is due to the resistance of that medium, whence we may fairly conclude,
that if the air was altogether absent, and no other resisting medium
occupied its place, all bodies of whatever size, and of whatever weight,
must descend with the same speed. Under such circumstances, a balloon
and the smoke of the fire would descend, instead of ascending as they
do, by the pressure of the air, which, bulk for bulk, is heavier than
themselves. A dry leaf falls very slowly, and a piece of gold very
rapidly, but if the gold be beaten into a thin leaf, the time of its
descent is greatly prolonged. If a piece of metal and a feather are let
fall at the same instant from the top of a tall exhausted receiver, it
will be found that these two bodies, so dissimilar in weight, will
strike the table of the air-pump, on which the receiver stands, at the
same instant. Supposing the air did not offer any resistance to the
onward course of a projectile, and that the instantaneous force
communicated to a bullet, from the explosion of the gunpowder, were to
project it in the line A.B. (plate 21, fig. 4.) from the point A., with
a velocity that will send it in the first second of time as far as C.,
then if there were no other force to affect it, it would continue to
move in the same direction B., and with the same velocity, and at the
next second it would have passed over another space, C.D., equal to
A.C., so that in the third second it would have reached E., keeping
constantly in the same straight line.
~Bullet under two forces, powder and gravity.~
But no sooner does the bullet quit the muzzle, than it immediately comes
under the influence of another force, called the force of gravity, which
differs from the force caused by the explosion of the powder, which
ceases to influence the bullet, after it has once communicated to it its
velocity.
~An accelerating force.~
~Effect of gravity.~
Gravity is an accelerating force, acting constantly upon, and causing
the bullet to move towards the earth, with a velocity increasing with
the length of time the bullet is exposed to its influence. It has been
found from experiment that this increase of velocity will cause a body
to move through spaces, in proportion to the squares of the time taken
to pass over the distance. Thus, if a body falls a given space in one
second, in two it will have fallen over a space equal to four times what
it fell through in the first second, and in the three first seconds it
will have fallen through a space equal to nine times that which it fell
through in the first second.
~Result of gravity.~
~Course of the bullet.~
The consequence of this principle is, that all bodies of similar figure,
and equal density, at equal distances from the earth, fall with equal
velocity; and if a body describes a space of 16ft. in the first second
of time, it will, in the next second of time, fall _three_ times 16, or
48 feet, and thus will have fallen, from the time it first dropped, four
times 16 feet, or 64 feet, because 4 is the square of 2, the time the
body was falling. In the third second, it will fall 5 times 16 feet, or
80 feet, and these sums collectively, viz., 16 + 48 + 80 = 144 feet, the
whole distance described by the falling body in three seconds of time.
From this it is evident, that instead of moving in a straight line A.
B., (plate 21, fig. 5.), the bullet will be drawn from that course.
~Parabolic theory.~
From the point C., draw C. F., equal to the space that the bullet may be
supposed to fall in one second of time, then at the end of the first
second of time the bullet will be at F., instead of at C., and will have
moved in the direction A. F., instead of A. C.; at the end of the next
second it will have fallen a total distance D. G., equal to four times
C. F., thus the bullet will have fallen at the end of the third second a
distance E. H., equal to nine times C. F., and it will have moved in the
line A. F. G. H. instead of the straight line A. B., in which it would
have moved, had it not been affected by the force of gravity. The curve
A. H., is of the form called a Parabola, and hence the theory is called
the “Parabolic Theory.” It is founded on the principle that the velocity
given to the bullet by the explosion of the gunpowder is continued
throughout its course, but this would only be true in vacuo, and is
therefore of little value in calculating the real course of the bullet
in the air.
ON THE TIME TAKEN TO DRAW A BALL TO THE GROUND BY THE FORCE OF GRAVITY.
~If fired with axis parallel to the ground.~
1st Case. Supposing a ball to be fired when the axis of the piece is
parallel to the ground and 16 feet above it, then the projectile will
strike the earth in the same length of time that it would have done, had
it been rolled out of the muzzle, quite irrespective of the velocity
with which it may have been propelled, or the consequent extent of
range; that is to say the ball will have reached the point B., (plate
22, fig. 1.), in the same length of time that it would require to fall
from the muzzle A., to the earth C.; _i. e._, in one second.
2nd Case. Were three guns to be fired at the same instant, with their
three axes parallel to the horizon as before, and loaded respectively
with ¹⁄₂ drm., 1 drm., and 1¹⁄₂ drm. of powder of the same strength,
then, although the three initial velocities and three ranges would
consequently all be different, yet the three balls would strike the
ground at the same time, _i. e._ at the points B. B. B. in one second.
(Plate 22, fig. 2.)
