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The meaning of the second law of motion is this: If a certain force be sufficient to move a body with such a velocity as to carry it over any given space in a given time, double that force applied to it would carry it over double the space in the same time; and treble the force, over three times the space. This is evident enough when only one force is applied to move the body; but the application of the law, when two or more forces are applied at the same time in different directions, will need some explication. Suppose on a body at A, a force were exerted that would carry it over the space AB in a given time, in the direction of that force, and at the same time, another force exerted its influence on the same body in the direction AC, with sufficient efficacy to carry it over the space AC in the same time. Now these two forces in the given time must produce their separate effects: that is, the body must be carried the distance of AB from the point A, and also the distance of AC from the same point, and in the directions of the forces. But this can be no otherwise effected, than by their joint energy causing the body to move in the diagonal of a parallelogram, whose sides are in the direction of the two forces, and proportional to their strength. So that at the end of the time the body will be found at D, having passed over the space AD, in the same time, and by their combined influence, that would have been required to pass over either of the spaces AB or AC by the respective forces alone, which acted in these directions.
This second law of motion, therefore, teaches us how to compound and resolve various forces, and to estimate their joint or separate effects, namely, by describing a parallelogram, whose sides are in the direc* See Plate I. Fig. 3.
tions of two of the forces and proportional to their separate influence, and taking the diagonal as the measure of their joint effect and the direction of the motion produced. This diagonal, being considered as the measure and direction of two forces, may be compounded with a third, by making it the side of a new parallelogram, of which the third force constitutes the measure and direction of the other side; so that the diagonal of this second parallelogram will exhibit the joint effect of the three forces, acting on the body at the same time. And thus may any number of forces be compounded into one.
As we are now on the subject of the forces requi site to give motion to bodies, it is necessary that we be very precise in our ideas of motion; by which we are not to understand the velocity, with which a body is moved, but the force, with which it moves. As if two bodies A and B, being in motion, when twice the force is required to stop A as to stop B, then the mo. tion of A is said to be double the motion of B.
In moving bodies, we must carefully distinguish between the velocity with which they move over a given space in a given time, and the quantity of motion, with which they press upon any obstacle in their way. The quantity of their motion is sometimes called their momentum; and depends upon the velocity of the body and the quantity of matter it contains. If M represent the quantity of motion, Q the quantity of matter, and V the velocity of a body in motion; then M=QV; or, the quantity of motion is proportional to the rectangle of the quantity of matter, and the velocity of the motion.
The third law of motion is evident both from sense and experiment. A stretched cord, being cut asunder in the middle, flies towards both ends. If a boat and ship be connected by a rope, they approach each other with an equal quantity of motion, let either end of the rope be drawn. When a horse draws a load, the harness that is stretched between them, presses as much upon the breast of the horse as upon the load; and the progressive motion of the horse is as much retarded or hindered, as the motion of the load is promoted.
OF NONELASTIC BODIES.
If a body that is perfectly hard or soft strike against any other body, it will give some degree of motion to it, and lose just the same quantity. Suppose the two bodies equal: from the time they meet, both are to be moved by the single motion of the first; but now, as both are to be moved by that force, which moved the first only, the ensuing velocity will be the same, as if the power which moved the first had been applied to both at the same time; hence they will proceed with half the velocity which the first moving body had; that is, the first body will have lost half its motion, and the other will have gained just as much. Therefore, if the bodies be both in motion, and moving the same way, the sum of their motions towards the same part, or in the same direction, will be M+m=QV+qv. But if they move in contrary directions, the sum of the motion in the direction of the first will be M-m-QV-qv. M and m representing the momenta, Q and q, the quantities of matter, and V and v, the velocities of the bodies. Hence supposing the bodies to remain contiguous after their impact, we have the following rule for their common velocity, x. Divide these momenta by the quantity of
matter in both, and you have the velocity after col
BODIES upon striking each other suffer an alteration in their figure, having their parts pressed inwards by the stroke, which recoil again as the bodies endeavour to recover their former shape. This power, whereby bodies recover their former figure, is usually called their elasticity; and when it acts it forces the bodies from each other and causes them to separate. Now the effect of this elasticity, when it is perfect, is such, that they recover their figure with as much force as that with which they were compressed, and in the same time; hence this power will separate the bodies, as swiftly as they before approached; and acting on both equally on the one in a direction contrary to its motion, and on the other in the direction of its motion, it will take from the former, and add to the latter, equal degrees of velocity, when the two bodies are moving in the same direction.
The change of velocity which is made in any case of congress, when the bodies are perfectly elastic, is: double the change which would be made, in like circumstances, where the bodies are void of elasticity. For if two bodies A and B strike one another, whatever motion A loses, by the stroke, B gains, if they be void of elasticity. But by the stroke the sides of both are pressed inwards, and, as they recover their former shape with the same velocity and in the same time that was required for their compression; while the side of A next to B is returning to its former situation it must impress B with a force equal to which
relative velocities with which they separate, will still be the same as that by which they approached each other; the body that moved fastest and overtook the other, will, after collision, move slowest; and the other having received an increment of velocity from the stroke, will now move forward and leave the other behind as fast as it was overtaken by it before: so that their velocities will be interchanged. Hence if one body strike an equal body at rest, the percutient body will communicate all its motion to the other, and remain at rest.
From what has been said we may determine the velocity of the bodies after impact, in the following manner. Let Q= the quantity of matter in the body A, and q that in B; V and v = their respective velocities, and let x = the quantity of motion lost by A and gained by B. Then QV-x, and qv+x will be their respective motions after the stroke. ConsequentQV-x qv+x will be their velocities. Now the
difference of these velocities, making that the least, which was greatest before collision, will still be equal to the difference of their velocities before they met; supposing the bodies moved the same way.
Now substituting this value of x in the expressions for the velocities of the bodies A and B, we have