For the magnitudes are as the cubes of the diameters, or other like dimensions. Cor. 2. The masses, or quantities of matter, are as the magnitudes and specific gravities. For, by Art. 4 and 13, the densities of bodies are as the specific gravities. Therefore, if B denote the quantity of matter in a body, M its magnitude, D its density, S its specific gravity, and Pits diameter; and let b denote the quantity of matter in any other body, m its magnitude, d its density, s its specific gravity, and p its diameter, or other like dimension; then, B: b:: MxD: mxd:: MxS: mxs :: P3 x D: p3 x d 101. The velocity of a moving body is said to be uniform, when the body passes over equal spaces in equal times. 102. The space which a body moving with an uniform velocity passes over in any time, is found by multiplying the time by the velocity; that is, by multiplying the number of seconds the body has been in motion, by the space moved over in one second. Thus, if a body moves uniformly at the rate of 4 feet per second, and is 30 seconds in motion, then 4 x 30= 120 feet, the space or length of line passed over by the body. Now, if S be the length of the line described, in the time T, with the uniform velocity V, then, If two bodies move uniformly for 4 and 6 seconds, the former with a velocity of 10 feet, and the latter with a velocity of 5 feet per second; then the spaces described by them will be in proportion to each other as 10 x 4 to 5 x 6, or as 4: 3. MOMENTUM. 103. When the motion of a body is considered with respect to the mass, or quantity of matter, moved, as well as its velocity, it is called its momentum, or quantity of motion. 104. The momentum of a body is in the compound ratio of its quantity of matter and velocity. For the momentum of the whole body is the sum of the motions of all its parts; therefore the quantity of motion. depends on the number of parts and the velocity of each. = Let M the momentum of a body, W its quantity of matter or weight, and V its velocity; and let m = the momentum, w = the weight, and v = the velocity of any other body, expressed in the same terms; then the relation which exists between M, W, V, and m, w, v, may be expressed as follows: M: m :: W x V : w x v If the bodies move with equal velocities, then, If two bodies move with velocities which are inversely as their quantities of matter, their momenta will be equal. For then, M: m :: v: V And since the product of the extreme terms of a proportion is equal to the product of the mean terms, we have MV = m v If a body, the weight of which is 8 lbs. moves with a velocity of 10 feet per second; and another body, the weight of which is 10 lbs. moves with a velocity of 5 feet per second; then, M: m :: 8 x 10 : 10 x 5 :: 8:5 That is, the momentum of the former is to the momentum of the latter as 8 to 5. Or, if we take the momentum of the former to that of the latter as 8 to 5, and their velocities as 2 to 1, then their weights are to each other as 8 5 that is as 4 to 5. And if the momenta of four bodies are as 1, 2, 3, 4, and their weights as 3, 4, 5, 6, then their velocities are as 1 2 3 4 1 1 3 2 ; or, which 3' 4' 5 is the same, 10, 15, 18, 20. The battering-ram of Vespasian weighed, suppose, 100,000 lbs.; and was moved, let us admit, with such a velocity, by strength of hands, as to pass through 20 feet in one second of time; and this was sufficient to demolish the walls of Jerusalem: with what velocity must a bullet, which weighs but 30 lbs. be moved, in order to do the same execution? = Here, 100,000 x 20 2,000,000 lbs. the momentum of the ram; and, to produce the same effect, the momentum of the bullet must be equal to the momentum of the ram. Therefore, 2,000,000 ÷ 30 66,6663 feet per second. = ACCELERATED MOTION. 105. A force acting incessantly upon a body is called a constant or uniformly accelerating force, when the velocity increases equally in equal times. Thus, the force of gravity near the earth's surface is of this kind; for it generates a velocity of 321 feet in each second of time. That is, a body, after falling one second, acquires a velocity of 32 feet; after falling 2 seconds, it will acquire a velocity of 2 × 32 feet; after 3 seconds, a velocity of 3 × 321 feet; and so on. 106. It is called a variable force when the variation of the acquired velocity is not the same in each succeeding instant. Universal Gravitation furnishes us with an example of this kind; for the variation of its law is inversely as the square of the distance from the earth's centre. Thus, a body at the distance of two semi-diameters from the earth's centre, will be acted on by a force which is only one-fourth part of the force which acts upon a body at the earth's surface, and which will therefore generate only one-fourth part of the velocity. At three semi-diameters, the force will be one-ninth; at four semidiameters, one-sixteenth; and so on, always decreasing inversely as the square of the distance. 107. The momentum, or quantity of motion, which is generated by an uniformly accelerating force, is in the compound ratio of the force and time of acting. For, in any given time, the momentum which is generated will be proportional to the force which generates it. And since the force has the same efficacy in each instant of time, the whole momentum will be as the sum of these instants, or whole time. Consequently, the whole momentum, or quantity of motion generated, is in the compound ratio of the force and time of acting. Cor. 1. The quantity of motion lost or destroyed in any time, by a force acting in an opposite direction, is also in the compound ratio of the force and time. Cor. 2. The velocity generated or destroyed in any time, is as the force and time directly, and quantity of ft. matter reciprocally; that is, v is as where denotes the velocity, f the force, and b the body, or quantity of matter. For, by Art. 104, the momentum is as the quantity of matter and velocity; therefore, the velocity is as the momentum directly and quantity of matter reciprocally; that is, by this Art. as the force and time directly and quantity of matter reciprocally. Cor. 3. Hence, if the body be given, the velocity will be in the compound ratio of the force and time; and if the force be given, the time is in the compound ratio of the quantity of matter and the velocity, or as the momentum. 108. If a given body be urged by a constant and uniform force, the space which is described by the body from the beginning of the motion is as the force and square of the time. For, suppose the time to be divided into an indefinite number of equal parts. Then, in each of these equal parts of time, the space described will be as the velocity gained; that is, by Art. 107, Cor. 3, as the force and time from the beginning. And the sum of all the spaces, or the whole space described, will be as the force and the sum of all the equal parts of time from the beginning. If we put n the whole time, the whole space described will be as the sum of the times 1 + 2 + 3 + 4, &c. to n. But the sum of the arithmetical series, 1 + 2 + 3 + 4, 1+n 1 + n n; but n is infinite, therefore, 2 &c. to n = 2 x n becomes Hence, the whole space described will be as the force and n2, or force and square of the time; for is a constant quantity. |