Motion--absolute and relative.--Uniform, accelerated, and retarded Velocity.--The times of Ascent and Descent are equal.--Vis inertiÆ.--Friction.--Action and Reaction are equal and in opposite directions.--Momentum defined and explained.--The Three Great Laws of Motion.

“The table-cloth is removed,” cried Tom, as he cast a sly glance through the open window of the dining-room.

“It is, my boy,” replied Mr. Twaddleton; “Diffugere nives, as the poet has it.”

“Et redeunt jam gramina campis,” added Mr. Seymour, archly, as he pointed to the green cloth with which the table was covered.

“Et decrescentia flumina,” continued the vicar, with a smile; “but, psha! enough of wine and quotation. Come, let us join the children.”

Mr. Twaddleton, accompanied by Mr. and Mrs. Seymour, and Louisa, rose from the table, and proceeded to the lawn.

“The gravitation of Tom’s ball,” said Mr. Seymour, “furnished an ample subject for our morning’s diversion; let us try whether its other motions will not suggest further objects of enquiry.”

“I well remember,” observed Louisa, “that Mrs. Marcet extols that apple, the fall of which attracted the notice of Sir Isaac Newton, above all the apples that have ever been sung by the poets: and she declares, that the apple presented to Venus by Paris; the golden apples by which Atalanta won the race; nay, even the apple which William Tell shot from the head of his own son, cannot be brought into comparison with it.”

“Well said! Mrs. Marcet,” exclaimed Mr. Seymour; “upon my word, had the mother of mankind used but half such eloquence in praise of an apple, we cannot wonder at its influence.”

“What honours, then,” continued Louisa, “shall we decree to Tom’s ball, if it instructs us in the first principles of philosophy?”

“We are trifling,” observed Mr. Seymour, and so saying, he took the ball from Tom’s hand, and rolling it along the ground, exclaimed, “there it goes, performing, as you may perceive, two different kinds of motion at the same time; it turns round, or revolves on its axis; and goes straight forward, or, to speak more philosophically, performs a rectilinear motion.”

Tom said that he did not exactly comprehend what was meant by the axis.(7) His father, therefore, informed him that the axis of a revolving body was an imaginary line, which was itself at rest, but about which all its other parts turned, or rotated: “But,” continued he, “can you tell me whether you understand what is meant by the word motion?”

“If he can,” exclaimed the vicar, “he is a cleverer fellow than the wisest philosopher of antiquity, who, upon being asked the very same question, is said to have walked across the room, and to have replied, ‘You see it, but what it is I cannot tell you.’”

“Your ancient acquaintances,” observed Mr. Seymour, “entertained some very strange notions touching this said subject of motion. If I remember right, Diodorus denied its very existence; but we are told that he did not himself remain unmoved, when he dislocated his shoulder, and the surgeon kept him in torture while he endeavoured to convince him, by his own mode of reasoning, that the bone could not have moved out of its place: we have, however, at present, nothing to do with the ancients; the philosophers of our own times agree in defining motion to be ‘the act of a body changing its situation with regard to any other;’ and you will therefore readily perceive, that this may actually happen to a body while it remains absolutely at rest.”

“Well, that beats all the paradoxes I ever heard,” cried Tom; “a body then may be in motion, while it is at rest?”

“Certainly,” replied Mr. Seymour; “it may be relatively in motion, while it is absolutely at rest.”

“How can a body change its place,” said Louisa, “except by moving?”

“Very readily,” answered her father; “it may have its relative situation changed with respect to surrounding objects; there is your ball, and here is a stone, has not each of them a particular situation with respect to the other; and by moving one, do I not change the relative situation of both?”

“I perceive your meaning,” said Tom.

“To prevent confusion, therefore, in our ideas, it became necessary to distinguish these two kinds of motion from each other by appropriate terms; and, accordingly, where there has been an actual change of place, in the common meaning of the term, the motion which produced it is termed ABSOLUTE motion; whereas, on the contrary, when the situation has been only relatively changed, by an alteration in the position of surrounding bodies, the motion is said to be RELATIVE.”

