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(54.) The effects of inertia or inactivity, considered in the last chapter, are such as may be manifested by a single insulated body, without reference to, or connection with, any other body whatever. They might all be recognised if there were but one body existing in the universe. There are, however, other important results of this law, to the development of which two bodies at least are necessary.

(55.) If a mass A, fig.4., moving towards C, impinge upon an equal mass, which is quiescent at B, the two masses will move together towards C after the impact. But it will be observed, that their speed after the impact will be only half that of A before it. Thus, after the impact, A loses half its velocity; and B, which was before quiescent, receives exactly this amount of motion. It appears, therefore, in this case, that B receives exactly as much motion as A loses: so that the real quantity of motion from B to C is the same as the quantity of motion from A to B.

Now, suppose that B consisted of two masses, each equal to A, it would be found that in this case the velocity of the triple mass after impact would be one-third of the velocity from A to B. Thus, after impact, A loses two-thirds of its velocity and, B consisting of two masses each equal to A, each of these two receives one-third of A’s motion; so that the whole motion received by B is two-thirds of the motion of A before impact. By the impact, therefore, exactly as much motion is received by B as is lost by A.

A similar result will be obtained, whatever proportion may subsist between the masses A and B. Suppose B to be ten times A; then the whole motion of A must, after the impact, be distributed among the parts of the united masses of A and B: but these united masses are, in this case, eleven times the mass of A. Now, as they all move with a common motion, it follows that A’s former motion must be equally distributed among them; so that each part shall have an eleventh part of it. Therefore the velocity after impact will be the eleventh part of the velocity of A before it. Thus A loses by the impact ten-eleventh parts of its motion, which are precisely what B receives.

Again, if the masses of A and B be 5 and 7, then the united mass after impact will be 12. The motion of A before impact will be equally distributed between these twelve parts, so that each part will have a twelfth of it; but five of these parts belong to the mass A, and seven to B. Hence B will receive seven-twelfths, while A retains five-twelfths.

(56.) In general, therefore, when a mass A in motion impinges on a mass B at rest, to find the motion of the united mass after impact, “divide the whole motion of A into as many equal parts as there are equal component masses in A and B together, and then B will receive by the impact as many parts of this motion as it has equal component masses.”

This is an immediate consequence of the property of inertia, explained in the last chapter. If we were to suppose that by their mutual impact A were to give to B either more or less motion than that which it (A) loses, it would necessarily follow, that either A or B must have a power of producing or of resisting motion, which would be inconsistent with the quality of inertia already defined. For if A give to B more motion than it loses, all the overplus or excess must be excited in B by the action of A; and, therefore, A is not inactive, but is capable of exciting motion which it does not possess. On the other hand, B cannot receive from A less motion than A loses, because then B must be admitted to have the power by its resistance of destroying all the deficiency; a power essentially active, and inconsistent with the quality of inertia.

(57.) If we contemplate the effects of impact, which we have now described, as facts ascertained by experiment (which they may be), we may take them as further verification of the universality of the quality of inertia. But, on the other hand, we may view them as phenomena which may certainly be predicted from the previous knowledge of that quality; and this is one of many instances of the advantage which science possesses over knowledge merely practical. Having obtained by observation or experience a certain number of simple facts, and thence deduced the general qualities of bodies, we are enabled, by demonstrative reasoning, to discover other facts which have never fallen under our observation, or, if so, may have never excited attention. In this way philosophers have discovered certain small motions and slight changes which have taken place among the heavenly bodies, and have directed the attention of astronomical observers to them, instructing them with the greatest precision as to the exact moment of time and the point of the firmament to which they should direct the telescope, in order to witness the predicted event.

(58.) Since by the quality of inertia a body can neither generate nor destroy motion, it follows that when two bodies act upon each other in any way whatever, the total quantity of motion in a given direction, after the action takes place, must be the same as before it, for otherwise some motion would be produced by the action of the bodies, which would contradict the principle that they are inert. The word “action” is here applied, perhaps improperly, but according to the usage of mechanical writers, to express a certain phenomenon or effect. It is, therefore, not to be understood as implying any active principle in the bodies to which it is attributed.

(59.) In the cases of collision of which we have spoken, one of the masses B was supposed to be quiescent before the impact. We shall now suppose it to be moving in the same direction as A, that is, towards C, but with a less velocity, so that A shall overtake it, and impinge upon it. After the impact, the two masses will move towards C with a common velocity, the amount of which we now propose to determine.

