ON THE LAWS OF MOTION. 18. Law I. Any body at rest will continue at rest ; and if it be in motion, it will continue to move uniformly forward in a right line, till it is acted upon by some external force. 19. LAW II. Motion, or change of motion, is always proportional to the force impressed, and takes place in the direction in which the force acts. 20. Law III. Action and re-action, letween any two lodies, are equal and contrary. That is, by action and re-action, equal changes of motion are produced in bodies acting on each other; and these changes are directed towards opposite or contrary parts. Scholium. These laws are the simplest principles to which motion can be reduced, and upon thein the whole thicory depends. They are not, indeed, self-evident; but they are, however, constantly and invariably suggested to our senses, and they agree with experiment as far as experiment can go ; and the more accurately the experiments are made, and the greater care we take to renove all those impedimients which tend to render the conclusions erroneous, the more nearly do the experiments coincide with these laus. OF UNIFORM MOTIONS. 21. The momentum, or quantity of motion, generated ly a single impulse, or by the action of any momentary force, is as the generating force. For every effect is proportional to its adequate cause. So that a double force will impress a double quantity of motion ; a triple force, a triple motion; and so on. Hence, calling m the moinentum, or quantity of motion, and f the force, we shall have ma f. 22. The quantities of motion, or momenta, in moving bodies, are in the compound ratio of the masses and velocities. For, since the motion of any body is made up of the motions of all its parts, if the velocities of the bodies be equal, the momenta will be as the masses ; for a double mass will strike with a double force, a triple mass with a triple force, and so on. Again, if the masses of any two bodies be supposed the same, it will require a double force to move either body with a double velocity, a triple force with a triple velocity, and so on; that is, the motive force is as the velocity ; but, by the preceding article, the momentum impressed is as the force which produced it; and, therefore, the momentum is as the velocity when the mass is the same. But the momentum was found to be as the mass when the velocity was supposed constant; and, therefore, when neither are the same, the momentum is in the ratio compounded of both mass and velocity. Or, calling 6 the mass or body, v the velocity, and m the momentum, as before, mabu and this expression is true, independently of the nature of the force by which the velocity is generated. 23. In uniform motions, the spaces described are in the compound ratio of the velocities and the times of their description. For, by the nature of uniform motion, the greater the velocity, che greater is the space described in any given portion of time ; that is, the space is proportional to the velocity. And when the velocity is the same, or constant, the space will be as the time ; that is, in a double time a double space will be described ; in a triple time a triple space ; and so on. Hence, when both the time and velocity vary, the space will be in the ratio compounded of both. Or, calling s the space, t the time of description, and v the velocity, as before, we shall have satu. 24. Cur, 1.t 10,0 or the time is as the space directly, and velocity recipro cally; or as the space divided by the velocity. Hence, if we suppose the velocity to be the same in uniform motions, the time will be as the space; bat if we suppose the space to be the same, the time will be as or reciprocally as the velocity. 25. Cor. 2. o a, that is, the velocity is as the space directly, and the time reciprocally, or as the space divided by the time. Therefore, when the time is the same, the velocity is as the space; but when the space is the same, the velocity is reciprocally as the time. 26. Cor. 3. The space is not only proportional to to, but it is also equal to it. If v represents the rate of motion, or space passed over in a given time, as one second, then tv will evidenily represent the space passed over in econds, and therefore setv. Consequently or and t= Scholium. 27. In uniform motions, or such as are generated by momentary impulse, let b = any body, or quantity of matter to be moved. 6 = the time of describing the space s with the velocity : Then, from the last three articles and corollaries we have these three general proportions. fam, mabv, and satv. combining these, we easily derive the following table of the general relations of those six quantities. Table for uniform Motions and Impulsive Forces. bs famalva 8 8 m be bs τα 2α of By means of these formula, we are enabled to resolve all questions relating to uniform motions, and the effects of momentary or impulsive forces. Examples of Uniform Motions. Let V and v be the velocities of two bodies moving with different uniform motions ; T and t the times they have been in motion ; S and s the spaces described in those times. Then, because (23) sotv, we shall always have S:s :: VT : vt. and the same with any other relation in the table. Ex. 1. Let A and B nove uniformly, and suppose the times they have been in motion to be as 6 : 5, and their velocities as 2:3; required the ratio of the spaces described. Here S:s :: VT:vi:: 6x2:5X3 :: 12:15 :: 4:5; that is, the spaces are as 4 to 5. Ex. 2. Let A move through 5 feet in 3 seconds, anil B through 9 feet in 7 seconds; to find the ratio of their velocities. By 25, vac ; and, therefore, 5 :: 35 : 27. 3° 7 Ex. 3. Let the velocities of A and B be as 5:4; required the ratio of the times in which they will describe 9 and 7 feet respectively. Here toc ; and, therefore, T:t:: 9 7 :: 36 : 35. 4 V:v :: 5 ON THE MOTION OF SOUND. 