“Surely, papa,” said Louisa, “no person can ever mistake relative for absolute motion; what then is the use of such frivolous distinctions? When a body really moves, we can observe it in the act of changing its place, and no difficulty can arise about the matter.”

“Nothing, my dear, is more fallacious than our vision; the earth appears motionless, and the sun and stars seem as if they revolved round it; but it is scarcely necessary for me to inform you that our globe is constantly moving with considerable velocity, while the sun remains at rest.--Mr. Sadler, the famous aËronaut,” continued Mr. Seymour, “informed me, that he was never sensible of the motion of the balloon in any of his excursions, but that, as he ascended into the air, the earth always appeared as if sinking beneath him, and as he descended, as if rising to meet him.”

Mr. Twaddleton here observed, that he had heard a very curious anecdote, when he was last in London, which fully confirmed the truth of Mr. Sadler’s statement. “An aËronaut,” said he, “whose name I cannot at this moment recollect, had recently published a map of his voyage, and, instead of proceeding in any one line of direction, his track absolutely appeared in the form of circles, connected with each other like the links of a chain: this occasioned considerable astonishment, and, of course, some speculation, until it was at length discovered, that his apparent journey was to be attributed to the rotatory motion of the balloon, which the voyager, not feeling, had never suspected.”

“And what,” asked Tom, “could have been the reason of his not having felt the motion?”

His father explained to him, that we are only conscious of being in motion when the conveyance, in which we are placed, suffers some impediment in its progress. “If,” said he, “you were to close your eyes, when sailing on calm water, with a steady breeze, you would not perceive that you were moving: for you could not feel the motion, and you could only see it by observing the change of place in the different objects on the shore; and then it would be almost impossible, without the aid of reason and experience, to believe that the shore itself was not in motion, and that you were at rest; I shall, however, be able to explain this subject more clearly by an optical toy which I have in preparation.”

Mrs. Seymour here repeated the following passage from that interesting novel “Anastasius,” which she observed was beautifully descriptive of the illusive appearance to which their papa had just referred:--

“The gradually increasing breeze carried us rapidly out of the Straits of Chio. The different objects on the shore,--mountains,--valleys,--villages,--and steeples,--seemed in swift succession, first advancing to meet us, then halting an instant alongside our vessel, as if to greet us on our passage, and, lastly, again gliding off with equal speed; till, launched into the open main, we saw the whole line of coast gradually dissolve in distant darkness.”

“That is indeed a beautiful and very apposite illustration,” said Mr. Seymour; “and I think Louisa will now admit, that it is not quite so easy, as she at first imagined, to distinguish between Absolute and Relative motion.”

As the children now understood what was meant by the term Motion, their father asked them whether they could tell him what produced it.

“I can make a body move by various means,” answered Tom.

“But they may all be reduced to one,” said Mr. Seymour; “viz. some exertion which is called Force; thus the force of my hand put your ball in motion; while gravitation was the force which made it fall to the earth; and I must, moreover, inform you, that a body always moves in the direction of the force which impels it, and with a velocity, or rate of motion, which is proportional to its degree, or strength; and, were there no other forces in action but that which originally produced the motion, the body would proceed onwards in a right line, and with a uniform velocity for ever.”

“For ever!” exclaimed Louisa.

“Ay, my dear, for ever: but we will discuss that question presently; you must first tell me whether you understand what is meant by uniform velocity.”

“I suppose that uniform velocity is that which is regular, and of an equal rate throughout.”

“Philosophers,” replied her father, “call the motion of a body uniform, when it passes over equal spaces in equal times.--Now, Tom, it is your turn to answer a question. Can you describe the meaning of the terms Accelerated and Retarded motion?”

“I conclude that motion is said to be accelerated when it moves every moment quicker and quicker; and to be retarded when it moves slower and slower.”