If the masses A and B be equal, then their motions or velocities added together must be the motion of the united mass after impact, since no motion can either be created or destroyed by that event. But as A and B move with a common motion, this sum must be equally distributed between them, and therefore each will move with a velocity equal to half the sum of their velocities before the impact. Thus, if A have the velocity 7, and B have 5, the velocity of the united mass after impact is 6, being the half of 12, the sum of 7 and 5.

If A and B be not equal, suppose them divided into equal component parts, and let A consist of 8, and B of 6, equal masses: let the velocity of A be 17, so that the motion of each of the 8 parts being 17, the motion of the whole will be 136. In the same manner, let the velocity of B be 10, the motion of each part being 10, the whole motion of the 6 parts will be 60. The sum of the two motions, therefore, towards C is 196; and since none of this can be lost by the impact, nor any motion added to it, this must also be the whole motion of the united masses after impact. Being equally distributed among the 14 component parts of which these united masses consist, each part will have a fourteenth of the whole motion. Hence, 196 being divided by 14, we obtain the quotient 14, which is the velocity with which the whole moves.

(60.) In general, therefore, when two masses moving in the same direction impinge one upon the other, and after impact move together, their common velocity may be determined by the following rule: “Express the masses and velocities by numbers in the usual way, and multiply the numbers expressing the masses by the numbers which express the velocities; the two products thus obtained being added together, and their sum divided by the sum of the numbers expressing the masses, the quotient will be the number expressing the required velocity.”

(61.) From the preceding details, it appears that motion is not adequately estimated by speed or velocity. For example, a certain mass A, moving at a determinate rate, has a certain quantity of motion. If another equal mass B be added to A, and a similar velocity be given to it, as much more motion will evidently be called into existence. In other words, the two equal masses A and B united have twice as much motion as the single mass A had when moving alone, and with the same speed. The same reasoning will show that three equal masses will with the same speed have three times the motion of any one of them. In general, therefore, the velocity being the same, the quantity of motion will always be increased or diminished in the same proportion as the mass moved is increased or diminished.

(62.) On the other hand, the quantity of motion does not depend on the mass only, but also on the speed. If a certain determinate mass move with a certain determinate speed, another equal mass which moves with twice the speed, that is, which moves over twice the space in the same time, will have twice the quantity of motion. In this manner, the mass being the same, the quantity of motion will increase or diminish in the same proportion as the velocity.

(63.) The true estimate, then, of the quantity of motion is found by multiplying together the numbers which express the mass and the velocity. Thus, in the example which has been last given of the impact of masses, the quantities of motion before and after impact appear to be as follow:

Before Impact.	After Impact.
Mass of A		8	Mass of A		8
Velocity of A		17	Common velocity		14
Quantity of motion of A		8×17* or 136	Quantity of motion of A		8×14 or 112
Mass of B		6	Mass of B		6
Velocity of B		10	Common velocity		14
Quantity of motion of B		6×10 or 60	Quantity of motion of B		6×14 = 84

* The sign × placed between two numbers meant that they are to be multiplied together.

By this calculation it appears that in the impact A has lost a quantity of motion expressed by 24, and that B has received exactly that amount. The effect, therefore, of the impact is a transfer of motion from A to B; but no new motion is produced in the direction AC which did not exist before. This is obviously consistent with the property of inertia, and indeed an inevitable result of it.

These results may be generalised and more clearly and concisely expressed by the aid of the symbols of arithmetic.

Let a express the velocity of A.

Let b express the velocity of B.

Let x express the velocity of the united masses of A and B after impact, each of these velocities being expressed in feet per second, and the masses of A and B being expressed by the weight in pounds.

We shall then have the momenta or moving forces of A and B before impact, expressed by A × a and B × b, and the moving force of the united mass after impact will be expressed by (A + B)×x.

The moving force of A after impact is A × x, and therefore the force it loses by the collision will be (A × a - A × x). The force of B after impact will be B × x, and therefore the force it gains will be B × x - B × b. But since the force lost by A must be equal to the force gained by B, we shall have

A × a - A × x = B × x - B × b

from which it is easy to infer

(A + B)×x = A × a + B × b

and if it be required to express the velocity of the united masses after impact, we have

x = A × a + B × b/A + B

When it is said that A × a and B × b express the moving forces of A and B, it must be understood that the unit of momentum or moving force is in the case here supposed, the force with which a mass of matter weighing 1lb. would move if its velocity were 1 foot per second, and accordingly the forces with which A and B move before impact are as many times this as there are units respectively in the numbers signified by the general symbols A × a and B × b.