28. From numerous experiments it has been ascertained, that sound dies uniformly at the rate of 1142 feet in a second of time, or a mile in 4 seconds. Therefore, since by art. 23, s=tv, if we multiply the observed tiine in seconds by 1:42, we shall obtain the distance in feei through which sound flies in that time. Ez. Arter observing a flash of lightning, it was 12 seconds before the thunder was heard, at what distance was the cloud from whenca it came ? Here, as 4.: 1 mile :: 12 : 24 miles, the distance required Miscellaneous Eramples on Uniform Motions, &c. 1 If I see the flash of a cannon, fired by a ship in distress at sea. and hear the report 33 seconds afterwards, how far is she from me: Ans. 717 nuiles. 2 The diameter of an iron shot is 675 inches; what is its weight, it being known from experiment, that a cast-iron ball of 4 inches diameter weighs 9 lbs ? Ans. 42,294 lbs. 3. What is the weight of a leaden ball of 6 š inches in diameter, the weight of a leaden ball of 4 inches in diameter being 17 lbs. Ans. 63,888 lbs. 4. It is proposed to determine the proportional quantities of matter in the earth and moon, the density of the former being to that of the latter as 10 to 7, and their diameters 7930 and 2100 miles respectively. Ans. as 71 to I nearly. 5. A body weighing 20 lbs. is impelled by such a force as to send it through 100 feet in a second ; with what velocity would a body of g lbs. move if it were impelled by the same force ? Aus. 250 feet per second. 6. The body A weighs 100 lbs., another body B weighs 60 lbs., but the body B is impelled by a force 8 times greater than A ; required the proportion of the velocities with which they move ? Aps. the velocity of A is to that of B as 3 to 40. 7. The body A has passed over 50 miles, the body B only 5, but A moves with 5 times the velocity of B ; what is the ratio of the times that they have been in motion ? Ans. 2 to 1. 8. The body A moves 30 times swifter than B, and A has moved 12 minutes but B only 1, wbat difference will there be between the spaces described by them. supposing B to have moved over a space 5 feet? Ans. 1795 feet. 9. The hour and minute hands of a clock are together at 12 o'clock; when are they next together ? Ans. 53 minutes past 1. of ON THE MOTIONS OF BODIES ACCELERATED OR RETARDED BY THE ACTION OF CONSTANT AND UNIFORM FORCES. 28. The momentum generated by a constant and uniform force, acting for any time, is in the compound ratio of the force and time of acting. For, suppose the time divided into very small parts, then (by article 21) the momentum generated in cach particle of time is the same, and therefore the whole momentum will be as the whole time, or sum of all the small parts. But by the sanie article the momentum for each small time, is also as the motive force. Consequently the whole momentum generated, is in the compound ratio of the force and tiine of acting. Or, retaining the same notation as before, .mft. 29 Cor. 1. The momentum, or motiou lost or destroyed in any time is also in the compound ratio of the force and time. For, whatever momentum any force generates in a given time, the same momentun will an equal force destroy in the same or equal time, if acting in a contrary direction. And the same is true of the increase or decrease of motion, by forces that conspire with, or oppose the motion ot' bodies. 30. Cor. 2. The relocity generated, or destroyed in any time, is as the force and time directly, and the body or mass of matter reciprocally. For by the present article mast, and, by 22, in any sort of motion mocbo; therefore fuocor, or omen And if we suppose b and s constant, the velocity is simply as the time. 31. If a lody be moved from a state of rest by an uniform force, the space described, reckoning from the beginning of the motion, varies as the square of the time, or as the square of the last acquired velocity. Let AB. fig. 1, represent the tiine of the body's motion ; draw BC at right angles to AB, and let BC represent the last acquired velocity ; join AC; divide the time AB into small equal portions AD, DE, EF, FG, &c. and from the poivts D, E, F, G &c. draw DK, EL, FM, GN, &c. parallel to BC, meeting AC in the points K, L, M, N, &c. complete the parallelograms DX, EW, FV, GT, &c. Then, in the similar triangles ABC, ADK, we have AB : AD :: BC: DK; and, sincc BC represents the velocity acquired in the line AB, DK will represent the velocity acquired in the time AD; because, (by Cor. 2, Art. 30,) when the mass and force are constant, the velocity is as the time; in the same manner, it appears that EL, FM, GN, &c. represent the velocities generated in the times A:E, AF, AG, &c. Now, if the body move with the uniform velocity DK, during the time AD, and with the uniform velocities EL, FM, GN, &c. during the times DE, EF, FG, &c. respectively, the spaces described, may be properly represented by the rectangles DX, EW, FV, GT, &c. (because in uniform inotion s is always as vt); therefore, the whole space described, on this supposition, will be represented by the sum of these rectangles, or by the triangle ABC, together with the sum of the triangles AXK, KWL, LVM, MTN, &c. or because the bases of these small triangles are respectively equal to IB, and the sum of their altitudes is equal to BC, the whole space described may be represented by the triangle ABC, together with half the rectangle BQ. Let now the intervals AD, DE, EF, FG, &c. be diminished without limit with respect to AB, and the rectangle BQ is diminished without limit with respect to the triangle ABC; or, in other words, ABC+{BQ approaches to ABC as its limit; therefore, when the motion of the body is constantly accelerated, the space described is represented by the area of the triangle ABC. The space described in any other time AG, reckoning from the beginning of the motion, is represented on the same scale by the area of the triangle AGN, and because these triangles are similar, the space described in the time AB : the space described in the time AG :: AB? : AG”. And therefore, generally, sa 1. 32. Again BG, GN, represent the velocities generated in the times AB, AC; and therefore from the same similar triangles the space described in the time AB : the space describ in the time AG :: BGR : GN”. And consequently we have, in general, socvs. |