~If axis at an angle to the ground.~
3rd Case. When a ball is fired at an angle of elevation it will reach
the earth in the same length of time which it would occupy in falling
the length of the tangent of the angle of projection; hence supposing F.
G. (plate 22, fig. 3.) to be 16 feet, the ball would reach the point G.
in one second, irrespective of the distance from D. to G.
ATMOSPHERE.
Let us now take into our consideration the course of a projectile while
under the influence of _three_ forces, viz., powder, gravity, and air.
~Why named.~
The atmosphere, or sphere of gases, is the general name applied to the
whole gaseous portion of this planet, as the term ocean is applied to
its liquid, and land to its solid portions.
Being much lighter than either land or water, it necessarily floats or
rests upon them, and is in sufficient quantity to cover the highest
mountains, and to rise nine or ten times their height, to about 45 miles
above the sea level, so as to form a layer over the whole surface,
averaging probably between forty and fifty miles in thickness, which is
about as thick, in proportion to the globe, as the liquid layer adhering
to the surface of an orange, after it had been dipped in water.
~Composition of air.~
It consists essentially of two gases, called oxygen and nitrogen, and
also contains a variable quantity of aqueous vapour.
~Qualities of air.~
In common with matter in every state, the air possesses impenetrability.
It can be compressed, but cannot be annihilated. It has weight, inertia,
momentum, and elasticity.
In consequence of its weight is its pressure, which acts uniformly on
all bodies, and is equal to between 14lbs. and 15lbs. on every square
inch of surface at the sea-level.
~Early idea of air’s resistance.~
~How air acts.~
The first experiments that were made on projectiles, were carried out on
the idea that the resistance of the air would not materially affect the
track of a bullet which had great velocity. But the moment a body is
launched into space, it meets with particles of the air at every instant
of its movement, to which it yields part of its velocity, and the air
being a constant force, the velocity of the body decreases at every
instant from the commencement of its motion.
RESULT OF THE AIR’S RESISTANCE.
~Robins, 1742, showed effect of air’s resistance.~
~Course of ball was not a parabola.~
~Why not a parabola.~
It remained for Robins, 1742, in a work then published, to show the real
effect of the atmosphere upon moving bodies. He proved by actual
experiment, that a 24lb. shot did not range the fifth part of the
distance it should have done according to the parabolic theory. If a
cannon shot moved in a parabolic curve, then from the known properties
of that curve, it was evident that when fired with elevation, the angle
of descent of the bullet should have been the same as the angle at which
it was projected, and this he showed was not the case in practice. Now
Robins acknowledged the opinion of Galileo, as regards the force of
gravity, to be correct; he could not therefore attribute to him any
miscalculation on the score of gravity. He therefore concluded, that the
error of the “parabolic theory” arose from the supposition that the
bullet continued to move at the same velocity throughout its course.
~Ballistic pendulum.~
Robins tried a series of experiments by firing at a ballistic pendulum
at different distances; the oscillation of this pendulum enabled him to
calculate the velocity of the bullet, at the time it struck the
pendulum, and by this means he ascertained, that according to his
expectations, the bullet moved slower in proportion as it became more
distant from the point at which it was fired. This diminution he
attributed to the resistance of the air.
~Trajectory more curved than a parabola.~
From these considerations it is evident that instead of moving over
equal spaces A. C., C. D., D. E., (plate 22, fig. 4), at each succeeding
second of time, it will require considerably longer to traverse each
succeeding distance, and the force of gravity will consequently have
longer time to act upon it, and will have the effect of lowering the
bullet much more than it would do according to the “parabolic theory;”
moreover it is evident, that as the velocity of the bullet diminishes,
the trajectory or path followed by the bullet, will become still more
incurvated.
Having now proved the error of the “parabolic theory,” Robins began his
endeavours to calculate the actual course of the bullet, according to
this new theory which he had demonstrated, but this calculation was
necessarily attended with great difficulties, for in so doing a number
of circumstances had to be considered.
~Resultant.~
The resultant of the three forces acting on a projectile, (plate 23,
fig. 1), viz., gunpowder, gravity, and the resistance of the air, is a
motal force, diminishing in velocity at every instant, causing the
projectile to describe a curved line in its flight, the incipient point
of the curve lying in the axis of the bore of the piece, and its
continuation diverging in the direction of the attraction of gravity,
till the projectile obeys the latter force alone.
EXPERIMENTS IN FRANCE.
~Angle for greatest range.~
~Velocity.~
It is stated by Captain Jervis, R.A., in the “Rifle Musket,” that “From
experiments made in France, it has been found that the greatest range of
the common percussion musket, with spherical bullet fired with the
regulation charge, was at 25°; yet, by theoretical calculation, it
should be 45°; also that the usual velocity was some 500 yards per
second, whilst in vacuum it would be 19,792 yards per second.