In like manner, the force of the united masses after impact is as many times greater than that of 1lb. moving through 1 foot per second as there are units in the numbers expressed by (A + B)×x.

(64.) These phenomena present an example of a law deduced from the property of inertia, and generally expressed thus—“action and reaction are equal, and in contrary directions.” The student must, however, be cautious not to receive these terms in their ordinary acceptation. After the full explanation of inertia given in the last chapter, it is, perhaps, scarcely necessary here to repeat, that in the phenomena manifested by the motion of two bodies, there can be neither “action” nor “reaction,” properly so called. The bodies are absolutely incapable either of action or resistance. The sense in which these words must be received, as used in the law, is merely an expression of the transfer of a certain quantity of motion from one body to another, which is called an action in the body which loses the motion, and a reaction in the body which receives it. The accession of motion to the latter is said to proceed from the action of the former; and the loss of the same motion in the former is ascribed to the reaction of the latter. The whole phraseology is, however, most objectionable and unphilosophical, and is calculated to create wrong notions.

(65.) The bodies impinging were, in the last case, supposed to move in the same direction. We shall now consider the case in which they move in opposite directions.

First, let the masses A and B be supposed to be equal, and moving in opposite directions, with the same velocity. Let C, fig.5., be the point at which they meet. The equal motions in opposite directions will, in this case, destroy each other, and both masses will be reduced to a state of rest. Thus, the mass A loses all its motion in the direction AC, which it may be supposed to transfer to B at the moment of impact. But B having previously had an equal quantity of motion in the direction BC, will now have two equal motions impressed upon it, in directions immediately opposite; and these motions neutralising each other, the mass becomes quiescent. In this case, therefore, as in all the former examples, each body transfers to the other all the motion which it loses, consistently with the principle of “action and reaction.”

The masses A and B being still supposed equal, let them move towards C with different velocities. Let A move with the velocity 10, and B with the velocity 6. Of the 10 parts of motion with which A is endued, 6 being transferred to B, will destroy the equal velocity 6, which B has in the direction BC. The bodies will then move together in the direction CB, the four remaining parts of A’s motion being equally distributed between them. Each body will, therefore, have two parts of A’s original motion, and 2 therefore will be their common velocity after impact. In this case, A loses 8 of the 10 parts of its motion in the direction AC. On the other hand, B loses the entire of its 6 parts of motion in the direction BC, and receives 2 parts in the direction AC. This is equivalent to receiving 8 parts of A’s motion in the direction AC. Thus, according to the law of “action and reaction,” B receives exactly what A loses.

Finally, suppose that both the masses and velocities of A and B are unequal. Let the mass of A be 8, and its velocity 9: and let the mass of B be 6, and its velocity 5. The quantity of motion of A will be 72, and that of B, in the opposite direction, will be 30. Of the 72 parts of motion, which A has in the direction AC, 30 being transferred to B, will destroy all its 30 parts of motion in the direction BC, and the two masses will move in the direction CB, with the remaining 42 parts of motion, which will be equally distributed among their 14 component masses. Each component part will, therefore, receive 3 parts of motion; and accordingly 3 will be the common velocity of the united mass after impact.

(66.) When two masses moving in opposite directions impinge and move together, their common velocity after impact may be found by the following rule:—“Multiply the numbers expressing the masses by those which express the velocities respectively, and subtract the lesser product from the greater; divide the remainder by the sum of the numbers expressing the masses, and the quotient will be the common velocity; the direction will be that of the mass which has the greater quantity of motion.”

It may be shown without difficulty, that the example which we have just given obeys the law of “action and reaction.”

Before Impact.	After Impact.
Mass of A		8	Mass of A		8
Velocity of A		9	Common velocity		3
Quantity of motion in direction AC		8×9 or 72	Quantity of motion in direction AC		8×3 or 24
Mass of B		6	Mass of B		6
Velocity of B		5	Common velocity		3
Quantity of motion in direction BC		6×5 or 30	Quantity of motion in direction AC		6×3 = 18

Hence it appears that the quantity of motion in the direction AC of which A has been deprived by the impact is 48, the difference between 72 and 24. On the other hand, B loses by the impact the quantity 30 in the direction BC, which is equivalent to receiving 30 in the direction AC. But it also acquires a quantity 18 in the direction AC, which, added to the former 30, gives a total of 48 received by B in the direction AC. Thus the same quantity of motion which A loses in the direction AC, is received by B in the same direction. The law of “action and reaction” is, therefore, fulfilled.