“You are perfectly right; and gravity may either act in occasioning the one or the other; our experiment at the well this morning afforded you an example of gravity producing a regularly accelerated motion. I did not fully explain the fact at the time, because I was desirous of avoiding too many new ideas at once; we must win our way slowly and cautiously through the mazes of philosophy: I will, however, now endeavour to give you as clear an explanation as the subject will allow.--It is, I think, evident, that if, at the moment you dropped the stone from your hand, the force of gravity could have been suspended, it would have descended to the bottom of the well with a uniform velocity; because there could have been nothing either to accelerate or retard its motion. But this was not the case, for the power of gravity was in constant operation; and, if you keep this fact in mind, you will readily understand how the velocity became accelerated: for, suppose the impulse given by gravity to the stone, during the first instant of its descent, be equal to one, the next instant we shall find that an additional impulse gives the stone an additional velocity equal to one, so that the accumulated velocity is now equal to two; the following instant, again, increases the velocity to three, and so on till the stone reaches the bottom.”

Mr. Twaddleton observed, the fact might be shortly expressed by saying, that “the effects of preceding impulses must be added to subsequent velocities.”

Mr. Seymour then remarked that the same explanation would apply to retarded velocity. “If,” said he, “you throw a stone perpendicularly upwards, the velocity will be as much retarded, as it was in the other case accelerated, by gravity; the consequence of which is, that it will be exactly the same length of time ascending that it was descending.”

“I should have thought the very reverse,” cried Louisa, “and that it would have fallen quicker than it rose.”

“You have forgotten to take into account the force with which the stone is projected upwards, and which is destroyed by gravity before it begins to descend.”

“Certainly,” answered Louisa; “but the force given to a stone in throwing it upwards, cannot always be equal to the force of gravity in bringing it down again; for the force of gravity is always the same, while the force given to the stone is entirely optional. I may throw it up gently or otherwise, as I please.”

“If you throw it gently,” said her father, “it will not rise high, and gravity will soon bring it down again; if you throw it with violence, it will rise much higher, and gravity will be longer in bringing it back again to the ground. Suppose, for instance, that you throw it with a force that will make it rise only sixteen feet; in that case, you know, it will fall in one second of time. Now it is proved by experiment, that an impulse requisite to project a body sixteen feet upwards, will make it ascend that height in one second of time; here, then, the times of ascent and descent are equal. But, supposing it be required to throw a stone twice that height, the force must be proportionally greater. You see, then, that the impulse of projection, in throwing a body upwards, is always equal to the action of the force of gravity during its descent; and that it is the greater or less distance to which the body rises that makes these two forces balance each other.”

“Thank you, dear papa, for the pains you have taken in explaining this subject to us.”

“Nay,” replied Mr. Seymour, “bestow your thanks upon those to whom they are more justly due; Mrs. Marcet is entitled to the merit of this explanation; for I obtained it from her ‘Conversations.’ Before I quit this subject, I would just observe that, when we come to the consideration of the bow and arrow, you will, by the application of the law I have endeavoured to expound, be enabled to ascertain the height to which your arrow may ascend, with the same facility as you discovered the depth of the well: for, since the times of ascent and descent are equal, you have only to determine the number of seconds which intervene between the instant at which the arrow quits the bow to that at which it falls to the ground, and halving them, to make the usual calculation.--But let us proceed to another subject. Roll the ball hither, Tom; roll the ball hither, I say! you stand as if you thought it would advance to us of its own accord.”

“I know a little better than that, too,” cried Tom; “no body can move without the application of some force.”

“Nor stop, either,” added Mr. Seymour, “when it is once in motion; for matter is equally indifferent to both rest and motion.”

“And yet, papa,” cried Louisa, “unfortunately for your assertion, the ball stopped just now, and I am sure that no force was used to make it do so.”

“And pray, Miss Pert, why are you so sure that no force was opposed to its progress? I begin to fear that my lesson has been thrown away upon you, or you would not, surely, have concluded so falsely.”

