Chapter II: Part 2

That it is the air alone which excludes the water from the goblet, in the preceding experiments, can easily be proved. When the goblet is sunk deep in the vessel of water, let it be inclined a little to one side until its mouth is presented towards the side of the vessel; let this inclination be so regulated, that the surface of the water in the goblet shall just reach its edge. Upon a slight increase of inclination, air will be observed to escape from the goblet, and to rise in bubbles to the surface of the water. If the goblet be then restored to its position, it will be found that the cork will rise higher in it than before the escape of the air. The water in this case rises and fills the space which the air allowed to escape has deserted. The same process may be repeated until all the air has escaped, and then the goblet will be completely filled by the water.

(35.) Liquids are compressible by mechanical force in so slight a degree, that they are considered in all hydrostatical treatises as incompressible fluids. They are, however, not absolutely incompressible, but yield slightly to very intense pressure. The question of the compressibility of liquids was raised at a remote period in the history of science. Nearly two centuries ago, an experiment was instituted at the Academy _del Cimento_ in Florence, to ascertain whether water be compressible. With this view, a hollow ball of gold was filled with the liquid, and the aperture exactly and firmly closed. The globe was then submitted to a very severe pressure, by which its figure was slightly changed. Now it is proved in geometry, that a globe has this peculiar property, that any change whatever in its figure must necessarily diminish its volume or contents. Hence it was inferred, that if the water did not issue through the pores of the gold, or burst the globe, its compressibility would be established. The result of the experiment was, that the water _did_ ooze through the pores, and covered the surface of the globe, presenting the appearance of dew, or of steam cooled by the metal. But this experiment was inconclusive. It is quite true, that if the water _had not_ escaped upon the change of figure of the globe, the _compressibility_ of the liquid would have been established. The escape of the water does not, however, prove its _incompressibility_. To accomplish this, it would be necessary first to measure accurately the volume of water which transuded by compression, and next to measure the diminution of volume which the vessel suffered by its change of figure. If this diminution were greater than the volume of water which escaped, it would follow that the water remaining in the globe had been compressed, notwithstanding the escape of the remainder. But this could never be accomplished with the delicacy and exactitude necessary in such an experiment; and, consequently, as far as the question of the compressibility of water was concerned, nothing was proved. It forms, however, a very striking illustration of the porosity of so dense a substance as gold, and proves that its pores are larger than the elementary particles of water, since these are capable of passing through them.

(36.) It has since been proved, that water, and other liquids, are compressible. In the year 1761, Canton communicated to the Royal Society the results of some experiments which proved this fact. He provided a glass tube with a bulb, such as that described in (28), and filled the bulb and a part of the tube with water well purified from air. He then placed this in an apparatus called a condenser, by which he was enabled to submit the surface of the liquid in the tube to very intense pressure of condensed air. He found that the level of the liquid in the tube fell in a perceptible degree upon the application of the pressure. The same experiment established the fact, that liquids are _elastic_; for upon removing the pressure, the liquid rose to its original level, and therefore resumed its former dimensions.

(37.) Elasticity does not always accompany compressibility. If lead or iron be submitted to the hammer, it may be hardened and diminished in its volume; but it will not resume its former volume after each stroke of the hammer.

(38.) There are some bodies which maintain the state of density in which they are commonly found by the continual agency of mechanical pressure; and such bodies are endued with a quality, in virtue of which they would enlarge their dimensions without limit, if the pressure which confines them were removed. Such bodies are called _elastic fluids_ or _gases_, and always exist in the form of common air, in whose mechanical properties they participate. They are hence often called _aeriform fluids_.

Those who are provided with an air-pump can easily establish this property experimentally. Take a flaccid bladder, such as that already described in (27.), and place it under the glass receiver of an air-pump: by this instrument we shall be able to remove the air which surrounds the bladder under the receiver, so as to relieve the small quantity of air which is inclosed in the bladder from the pressure of the external air: when this is accomplished, the bladder will be observed to swell, as if it were inflated, and will be perfectly distended. The air contained in it, therefore, has a tendency to dilate, which takes effect when it ceases to be resisted by the pressure of surrounding air.

