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× V-v= the velocity of A after collision.
2Q +Q+q N. B. If the bodies move in contrary directions, v must have a contrary sign, because B is supposed in these theorems to move in the same direction with A. If B be supposed at rest, then v=o. And if either of these expressions should be negative, then the motion of that body is contrary to what it was supposed to be before their congress, in these theorems.
x V-v- the velocity of B after collision.
Let Q=3, V=8, q=9 and v=2, when both bodies move in the direction of A forwards.
=8—9——1— the velocity of A backwards.
=5= the velocity of B forwards.
Let Q=1, V=5, q=3, v=11, when B moves contrary to A.
5-- =-19= the velocity of A backwards.
-11+ -3 the velocity of B forwards, or
contrary to the direction of A.
These theorems may be easily remembered by comparing the quantity of motion lost and gained, by bodies that have no elasticity, with what is lost or gained by elastic bodies, and remembering that the action of the latter is always double the action of the former.
These theorems therefore may be expressed in the following
Compute the motion lost and gained by the collision on the supposition that the bodies were nonelastic; double this quantity and add it to the first motion of the body that has gained motion by the stroke, and subtract the double loss of motion from the first motion of the body that has lost motion by the stroke, and you will have the motion of both after collision.
N. B. When a body loses its whole motion and acquires a contrary motion the other way, these two motions must be collected into one sum, in order to have the motion that is lost; and when the quantity to be subtracted exceeds the quantity of motion before the stroke, the lesser is still to be subtracted from the greater, and the difference is the motion, in the contrary direction.
From the foregoing theorems and computations it will always be found that the quantity of motion towards the same point is not altered by the stroke, either in elastic or nonelastic bodies: the motion in the contrary direction being always subtracted. In elastic bodies, if the body A of one pound, and 15 degrees of velocity strike against B at rest of four pounds weight, B will move forward with 6 degrees of velocity; so that its quantity of motion will be 24; which is nine more than the motion of A at first; but this 9 must be subtracted from 24; because the body A after the collision is reflected back with that quantity of motion. It will also be found that if the bodies be equal, they will either move forward, or be reflected backwards, with interchanged velocities. If one, two, three, or four bodies, all equal, strike against any number of equal bodies, the motion will be communicated through them, and as many will fly off.
while the others will remain at rest. If one strike against a number at rest, it will move the first with its own velocity and remain at rest; the second moves the third in like manner and remains at rest, and so on to the last; which, having none to strike, flies off with the velocity acquired. In like manner you may conceive of all the rest.
If a small body strike a larger body at rest, the greater body acquires more momentum than the smaller body had at first, viz. as much more as that with which it is reflected back: and the motion of the larger body is increased by the interposition of a body of an intermediate bigness; and this increase of motion is greatest when the intermediate body is a geometrical mean between the other two. Let the quantities of matter in the three bodies be as 1,2,4, and the first strike upon the second with the velocity 18, it will communicate to the second a velocity of 12, and a motion or momentum of 24; which motion will communicate to the third a velocity of 8, and a momentum of 32. But for the want of perfect elasticity, the error in experiments, where there is a double percussion, will be more sensible, and the momentum in the above case will in ivory balls, scarcely exceed
On the doctrine of the momenta of bodies, depends the operation of the mechanic powers: but before we can explain these, it will be necessary to premise a few things, concerning the centers of magnitude, of motion, and of gravity. The center of magnitude is that point of a body which is equally distant from all the external parts of it: and in uniform bodies, it is the same with the center of motion, which is that
point which remains at rest, while all the parts of the body revolve round it, in any direction. The center of gravity is that point, about which all the parts of the body exactly balance each other; those in opposite directions having equal momenta. So that if a body be suspended by this point, it will rest indifferently in any position; for then the momentum of any particles to carry the body round is counteracted by an equal momentum of other particles in the opposite direction. As long as this point is supported, the whole body is prevented from falling or descending; but if it be at liberty to descend, the body falls or tumbles downward, however to appearance it may seem to ascend, as in the case of the rolling cones, whose center of gravity really descends while the ends of the cones seem to roll up an inclined plane. Another property of this point is this: if a body be suspended by any other point, it cannot be at rest, in any position, excepting two, namely, when the center of gravity is directly above, or directly below the point of suspension. Hence the solution of many curious phenomena: as for example,-why some bodies stand more firmly on their bases than others,-why some stand better in an inclined position,-why a wagon loaded with iron or heavy materials is carried safely along a shelving road, where one loaded with hay would be overset,-why we stand firm, while the center of gravity of the human body (which is in the pelvis) is perpendicularly over the base line of the feet, but tumble down, when it is removed from that position, -why equilibrists and wiredancers perform such wonders, by a nice adjustment of this point over the wire-why our feet must be brought under this point before we can rise from our seats,-why fowls stretch
their heads forwards, when they walk up an hill,and how a bucket may be formed, that shall turn its mouth downwards when empty, and upwards when full. Hence also you see the reason of suspending a bucket of water upon a short stick laid on the edge of a table, &c.
You may easily find the center of gravity of any irregular body, that has a plane surface, by suspending it by any point, and hanging a plummet from the point of suspension; for the center of gravity is in this line: then suspend it by any other point, and the plumbline from this point must intersect the former line in the center of gravity.
In homogeneous bodies, such as circles, squares &c. it is the middle point of the line that connects any two opposite points: in a triangle, it is in a line drawn from the vertical angle bisecting the base, and one third of the length of this line from the base: in the axis of a cone, one third of the length from the base, if the cone be hollow, but one fourth, if the cone be solid. In two bodies connected together, their common center of gravity is between them, and at such a distance from each, that their respective distances multiplied into their weights shall produce equal momenta or products. If this point of an inflexible line, which connects the bodies be supported, it is evident, the bodies must be in equilibrio, and cannot descend, while it remains at rest. The force of each of the bodies, to cause the system to descend, depends upon its weight and velocity, which constitute their respective momenta; and their velocities, being measured by the spaces they pass over in equal times, while they librate up and down on their common center, are proportional to the arches of the circle they describe;