The vicar here interposed, observing that, simple as the question might appear to those who had studied it, the fact was so contrary to everything that passed before us, that Mr. Seymour ought not to feel any surprise at the scepticism of his daughter; he begged to remind him that the truth, apparent as it doubtless now was, lay hid for ages before the sagacity of Galileo brought it to light.

Mr. Seymour admitted the justice of this remark, and proceeded in his explanation.

“I think,” said he “you will readily allow that matter cannot, in itself, possess any power of changing its condition: it can, therefore, no more destroy, than it can originate its own motion; when it is at rest, it must ever remain so, unless some force be applied that can impart to it activity; and, when once in motion, it must continue to move until some counteracting force stops it. To believe otherwise you must suppose that matter possesses in itself a power to alter its condition, which is perfectly absurd.”

“And yet,” said Tom, “when I see my ball or marble stop of its own accord, how can you blame me for believing it possible?”

“Your difficulty arises from your ignorance of the existence of certain forces which act upon the rolling ball or marble. Its progress, as it rolls along, is impeded and ultimately stopped by the rubbing, or friction, occasioned by its passage over the ground; and this will be greater or less, according to the degree of roughness of the surface; if it be small, the ball will continue for a longer time in motion; you must have observed, that your marble has always rolled much farther on a smooth pavement than on a rough gravel walk.”

“Certainly,” said Tom, “and I well remember, that when we played at ring-taw last winter on the ice, we were obliged, for this very reason, to extend the usual boundaries.”

“Is it not evident, then, that the motion of a body is stopped by some opposing force; and that, if this could be entirely removed, the body would continue to move for ever?”

“What a provoking thing this friction is,” said Tom, “it is always interfering with our experiments.”

“Provoking, is it? I fancy,” said Mr. Seymour, “that you would be much more provoked by the loss of it: without it, you could not walk, nor even hold an object in your hands; and yet everything around you would be in perpetual motion, performing one universal and interminable dance.”

“I can readily understand, from what you have said, that, if friction were removed, motion might continue; but pray how is it that we should be unable to walk, or to hold anything in our hands?” enquired Louisa.

“It is the friction of the ground which, at every step we take, prevents the foot from sliding back, and thus enables us to push the body forwards. Everybody must have felt how difficult it is to walk on ice, where the friction is only diminished, not entirely removed,” answered her father; “and as to holding any object,” continued he, “it is the friction of the body to which we apply our hands that enables us to hold it firmly.”

“To be sure,” exclaimed the vicar; “why, my boy, you must surely remember, that in ancient combats it was the custom to rub the body with oil, that the adversary might not be able to keep his grasp.”

“Well,” said Tom, “our houses, I suppose, would remain firm, and we might sit quietly in our chairs, at all events.”

“Not so,” replied Mr. Seymour, “for even granting that you had houses and chairs, which, without the existence of friction would never exist, the stability of the structures could never be secured; the slightest breath would be sufficient to make the stones or bricks slide off from each other, and to reduce your dwellings into dancing ruins.”

Tom and Louisa, after some farther discussion, both admitted the justness of the argument; but, at the same time, would have been better satisfied if the fact could have been proved by actual experiment. Mr. Seymour told them that the perpetual revolution of the earth and heavenly bodies, where no friction whatever existed, afforded a proof which ought to satisfy them; and, especially, since it agreed with those views which were proved to be true by an examination of what took place on the surface of our own globe.

We will, therefore, with the permission of our readers, consider this point as settled, and proceed with the young philosophers to the investigation of some other topics connected with the doctrine of motion.

“Since a body at rest,” said Mr. Seymour, “can only be set in motion, or, when in motion, be brought to rest, by the impression of some force, it must follow, that it can only move in the direction in which such a force may act; and, moreover, that the degree of motion, or the velocity, must, other things being equal, be in proportion to the degree of force used.”

“Why, truly,” cried the vicar, “my young friends must of necessity admit that fact; for the body, not having any will of its own, as you say, must needs, if it move at all, go the road it is driven.”