(39.) It has been stated that the increase or diminution of temperature is accompanied by an increase or diminution of volume. Related to this, there is another phenomenon too remarkable to pass unnoticed, although this is not the proper place to dwell upon it: it is the converse of the former; viz. that an increase or diminution of bulk is accompanied by a diminution or increase of temperature. As the application of heat from some foreign source produces an increase of dimensions, so if the dimensions be increased from any other cause, a corresponding portion of the heat which the body had before the enlargement, will be absorbed in the process, and the temperature will be thereby diminished. In the same way, since the abstraction of heat causes a diminution of volume, so if that diminution be caused by any other means, the body will _give out_ the heat which in the other case was abstracted, and will rise in its temperature.

Numerous and well-known facts illustrate these observations. A smith by hammering a piece of bar iron, and thereby compressing it, will render it _red hot_. When air is violently compressed, it becomes so hot as to ignite cotton and other substances. An ingenious instrument for producing a light for domestic uses has been constructed, consisting of a small cylinder, in which a solid piston moves air-tight: a little tinder, or dry sponge, is attached to the bottom of the piston, which is then violently forced into the cylinder: the air between the bottom of the cylinder and the piston becomes intensely compressed, and evolves so much heat as to light the tinder.

In all the cases where friction or percussion produces heat or fire, it is because they are means of compression. The effects of flints, of pieces of wood rubbed together, the warmth produced by friction on the flesh, are all to be attributed to the same cause.

CHAP. III.

INERTIA.

(40.) The quality of matter which is of all others the most important in mechanical investigations, is that which has been called _Inertia_.

Matter is incapable of spontaneous change. This is one of the earliest and most universal results of human observation: it is equivalent to stating that mere matter is deprived of life; for spontaneous action is the only test of the presence of the living principle. If we see a mass of matter undergo any change, we never seek for the cause of that change in the body itself; we look for some external cause producing it. This inability for voluntary change of state or qualities is a more general principle than inertia. At any given moment of time a body must be in one or other of two states, rest or motion. _Inertia_, or _inactivity_, signifies the total absence of power to change this state. A body endued with inertia cannot of itself, and independent of all external influence, commence to move from a state of rest; neither can it when moving arrest its progress and become quiescent.

(41.) The same property by which a body is unable by any power of its own to pass from a state of rest to one of motion, or _vice versâ_, also renders it incapable of increasing or diminishing any motion which it may have received from an external cause. If a body be moving in a certain direction at the rate of ten miles per hour, it cannot by any energy of its own change its rate of motion to eleven or nine miles an hour. This is a direct consequence of that manifestation of inertia which has just been explained. For the same power which would cause a body moving at ten miles an hour to increase its rate to eleven miles, would also cause the same body at rest to commence moving at the rate of one mile an hour; and the same power which would cause a body moving at the rate of ten miles an hour to move at the rate of nine miles in the hour, would cause the same body moving at the rate of one mile an hour to become quiescent. It therefore appears, that to increase or diminish the motion of a body is an effect of the same kind as to change the state of rest into that of motion, or _vice versâ_.

(42.) The effects and phenomena which hourly fall under our observation afford unnumbered examples of the inability of lifeless matter to put itself into motion, or to increase any motion which may have been communicated to it. But it does not happen that we have the same direct and frequent evidence of its inability to destroy or diminish any motion which it may have received. And hence it arises, that while no one will deny to matter the former effect of inertia, few will at first acknowledge the latter. Indeed, even so late as the time of KEPLER, philosophers themselves held it as a maxim, that “matter is more inclined to rest than to motion;” we ought not, therefore, to be surprised if in the present day those who have not been conversant with physical science are slow to believe that a body once put in motion would continue for ever to move with the same velocity, if it were not stopped by some external cause.

Reason, assisted by observation, will, however, soon dispel this illusion. Experience shows us in various ways, that the same causes which destroy motion in one direction are capable of producing as much motion in the opposite direction. Thus, if a wheel, spinning on its axis with a certain velocity, be stopped by a hand seizing one of the spokes, the effort which accomplishes this is exactly the same as, had the wheel been previously at rest, would have put it in motion in the opposite direction with the same velocity. If a carriage drawn by horses be in motion, the same exertion of power in the horses is necessary to stop it, as would be necessary to _back_ it, if it were at rest. Now, if this be admitted as a general principle, it must be evident that a body which can destroy or diminish its own motion must also be capable of putting itself into motion from a state of rest, or of increasing any motion which it has received. But this latter is contrary to all experience, and therefore we are compelled to admit that a body cannot diminish or destroy any motion which it has received.

