2t 12 4s Ex. 1. How far will a body descend in 3 seconds by the force of gravity. Because s=gte, and in this case t=3" we taves=161'* x3!=1449 feet, the distance required. 2. Through what space inust a body fall from rest to acquire a velocity of 20 feet per second. 202 Hence we have s= =6.21 feet 4g 4x16'5 3. In what time will a body fall 90 feet, and what velocity will be be acquired. 90 First = -=N -=2.36 seconds nearly. 16ta Again v=2wgs=27(164 x 90)=76 feet per second nearly. Again, because the times are as the velocities, and the spaces as the squares of either; therefore If the times be as the puinbers 1, 2, 3, 4, 5, &c. And the spaces for each time as 1, 3, 5, 7, 9; namely, as the series of the odd numbers, which are the differences of the squares denoting the whole spaces. So that if the first series of natural numbers be seconds of time, Time in Seconds. 4", space for each second 1619, 481, 80's, 11215. 42. And because the laws for the destruction of motion are the same as those for its generation, by equal forces, but acting in a contrary direction; therefore, Ist. A body thrown directly upwards, with any velocity, will lose equal velocities in equal times. 2d. If a body be projected upwards, with the velocity it acquired in any time by descending freely, it will lose all its velocity in an equal time; and will ascend just to the same height from which it fell, and will describe equal spaces in equal times, in rising and fall. ing, but in an inverse order, and it will have equal velocities at any one and the same point of the line described, both in ascending and descending 43. If a body begin to move in the direction of gravity with any velocity, the whole space described in any time is equal to the space through which the first velocity would carry the body, together with the space through wbich it would fall by the force of gravity in that time. Er. If a body be projected perpendicularly downwards, with a velocity of 20 feet per second, to find the space described in 4 seconds. The space described in 4" with the first velocity is 4 x 20, or 80 feet; and the space falleu through in 4" by the action of gravity, is 1616 x 16 = 257; feet; therefore, the whole space described is 80+ 257=337$ feet. 44. If a body be projected perpendicularly upwards, the height to which it will ascend in any time is equal to the space through which it would move with the first velocity continued uniform, dininished by the space through which it would fall by the action of gravity in that time Ex. To what height will a body rise in 3" if projected perpendicularly upwards with a velocity of 100 feet per second ? The space which a body would describe in 3" with the first velocity, is 300 feet ; and the space through which the body would fall by the force of gravity in 3'', is 1619x9=1444 feet; therefore, the height required is 300-1441=1551 feet. Examples on Motions Uniformly varied. 1. What is the difference between the depths of two wells, into each of which, should a stone be dropped at the same instant, the one would strike the bottom at the end of 6, and the other at the end of 10 seconds ? Ans. 10293 feet. 2. Supposing Salisbury steeple to be 400 feet high, in what tine would a niusket-ball, let fall from the top of it, reach the ground ? Ans. in 5 seconds ncarly. 3. A heavy body was observed to fall 100 feet in the last second of time; from what height was it let fall, and how long was it in motion ? Ans. 342 seconds. 4. A stone being let fall into a well, it was observed that, after being dropped, it was 10 seconds before the sound of the fall at the bottom reached the ear ; what is the depth of the well ? Ans. 1270 feet nearly. 5. A drop of rain, in its descent towards the earth, was observed to fall through a space of 400 feet in the last two seconds; from what height did it fall : Ans. 837,83 feet. 6. If a body be projected perpendicularly upwards with a velocity of 80 fett per second, required its place at the end of 6 seconds ? Ans. 99 feet below the point from which it was projected. OF THE COMPOSITION AND RESOLUTION OF FORCES. 45. Composition of Forces, is the uniting of two or more forces into one, which shall have the same effect; or, it is the finding of one force that shall be equal to several others, taken together in different directions. Resolution of Forces, is the finding of two or more forces which, acting in any different directions, shall have the same effect as any given single force. 46. Two lines which represent the momenta communicated to the same or to equal bodies, will represent the spaces uniformly described by them in equal times ; and conversely, the lines which represent the spaces uniformly described by them in equal times, will also represent the momen.ta. The momenta of bodies may be represented by numbeis, thus, if the quantities of matter in two bodies be represented by 6 and 7, and their velocities by 9 and 8, their momenta will be represented by 6x9 and 7 x 8. But in many cases it will be much more convenient 10 represent momenta by lines, because lines will not only express the quantities of the momenta, but also the directions in which they are communicated. Any line, drawn in the proper direction, may be taken to represent one momentum, but to represent a second, a line in the direction of the latter motion must be taken in the same proportion to the former, that the second momentum has to the first. Let two lines thus taken, represent the momenta communicated to the same, or equal bodies; then, since (22) malv, and 6 is here given, mav; therefore, the lines which represent the momenta, will bs also represent the velocities; and since ma 7 (27), and in this case, b and t are constant a ms, and these lines will also represent the spaces uniformly described in equal times. Again, if the lines represent the spaces uniformly described in equal times, they represent the velocities, because sa vt and t is constant ; and since mabv, and 6 is constant mav, but we have shown that sav, therefore also sam, and these lines also represent the momenta. 