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successive portions of time, its motion is said to be uniform. When the successive portions of space, described in equal times, continually increase, the motion is said to be accelerated; and retarded, when those spaces continually decrease.
Again, motion is said to be uniformly accelerated or retarded, when the increments or decrements of the spaces, described in equal and successive portions of time, are always equal.
12. The Velocity of a body, or the rate of its motion, is measured by the space, uniformly described in a given time.
The given time, taken as a standard, is usually one second; and the space described is measured in feet. Thus, when v represents the velocity of a body, v is the number of feet which the body would uniformly describe in one second.
If the motion of a body be accelerated or retarded, the velocity at any point is not measured by the space actually described in a given time, but by the space which would have been described in the given time, if the motion had continued uniform from that point.
13. The Quantity of Motion, or Momentum, is that power or force in moving bodies, by which they continually tend from their present places, or with which they strike any obstacle that opposes their motion. It is measured by he velocity and quantity of matter jointly.
14. Whatever changes, or tends to change, the state of rest, or of uniform rectilinear motion of a body, is called force, or power.
Thus pressure, impact, gravity, &c. are called forces.
When a force produces its effect instantaneously, it is said to be impulsive. When it acts incessantly, it is called a constant force.
Constant forces are of two kinds, uniform and variable. A force is said to be uniform, when it always produces equal effects in equal successive portions of time; and variable, when the effects produced in equal time are unequal.
Forces which are known to us only by their effects, must be compared by estimating those effects under the same circumstances. Thus, impulsive forces must be measured by the whole effects produced; uniform forces, by the effects produced in equal times; and variable forces, by the effects which would be produced in equal times, were they to become and continue uniform during those times.
The effects produced by the actions of forces are of two kinds, velocity and momentum; and thus we have two methods of comparing them, according as we conceive them to be the causes of velocity or momentum. Under this point of view, forces are distinguished into motive, and accelerative, or retarding.
15. A motive, or moving force, is the power of an agent to produce motion; and it is measured by the momentum uniformly generated in a given time. If the momenta thus generated, in any two cases, be as 14 to 15, the moving forces are said to be in that ratio.
16. An accelerative, or retarding force, is understood to affect the velocity only, and is measured by the velocity uniformly generated or destroyed in a given time, without regard to the quantity of matter moved.
Thus, if the velocities uniformly generated in two cases, in equal times, be as 6 to 7, the accelerating forces are said to be in that ratio. Other qualities and properties of bodies, as Solidity, Fluidity, Hardness, Softness, Elasticity, &c. will be defined in the succeeding articles.
The relations between the quantities of matter, magnitudes, or bulks, densities of bodies, &c. may be expressed as follows.
17. The quantity of matter in all bodies is in the compound ratio of their magnitude and densities.
For when their magnitudes are equal, the quantity of matter which they contain is evidently as their densities, and when their densities are the same, the quantity of matter which they contain is as their magnitudes.
When, therefore, neither their magnitudes nor densities are equal, the quantity of matter which they contain is in a ratio compounded of both. Cor. 1. When the bodies are similar, then masses, or quantities of matter, are as their densities and the cubes of their diameters.
For the magnitudes of bodies is shown, in Geometry, to be as the cubes of their diameters, or other homologous dimensions.
2. The specific gravity of a body, is the weight of a certain magnitude of that substance, such as a cubic foot, a cubic inch, &c. Hence the specific gravities of bodies are proportional to their densities; and hence, again, the masses are as the magnitudes of the bodies and their specific gravities. Scholium. Let b denote any body, or quantity of matter m its magnitude,
d its density,
s its specific gravity,
a its diameter.
Then, from the last Articles and Corollaries, we obtain the following general Table of Proportions.
As an example of the application of this Table :-Required the proportion of the diameters of two spheres, in which the quantities of matter are as 6 to 5, and the densities as 4: 3.
Let A, a represent the respective diameters; B, b the quantities
of matter ; and D, d the densities; then, since by the Table a3x, w
the cubes of the diameters are as 9: 10; and the diameters-them
selves as 9: 10.
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, between any two bodies, 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 them the whole theory 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 remove all those impediments which tend to render the conclusions erroneous, the more nearly do the experiments coincide with these laws.
OF UNIFORM MOTIONS.
21. The momentum, or quantity of motion, generated by 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 momentum, or quantity of motion, and f the force, we shall have
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 & the mass or body, v the velocity, and m the momentum, as before,
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, the 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
24. Cur. 1. t 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; but if we suppose the space to be the same, the time will be as, or reciprocally as the velocity.
25. Cor. 2. v, 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 tv, 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 evidently represent the space passed over in t Weconds, and therefore stv. Consequently v= and t=
27. In uniform motions, or such as are generated by momentary impulse, let
bany body, or quantity of matter to be moved,
the force or impulse acting on the body b.
the uniform velocity generated in b
m = the momentum generated in b.
s = the space described by the body b.
t = 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 sotv.
By combining these, we easily derive the following table of the general relations of those six quantities.
Table for uniform Motions and Impulsive Forces.
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 move 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 Ss :: VT: vt :: 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, and B through 9 feet in 7 seconds; to find the ratio of their velocities.
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.
ON THE MOTION OF SOUND.
28. From numerous experiments it has been ascertained, that sound flies 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 time in seconds by 1142, we shall obtain the distance in feet through which sound flies in that time.
Ex. After observing a flash of lightning, it was 12 seconds before the thunder was heard, at what distance was the cloud from whenc it came ?
Here, as 44 : 1 mile :: 12: 24 miles, the distance required