Let us enquire why we are more disposed to admit the inability of matter to produce than to destroy motion in itself. We see most of those motions which take place around us on the surface of the earth subject to gradual decay, and if not renewed from time to time, at length cease. A stone rolled along the ground, a wheel revolving on its axis, the heaving of the deep after a storm, and all other motions produced in bodies by external causes, decay, when the exciting cause is suspended; and if that cause do not renew its action, they ultimately cease.

But is there no exciting cause, on the other hand, which thus gradually deprives those bodies of their motion?--and if that cause were removed, or its intensity diminished, would not the motion continue, or be more slowly retarded? When a stone is rolled along the ground, the inequalities of its shape as well as those of the ground are impediments, which retard and soon destroy its motion. Render the stone round, and the ground level, and the motion will be considerably prolonged. But still small asperities will remain on the stone, and on the surface over which it rolls: substitute for the stone a ball of highly-polished steel, moving on a highly-polished steel plane, truly level, and the motion will continue without sensible diminution for a very long period; but even here, and in every instance of motions produced by art, minute asperities must exist on the surfaces which move in contact with each other, which must resist, gradually diminish, and ultimately destroy the motion.

Independently of the obstructions to the continuation of motion arising from friction, there is another impediment to which all motions on the surface of the earth are liable--the resistance of the air. How much this may affect the continuation of motion appears by many familiar effects. On a calm day carry an open umbrella with its concave side presented in the direction in which you are moving, and a powerful resistance will be opposed to your progress, which will increase with every increase of the speed with which you move.

(43.) We are not, however, without direct experience to prove, that motions when unresisted will for ever continue. In the heavens we find an apparatus, which furnishes a sublime verification of this principle. There, removed from all casual obstructions and resistances, the vast bodies of the universe roll on in their appointed paths with unerring regularity, preserving without diminution all that motion which they received at their creation from the hand which launched them into space. This alone, unsupported by other reasons, would be sufficient to establish the quality of inertia; but viewed in connection with the other circumstances previously mentioned, no doubt can remain that this is an universal law of nature.

(44.) It has been proved, that inability to change the _quantity_ of motion is a consequence of _inertia_. The inability to change the _direction_ of motion is another consequence of this quality. The same cause which increases or diminishes motion, would also give motion to a body at rest; and therefore we infer that the same inability which prevents a body from moving itself, will also prevent it from increasing or diminishing any motion which it has received. In the same manner we can show, that any cause which changes the direction of motion would also give motion to a body at rest; and therefore if a body change the direction of its own motion, the same body might move itself from a state of rest; and therefore the power of changing the direction of any motion which it may have received is inconsistent with the quality of inertia.

(45.) If a body, moving from A, _fig. 3._ to B, receive at B a blow in the direction C B E, it will immediately change its direction to that of another line B D. The cause which produces this change of direction would have put the body in motion in the direction B E, had it been quiescent at B when it sustained the blow.

(46.) Again, suppose G H to be a hard plane surface; and let the body be supposed to be perfectly inelastic. When it strikes the surface at B, it will commence to move along it in the direction B H. This change of direction is produced by the resistance of the surface. If the body, instead of meeting the surface in the direction A B, had moved in the direction E B, perpendicular to it, all motion would have been destroyed, and the body reduced to a state of rest.

(47.) By the former example it appears that the deflecting cause would have put a quiescent body in motion, and by the latter it would have reduced a moving body to a state of rest. Hence the phenomenon of a change of direction is to be referred to the same class as the change from rest to motion, or from motion to rest. The quality of inertia is, therefore, inconsistent with any change in the direction of motion which does not arise from an external cause.

(48.) From all that has been here stated, we may infer generally, that an inanimate parcel of matter is incapable of changing its state of rest or motion; that, in whatever state it be, in that state it must for ever continue, unless disturbed by some external cause; that if it be in motion, that motion must always be _uniform_, or must proceed at the same rate, equal spaces being moved over in the same time: any increase of its rate must betray some impelling cause; any diminution must proceed from an impeding cause, and neither of these causes can exist in the body itself; that such motion must not only be constantly at the same uniform rate, but also must be always in the same direction, any deflection from one uniform direction necessarily arising from some external influence.

The language sometimes used to explain the property of inertia in popular works, is eminently calculated to mislead the student. The terms resistance and stubbornness to move are faulty in this respect. Inertia implies absolute passiveness, a perfect indifference to rest or motion. It implies as strongly the absence of all resistance to the reception of motion, as it does the absence of all power to move itself. The term _vis inertiæ_ or _force of inactivity_, so frequently used even by authors pretending to scientific accuracy, is still more reprehensible. It is a contradiction in terms; the term _inactivity_ implying the absence of all force.

