_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.”_
[Illustration: Two men on the ice, falling in opposite directions.]
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