“Yes,” added Mr. Seymour, “and it must go with a velocity in proportion to the force with which it is driven.”

“Doubtless, doubtless,” cried the vicar, “you admit that also; do you not, my young friends and playmates?”

It is hardly necessary to state, that the children instantly assented to these propositions. The vicar had placed them in so clear and popular a point of view, as to be intelligible to the lowest capacities.

“With these admissions, then, my dear children,” said their father, “I shall have but little difficulty in convincing you of the truth of the other laws by which the direction of moving bodies is governed. At present, however, it is not my intention to enter upon this subject; you have some preliminary knowledge to acquire before you can understand what is termed the Composition and Resolution of Forces.”

“I shall not easily forget,” said Louisa, “that matter is perfectly passive, and that it can neither put itself in motion when at rest, nor stop itself when in motion.”

“This indifference to rest or motion,” replied Mr. Seymour, “has been termed the Vis InertiÆ of matter.”

“A very objectionable term,--a very puzzling expression,” exclaimed the vicar;--“to denote a mere state of passive indifference by the term Vis, or power, does appear to me, who have been in the habit of connecting words with ideas, as excessively absurd.”

“I allow,” said Mr. Seymour, “that the simple word Inertia would have been more correct; but we are bound to receive an expression which has been long current. I suppose, however, you know that the addition of Vis originated with Kepler, who, like my boy Tom, could not help thinking that the disposition of a body to maintain its motion, or state of rest, indicated something like power; but we will not waste our time upon verbal disquisitions. It is clear, that matter, at rest, resists being put in motion; the degree of that resistance is always in proportion to the degree of force applied to put it in motion; or, to speak more philosophically, that Action and Reaction are equal, and in opposite directions.”

“You, surely, do not mean to say,” exclaimed Tom, “that if I strike my marble, the marble strikes my hand, with the same force in return?”

“Precisely; that is my meaning.”

“What!” cried Louisa, “if a man strikes another on the face with his hand, do you seriously maintain that both parties suffer the same pain?”

“Oh, no, no,” said Tom, “papa can never intend to say that; I am quite sure, if it were the case, Mr. Pearson would not be so fond of boxing our ears.”

Mr. Seymour answered this question, by observing that, if the hand possessed the same degree of feeling as the face, they would both suffer equally under the conflict. “If,” continued he, “you strike a glass bottle with an iron hammer, the blow will be received by the hammer and the glass; and it is quite immaterial whether the hammer be moved against the bottle at rest, or the bottle be moved against the hammer at rest, yet the bottle will be broken, though the hammer be not injured; because the same blow which is sufficient to shiver the glass is not sufficient to break or injure a lump of iron. In like manner, the blow that is sufficient to pain your sensitive face, and make your ears tingle, will not occasion the least annoyance to the obtuse hand of your preceptor. The operation of this law,” continued Mr. Seymour, “will be exemplified in every step of our progress. When the marble, as it rolls along, strikes any obstacles, it receives, in return, a corresponding blow, which will be found to influence its subsequent direction. The peg of the top, as it rubs on the ground, is as much influenced by the friction, as if a force were actually applied to it when in a state of rest; and when we consider the forces by which the kite is made to ascend into the air, you will learn, from the same law, the nature of that advantage which you derive from running with it.”

The vicar observed that the subject of Momentum might be introduced, and advantageously explained, upon this occasion.

“Momentum,” said Tom; “and pray what is that?”

“It is a power,” replied his father, “intimately connected with motion; and, therefore, as your friend, the vicar, justly remarks, may be very properly introduced before we quit that subject.--It is the force with which a body in motion strikes against another body.”

“That,” observed Tom, “must of course depend upon the velocity of the body’s motion.”

