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 }
------------+ -----------