47. Two uniform motions, which, when communicated separately to a body, would cause it to describe the adjacent sides of a parallelogram in a given time, will, when they are communicated at the same instant, cause it to describe the diagonal in that time ; and the motion in the diagonal will be uniform. Let a motion be communicated to a body at A, fig. 3, which would cause it to move uniformly from A to B in t", and at the same instant another motion, which alone would cause it to move uniforrily from A to C also in t"; complete the parallelogram BC, and draw the diagonal AD; then the body will arrive at the point D in t", having described AD with an uniform motion. For the motion in the direction AC can neither accelerate nor retard the approach of the body to the line BD, which is parallel to AC, hence the body will arrive at BD in the same time that it would a have done, had no motion been communicated to it in the direction AC, that is, in t". In the same manner, the motion in the direction AB can neither make the body approach to, nor recede from CD; therefore, in consequence of the motion in the direction AC, it will arrive at CD at the same time that it would have done, had no motion been communicated in the direction AB, that is, in t". Hence it follows, that in consequence of the two motions, the body will be found both in BD and CD at the end of t", and it will therefore be found in D, the point of their intersection. And since a body in motion continues to move uniformly forwara in a right line, till it is acted upon by some external force (18), the body A must have described the right line AD with an unitorin motion. 48. Cor. 1. If two sides of a triangle AB, BD, fig. 3, taken in order, represent the spaces over which two uniform motions would separately carry a body in a given time; when these motions are communicated at the same insiant to the body at A, it will describe the third side AD uniforinly in that time. For if the parallelogram BC be completed, the same motion, which wonld carry a body uniformly from B to D, would, if conmunicated at A, carry it in the same manner from A to C; and, in conseqnence of this motion, and of the motion in the direction AB, the body would uniformiy describe the diagonal AD, which is the third side of the triangle ABD. 49. Cor. 2. In the same manner, if the lines AB, BC, CD, DE, fig. 4, taken in order, represent the spaces over which any uniform motions would, separately, carry a, body, in a given time, these notions, when communicated at the sanie instant, will cause the body to describe the line AE which completes the figure in that time ; and the inotion in this line will be uniform. 50. Cor. 3. If AD, fig. 5, represent the uniform velocity of a body, and any paralielogram ABDC be described about it, the velocity AD may be supposed to arise from the two uniform velocities AB, AC, or AB, BD, and if by any means one of them, as AB, be taken away, the velocity remaining will be represented by AC or BD. 51. If the adjacent sides of a parallelogram represent the quanti ies and directions of two forces acting at the same time upon a lody, the diagonal will represent one equivalent to them both. Let AB, AC, fig. 6, represent two forces acting upon a body at A, then they represent the momenta communicated to it in those directions (21), and, consequently, the spaces which it would uniformly describe in equal times (46). Complete the parallelogram CB, and draw the diagonal AD; then, by the last article, AD is ihe space uniformly described in the same time when the two motions are communicated to the body at the same instant ; and since AB, AC, and AD, represent the spaces uniformly described by the same body in equal times, they also represent the momenta, and therefore the forces acting in those directions ; that is, the forces AB, AC, acting at the same time, produce a force which is represented in quantity and direction by AD. The force represented by AD is said to be compounded of the two, AB, AC. 52. Cor. 1. If two sides of a triangle, tuken in order, represent the quantities and directions of tuo forces, ihe third side will represent a force equivalent to the both. For a force represented by BD, acting at A, will produce the same effect that the force AC will produce, which is equal to it, and in the same direc . tion; and AB, AC, are equivalent to AD; therefore AB, BD, are also equi. valent to AD 53. Cor. 4. If any lines AB, AC, CD, DE, fig. 4, tuken in order, represent the quantities and directions of forces conimunicated at the same time to a body at A, the line AE, which completes the figure, will represent a furce equivalent to them all. For the two AB, BC, are eqnivalent to AC; also AC, CD, that is, AB, BC, CD, are equivalent to AD; in the same manner AD, DE, that is, AB, BC, CD, and DE, are equivalent to AE. 54. Cor. 3. Let AB and AC, ng. 6, represent the quantities and directions of two forces, as before, theu the resulting force may be expressed analytically in terms of AB, AC, and the included angle BAC. Because of the parallels AB, DC, and BD, AC, the BDA= DAC, and, conseqnently, < BAC = <BADÍ <BDA. Again, the angle ABD=180°– (Z BAD + 2BDA) = 180°- BAC — supplement of BAC, by what was just shown. But, by Trigonometry (art. 43), it appears that and therefore by transposition, AD' ZAB.BD AB*+BD! — 2AB BD cos. ABD; but cos. ABD = = cos. (180°_BAC) = cos. BAC, and BD2AC, therefore AD=NAB?+ AC + 2AB.AC cos.BAC. If we call AB, ; AC, b; and the angle BAC, A, then the resultant NA+6+ 2ab cos. A. The sine of the augle which this diagonal makes with AB, may be found by the proportion AD: BD :: sin. B : siu. BAD; or b sin. A w(a® +62+ 2ab cos. A):6 :: sin. A : N(*+63 + zub cos. A) .54. Cor. 4. Two given forces produce the greatest effect when they act in the same direction, and the least when they act in opposite directions ; for, in the forner case, the diagonal AD becomes equal to the sum of the sides AB, BC; and, in the latter, to their difference. 55. Cor. 5. Two forces cannot keep a body at rest unless they are equal and in opposite directions. For this is the only case in which the diagonal, representing the compound forces, vanishes. 56. Cor. 6. In the composition of forces, force ie lost; for the forces represented by the two sides AB, BD, by composition, produce the force represented by AD; and the two sides AB, BD, of a triangle, are, together, greater than the third side. 57. If a body at rest, be acted upon at the same time by three forces, which are represented in quantity and direction by the three sides of a triangle, taken in order, it will remain at test. Let AB, BC, and CA, fig. 7, represent the quantities and directions of three forces acting at the same time upon a body at A ; then, since AB and BC are equivalent to AC (53); AB, BC, and CA, are equivalent to AC and CA ; but AC and CA, wbich are equal and in opposite directions, keep the body at rest; therefore AB, BC, and CA, will also keep the body at rest. And, conversely, 58. If a body be kept at rest by three forces, and two of them be represented in quantity and direction by two sides of e triangle AB, BC; the third side, taken in order, will represent the quantity and direction of the other force. Y y |