounded with a uni. formly accelerated motion in the direction AC. A stone falling from the mast head of a ship, while she sails uniformly forward in the direction AB, affords an example of this kind of motion; for the stone will be observed to fall parallel to a plummet hung from the mast head. But the real motion of the stone is a parabolic arch Abfg, which AB touches in A; for while the mast head describes the equal lines AB, BF, FG, the stone has fallen to ẞ and and y, and the line AC is in the positions BB', FF', GG', so that Ap is four times Aẞ; and Ay is nine times A s. Therefore AB, A, Ay, are as the squares of 3 b, of, vg, and the line A bƒg is a parabola.

66. Knowing the direction and velocities of each of of com- the simple motions in any instant, of which two motions disco- tions, however variable, are compounded, we may disvered from cover the direction and velocities of the compound that of the motions in that instant. For it may be supposed that simple mo- each motion at that instant proceeds unchanged: a parallelogram is then constructed; the sides of which have the direction and proportions of the velocities of the simple motions; and the diagonal of this parallelo. gram will express the direction and velocity of the compound motion. But, on the other hand, if the direction and velocity of the compound motion, with the directions of each of the simple motions, be known, we may discover their velocities.

Fig. 15.

Danger of mistakes about changes of motion.

Plate CLXXXV. fig. 16.

67. In cases where a curvilineal motion, as ABC (fig. 15.), is the result of two motions compounded, of which the direction is known to be AD and AE, we discover the velocities of the three motions in any point B, by drawing the tangent BF, and the ordinate BG, parallel to one of the simple motions, and from any point H in that ordinate drawing HF parallel to the other motion, and cutting the tangent in the point F. The three velocities are in the proportion of the three lines FH, HB, and FB.

68. As the motions which are observed in nature are very different from what they are taken to be, it is not easy to avoid mistakes with respect to the changes of motion, and consequently with respect to the inference of its cause. Without considering the real motion of any body, we are apt to judge only of the changes of distance and direction in relation to ourselves. Thus it is that our inferences with regard to the planetary motions are very different from the motions themselves, if the rapid motion of our earth be considered.

PROP. X.

69. The motion of one body in relation to another body, or as it is seen from another body, which is also in motion, is compounded of its own real motion, and the opposite of the real motion of the second body.

Let A (fig. 16.) be a body in motion from A to C, as seen from B, which is another body in motion from

B to D the motion of A is compounded of its own real Motions motion, and of the opposite to the real motion of B. continually Join AB, and draw AE equal and parallel to BD. deflected. Complete the parallelogram ACFE, and join ED and DC. Produce EA, and make AL equal to AE or BD. Complete the parallelogram LACK, and draw AK and BK. If then A had moved along AE while B moves along BD, the two bodies would have been at E and D, at the same time, and would have the same relative situation; they would have the same bearing and distance as before. And if the spectator in B is not sensible of his own motion, A will appear not to have changed its place. In the same way two ships becalmed in an unknown current, seem to the persons on board to be at rest. The real position, therefore, and distance DC, are the same with BK; and if a spectator in B imagines himself at rest, the line AK will be taken as the motion of A. And this motion, it is obvious, is composed of the motion AC its real motion, and the motion AL which is the equal and opposite motion to that of BD.

Again, if BH be drawn equal and opposite to AC, and the parallelogram BHGD be completed, and BG and AG be drawn, the diagonal BG will be the motion of B as it is seen from A. Now as KAGB is a parallelogram, the relative situation and distances of A and B at the end of the motion will appear to be the same as in the former case. For B appears to have moved along BG, which is equal and opposite to AK. Hence it follows that the apparent or relative motions of two bodies are equal and opposite, whatever their real motions may be; and therefore they do not afford any information of their real motions.

70. Suppose equal and parallel motions are compounded with all and each of the motions of any number of bodies, moving in any manner of way, then their relative motions are not consequently changed. For if it be compounded with the motion of any one of the bodies which may be called A, the real motion of this body is changed; but its apparent motion, as seen from another body B, is compounded of the real change, and of the opposite to the real change in A, which therefore destroys that change, and the relative motion of A is the same as before. Thus it is that the motions in the cabin of a ship are not affected by the ship's progressive motion; and the motion of the earth round the sun produces no perceptible effect on the relative motions on its surface. And indeed it is only by observing other bodies which are not affected by these common motions, and to which we refer as to fixed points, that we arrive at any knowledge of