This result may in like manner be generalised. Retaining the former symbols, the moving forces of A and B before impact will be A × a and B × b and their forces after impact will be A × x and B × x. The force lost by A will therefore be A × a - A × x. The mass B will have lost all the force B × b which it had in its former direction, and will have received the force B × x in the opposite direction. Therefore the actual force imparted to B by the collision will be B × b + B × x. But since the force lost by A must be equal to that imparted to B, we shall have

A × a - A × x = B × b + B × x

and therefore

(A + B)×x = A × a - B × b

and if the common velocity after impact be required, we have

x = A × a - B × b/A + B

As a general rule, therefore, to find the common velocity after impact. Multiply the weights by the previous velocities and take their sum if the bodies move in the same direction, and their difference if they move in opposite directions, and divide the one or the other by the sum of their weights. The greatest will be the velocity after impact.

(67.) The examples of the equality of action and reaction in the collision of bodies may be exhibited experimentally by a very simple apparatus. Let A, fig.6., and B be two balls of soft clay, or any other substance which is inelastic, or nearly so, and let these be suspended from C by equal strings, so that they may be in contact; and let a graduated arc, of which the centre is C, be placed so that the balls may oscillate over it. One of the balls being moved from its place of rest along the arc, and allowed to descend upon the other through a certain number of degrees, will strike the other with a velocity corresponding to that number of degrees, and both balls will then move together with a velocity which may be estimated by the number of degrees of the arc through which they rise.

(68.) In all these cases in which we have explained the law of “action and reaction,” the transfer of motion from one body to the other has been made by impact or collision. The phenomenon has been selected only because it is the most ordinary way in which bodies are seen to affect each other. The law is, however, universal, and will be fulfilled in whatever manner the bodies may affect each other. Thus A may be connected with B by a flexible string, which, at the commencement of A’s motion, is slack. Until the string becomes stretched, that is, until A’s distance from B becomes equal to the length of the string, A will continue to have all the motion first impressed upon it. But when the string is stretched, a part of that motion is transferred to B, which is then drawn after A; and whatever motion B in this way receives, A must lose. All that has been observed of the effect of motion transferred by impact will be equally applicable in this case.

Again, if B, fig.4., be a magnet moving in the direction BC with a certain quantity of motion, and while it is so moving a mass of iron be placed at rest at A, the attraction of the magnet will draw the iron after it towards C, and will thus communicate to the iron a certain quantity of motion in the direction of C. All the motion thus communicated to the iron A must be lost by the magnet B.

If the magnet and the iron were both placed quiescent at B and A, the attraction of the magnet would cause the iron to move from A towards B; but the magnet in this case not having any motion, cannot be literally said to transfer a motion to the iron. At the moment, however, when the iron begins to move from A towards B, the magnet will be observed to begin also to move from B towards A; and if the velocities of the two bodies be expressed by numbers, and respectively multiplied by the numbers expressing their masses, the quantities of motion thus obtained will be found to be exactly equal. We have already explained why a quantity of motion received in the direction BA, is equivalent to the same quantity lost in the direction AB. Hence it appears, that the magnet in receiving as much motion in the direction BA, as it gives in the direction AB, suffers an effect which is equivalent to losing as much motion directed towards C as it has communicated to the iron in the same direction.

In the same manner, if the body B had any property in virtue of which it might repel A, it would itself be repelled with the same quantity of motion. In a word, whatever be the manner in which the bodies may affect each other, whether by collision, traction, attraction, or repulsion, or by whatever other name the phenomenon may be designated, still it is an inevitable consequence, that any motion, in a given direction, which one of the bodies may receive, must be accompanied by a loss of motion in the same direction, and to the same amount, by the other body, or the acquisition of as much motion in the contrary direction; or, finally, by a loss in the same direction, and an acquisition of motion in the contrary direction, the combined amount of which is equal to the motion received by the former.

(69.) From the principle, that the force of a body in motion depends on the mass and the velocity, it follows, that any body, however small, may be made to move with the same force as any other body, however great, by giving to the smaller body a velocity which bears to that of the greater the same proportion as the mass of the greater bears to the mass of the smaller. Thus a feather, ten thousand of which would have the same weight as a cannon-ball, would move with the same force if it had ten thousand times the velocity; and in such a case, these two bodies encountering in opposite directions, would mutually destroy each other’s motion.