“Undoubtedly, my dear; the quicker a body moves, the greater must be the force with which it would strike against another body; but we also know that the heavier a body is, the greater also will be its force; so that momentum, you perceive, must have a relation to both these circumstances, viz. velocity, and weight; or, to speak more correctly, the momentum of a body is composed of its quantity of matter, multiplied by its quantity of motion: for example, if the weight of a body be represented by the number 3, and its velocity also by 3, its momentum will be represented by 3 x 3 = 9; so that, in producing momentum, increased velocity will always compensate for deficiency of matter, and a light body may thus be made a more effective force than a heavy one, provided that its velocity be proportionally increased; thus, a small ball weighing only two pounds, and moving at the rate of five hundred feet in a second, will produce as much effect as a cannon ball of ten pounds in weight, provided it moved only at the rate of one hundred feet in the same time.”

“Let me see,” cried Tom, “whether I understand your statement. We must multiply, as you say, the weight by the velocity; the weight of the small ball you state at two pounds, and it travels at the rate of five hundred feet in a second; then its momentum must be a thousand. The weight of the great ball is ten pounds, its velocity only a hundred feet, then its momentum must also be a thousand; because, in both cases, the sums multiplied into each other will give the same product.”

“Exactly: and thus you perceive that the small ball becomes an exact balance to the larger one; the first making out in motion what it wanted in matter, while the latter makes out in matter what it wanted in motion. I wish you to keep this law of Momentum in your remembrance; upon it depends the action of all the mechanical powers(8), as they are termed.”

“I have heard,” said Louisa, “that a feather might be made to produce as much havoc as a cannon shot, if you could give it sufficient velocity.”

“Unquestionably: but there is a practical difficulty in the attempt, from the resistance of the air, which increases, as you have already seen in the experiment of the paper and penny-piece (p. 30), as the weight of a body decreases. Were it not for this resistance of the air, a hailstone falling from the clouds would acquire such a momentum, from its accelerated velocity, as to descend like a bullet from a gun, and destroy every thing before it; even those genial showers which refresh us in the spring and summer months, would, without such a provision, destroy the herbage they are so well calculated to cherish. Had the elephant possessed the mobility of the beetle, it would have overturned mountains. From this view of the subject of Momentum,” continued Mr. Seymour, “you will easily understand why the immense battering rams, used by the ancients, in the art of war, should have given place to cannon balls, of but a few pounds in weight. Suppose, for example, that the battering ram of Vespasian weighed 100,000 pounds, and was moved, we will admit, with such a velocity, by strength of hands, as to pass through 20 feet in one second of time, and that this was found sufficient to demolish the walls of Jerusalem, can you tell me with what velocity a 32-pounder must move to do the same execution?”

“I will try,” said Tom, as he took out his pencil and pocket-book, to make the calculation. “The momentum of the battering ram must be estimated by its weight, multiplied into the space passed over in a second of time; which is 100,000 multiplied by 20; that will give 2,000,000. Now, if this momentum, which must also be that of the cannon ball, be divided by the weight of the ball, it will give the velocity required, which I make out to be 62,500 feet.”

“Admirably calculated,” said Mr. Seymour: “and I will take care, my dear Tom, that your intelligence shall be suitably rewarded.”

Mr. Twaddleton here observed, that he thought “his young friends and playmates” must have received, for that day, as much philosophy as they could conveniently carry away without fatigue. Mr. Seymour concurred in this observation; and the more readily, as the path they had to travel was rugged, and beset with difficulties. “I will, therefore,” said he, “not impose any farther burthen upon them; but I will assist them in tying, into separate bundles, the materials which they have collected in their progress, in order that they may convey them away with greater ease and security. Know then, my dear children,” said the affectionate parent, “that you have this day been instructed in the three great Laws of Motion, viz.

I. That every body will continue in a state of rest, until put into motion by some external force applied to it, and if that force be single, the motion so produced will be rectilinear, i.e. in the direction of a straight line.

II. Change of motion is always proportional to the moving force impressed, and is always made in the direction of the right line in which the force acts.

III. Action and Reaction are equal in equal quantities of matter, and act in contrary directions to each other.”