* * * * *

(49.) Before we close this chapter, it may be advantageous to point out some practical and familiar examples of the general law of inertia. The student must, however, recollect, that the great object of science is generalisation, and that his mind is to be elevated to the contemplation of the _laws_ of nature, and to receive a habit the very reverse of that which disposes us to enjoy the descent from generals to particulars. Instances, taken from the occurrences of ordinary life, may, however, be useful in verifying the general law, and in impressing it upon the memory; and for this reason, we shall occasionally in the present treatise refer to such examples; always, however, keeping them in subservience to the general principles of which they are manifestations, and on which the attention of the student should never cease to be fixed.

(50.) If a carriage, a horse, or a boat, moving with speed, be suddenly retarded or stopped, by any cause which does not at the same time affect passengers, riders, or any loose bodies which are carried, they will be precipitated in the direction of the motion; because by reason of their inertia, they persevere in the motion which they shared in common with that which transported them, and are not deprived of that motion by the same cause.

(51.) If a passenger leap from a carriage in rapid motion, he will fall in the direction in which the carriage is moving at the moment his feet meet the ground; because his body, on quitting the vehicle, retains, by its inertia, the motion which it had in common with it. When he reaches the ground, this motion is destroyed by the resistance of the ground to the feet, but is retained in the upper and heavier part of the body; so that the same effect is produced as if the feet had been tripped.

(52.) When a carriage is once put in motion with a determinate speed on a level road, the only force necessary to sustain the motion is that which is sufficient to overcome the friction of the road; but at starting a greater expenditure of force is necessary, inasmuch as not only the friction is to be overcome, but the force with which the vehicle is intended to move must be communicated to it. Hence we see that horses make a much greater exertion at starting than subsequently, when the carriage is in motion; and we may also infer the inexpediency of attempting to start at full speed, especially with heavy carriages.

(53.) _Coursing_ owes all its interest to the instinctive consciousness of the nature of inertia which seems to govern the measures of the hare. The greyhound is a comparatively heavy body moving at the same or greater speed in pursuit. The hare _doubles_, that is, suddenly changes the direction of her course, and turns back at an oblique angle with the direction in which she had been running. The greyhound, unable to resist the tendency of its body to persevere in the rapid motion it had acquired, is urged forward many yards before it is able to check its speed and return to the pursuit. Meanwhile the hare is gaining ground in the other direction, so that the animals are at a very considerable distance asunder when the pursuit is recommenced. In this way a hare, though much less fleet than a greyhound, will often escape it.

In racing, the horses shoot far beyond the winning-post before their course can be arrested.

CHAP. IV.

ACTION AND REACTION.

(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 } 8 × 17[1] or 136 | Quantity of } 8 × 14 or 112
motion of A } | motion of A }
-----------------+ --------------
Mass of B 6 | Mass of B 6
Velocity of B 10 | Common velocity 14
-----------------+ --------------
Quantity of } 6 × 10 or 60 | Quantity of } 6 × 14 = 84
motion of B } | motion of B }
-----------------+ --------------

* 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 A C 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 1 lb. 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 1 lb. 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 A C, 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 B C, 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 B C. The bodies will then move together in the direction C B, 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 A C. On the other hand, B loses the entire of its 6 parts of motion in the direction B C, and receives 2 parts in the direction A C. This is equivalent to receiving 8 parts of A’s motion in the direction A C. 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 A C, 30 being transferred to B, will destroy all its 30 parts of motion in the direction B C, and the two masses will move in the direction C B, 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 } 8 × 9 or 72 | Quantity of motion } 8 × 3 or 24
in direction A C } | in direction A C }
------------+ -----------
Mass of B 6 | Mass of B 6
Velocity of B 5 | Common velocity 3
------------+ -----------
Quantity of motion } 6 × 5 or 30 | Quantity of motion } 6 × 3 or 18
in direction B C } | in direction A C }
------------+ -----------

Hence it appears that the quantity of motion in the direction A C 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 B C, which is equivalent to receiving 30 in the direction A C. But it also acquires a quantity 18 in the direction A C, which, added to the former 30, gives a total of 48 received by B in the direction A C. Thus the same quantity of motion which A loses in the direction A C, 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 B C 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 B A, is equivalent to the same quantity lost in the direction A B. Hence it appears, that the magnet in receiving as much motion in the direction B A, as it gives in the direction A B, 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.