(70.) The consequences of the property of inertia, which have been explained in the present and preceding chapters, have been given by Newton, in his Principia, and, after him, in most English treatises on mechanics, under the form of three propositions, which are called the “laws of motion.” They are as follow:—

I.

“Every body must persevere in its state of rest, or of uniform motion in a straight line, unless it be compelled to change that state by forces impressed upon it.”

II.

“Every change of motion must be proportional to the impressed force, and must be in the direction of that straight line in which the force is impressed.”

III.

“Action must always be equal and contrary to reaction; or the actions of two bodies upon each other must be equal, and directed towards contrary sides.”

When inertia and force are defined, the first law becomes an identical proposition. The second law cannot be rendered perfectly intelligible until the student has read the chapter on the composition and resolution of forces, for, in fact, it is intended as an expression of the whole body of results in that chapter. The third law has been explained in the present chapter, as far as it can be rendered intelligible in the present stage of our progress.

We have noticed these formularies more from a respect for the authorities by which they have been proposed and adopted, than from any persuasion of their utility. Their full import cannot be comprehended until nearly the whole of elementary mechanics has been acquired, and then all such summaries become useless.

(71.) The consequences deduced from the consideration of the quality of inertia in this chapter, will account for many effects which fall under our notice daily, and with which we have become so familiar, that they have almost ceased to excite curiosity. One of the facts of which we have most frequent practical illustration is, that the quantity of motion or moving force, as it is sometimes called, is estimated by the velocity of the motion, and the weight or mass of the thing moved conjointly.

If the same force impel two balls, one of one pound weight, and the other of two pounds, it follows, since the balls can neither give force to themselves, nor resist that which is impressed upon them, that they will move with the same force. But the lighter ball will move with twice the speed of the heavier. The impressed force which is manifested by giving velocity to a double mass in the one, is engaged in giving a double velocity to the other.

If a cannon-ball were forty times the weight of a musket-ball, but the musket-ball moved with forty times the velocity of the cannon-ball, both would strike any obstacle with the same force, and would overcome the same resistance; for the one would acquire from its velocity as much force as the other derives from its weight.

A very small velocity may be accompanied by enormous force, if the mass which is moved with that velocity be proportionally great. A large ship, floating near the pier wall, may approach it with so small a velocity as to be scarcely perceptible, and yet the force will be so great as to crush a small boat.

A grain of shot flung from the hand, and striking the person, will occasion no pain, and indeed will scarcely be felt, while a block of stone having the same velocity would occasion death.

If a body in motion strike a body at rest, the striking body must sustain as great a shock from the collision as if it had been at rest, and struck by the other body with the same force. For the loss of force which it sustains in the one direction, is an effect of the same kind as if, being at rest, it had received as much force in the opposite direction. If a man, walking rapidly or running, encounters another standing still, he suffers as much from the collision as the man against whom he strikes.

If a leaden bullet be discharged against a plank of hard wood, it will be found that the round shape of the ball is destroyed, and that it has itself suffered a force by the impact, which is equivalent to the effect which it produces upon the plank.

When two bodies moving in opposite directions meet, each body sustains as great a shock as if, being at rest, it had been struck by the other body with the united forces of the two. Thus, if two equal balls, moving at the rate of ten feet in a second, meet, each will be struck with the same force as if, being at rest, the other had moved against it at the rate of twenty feet in a second. In this case one part of the shock sustained arises from the loss of force in one direction, and another from the reception of force in the opposite direction.

For this reason, two persons walking in opposite directions receive from their encounter a more violent shock than might be expected. If they be of nearly equal weight, and one be walking at the rate of three and the other four miles an hour, each sustains the same shock as if he had been at rest, and struck by the other running at the rate of seven miles an hour.

This principle accounts for the destructive effects arising from ships running foul of each other at sea. If two ships of 500 tons burden encounter each other, sailing at ten knots an hour, each sustains the shock which, being at rest, it would receive from a vessel of 1000 tons burden sailing ten knots an hour.

It is a mistake to suppose, that when a large and small body encounter, the small body suffers a greater shock than the large one. The shock which they sustain must be the same; but the large body may be better able to bear it.

When the fist of a pugilist strikes the body of his antagonist, it sustains as great a shock as it gives; but the fist being more fitted to endure the blow, the injury and pain are inflicted on his opponent. This is not the case, however, when fist meets fist. Then the parts in collision are equally sensitive and vulnerable, and the effect is aggravated by both having approached each other with great force. The effect of the blow is the same as if one fist, being held at rest, were struck by the other with the combined force of both.