~Elevation giving certain range.~
“At an angle of from 4° to 5°, the real range was about 640 yards;
without the resistance of the air, and at an angle of 4¹⁄₂°, it would be
3,674, or six times greater.”
ON THE EFFECT OF THE RESISTANCE OF THE AIR UPON THE MOTION OF A
PROJECTILE.
~The effect of the air’s resistance upon the motion of a projectile.~
The effect of the resistance of the atmosphere to the motion of a
projectile, is a subject of the greatest importance in gunnery. It has
engaged the attention of the most eminent philosophers, and on account
of the great difficulty of determining by experiment, the correctness of
any particular hypothesis, much difference of opinion is entertained as
to the absolute effect of this retarding force upon bodies moving in the
atmosphere with great velocities; and although sufficient is known to
guide the practical artillerist in that art to which he is devoted,
still as a scientific question, it is one of considerable interest, but
more on account of the difficulty of its solution, than from its
practical importance.
~Mr. Robins’ discoveries.~
To our distinguished countryman, Mr. Benjamin Robins, is due the credit
of not only being the first practically to determine the enormous effect
of the resistance of the air in retarding the motions of military
projectiles, but also of pointing out and experimentally proving other
facts with regard to this resistance, which will be noticed when
considering the subject of the deviation of shot from the intended
direction.
~Result of Dr. Hutton’s experiments.~
After him, Dr. Hutton made a great number of experiments upon the same
point, viz., the effect of the resistance of the air upon bodies moving
in that medium, both with great and small velocities; and the inferences
which he drew from these experiments, although not absolutely true, are
sufficiently correct for all practical purposes.
ON THE RESISTANCE OF A FLUID TO A BODY IN MOTION.
~Circumstances affecting the resistance which a body meets with in its
motion in a fluid.~
The resistance which a body meets with in its motion through a fluid
will depend upon three principal causes, viz:--
1st. Its velocity, and the form and magnitude of the surface opposed to
the fluid.
2nd. Upon the density and tenacity of the fluid or cohesion of its
particles, and also upon the friction which will be caused by the
roughness of the surface of the body.
3rd. Upon the degree of compression to which this fluid, supposed to be
perfectly elastic, is subjected, upon which will depend the rapidity
with which it will close in and fill the space behind the body in
motion.
~The resistance of a fluid to a body as the squares of the
velocities.~
Firstly, with regard to the velocity of the body. It is evident that a
plane moving through a fluid in a direction perpendicular to its
surface, must impart to the particles of the fluid with which it comes
in contact, a velocity equal to its own; and, consequently, from this
cause alone, the resistances would be as the velocities; but the number
of particles struck in a certain time being also as the velocities, from
these two causes combined, the resistance of a fluid to a body in
motion, arising from the inertia of the particles of the fluid, will be
as the square of the velocity.
~Cohesion of the particles of a fluid, and friction.~
Secondly, a body moving in a fluid must overcome the force of cohesion
of those parts which are separated, and the friction, both which are
independent of the velocity. The total resistance then, from cohesion,
friction, and inertia, will be partly constant and partly as the square
of the velocity.
~Result.~
The resistances therefore are as the squares of the velocities in the
same fluid, and as the squares of the velocities multiplied by the
densities in different fluids.
Hitherto, however, we have imagined a fluid which does not exist in
nature; that is to say, a _discontinued_ fluid, or one which has its
particles separated and _unconnected_, and also perfectly non-elastic.
~Atmosphere, and its properties bearing on the question of its
resistance.~
Now, in the atmosphere, no one particle that is contiguous to the body
can be moved without moving a great number of others, some of which will
be distant from it. If the fluid be much compressed, and the velocity of
the moving body much less than that with which the particles of the
fluid will rush into vacuum in consequence of the compression, it is
clear that the space left by the moving body will be almost
instantaneously filled up, (plate 23, fig. 2); and the resistance of
such a medium would be less the greater the compression, provided the
density were the same, because the velocity of rushing into a vacuum
will be greater the greater the compression. Also, in a greatly
compressed fluid, the form of the fore part of the body influences the
amount of the retarding force but very slightly, while in a
non-compressed fluid this force would be considerably affected by the
peculiar shape which might be given to the projectile.
~Resistance increased when the body moves so fast that a vacuum is
formed behind it.~
Thirdly. If the body can be moved so rapidly that the fluid cannot
instantaneously press in behind it, as is found to be the case in the
atmosphere, the resisting power of the medium must be considerably
increased, for the projectile being deprived of the pressure of the
fluid on its hind part, must support on its fore part the whole weight
of a column of the fluid, over and above the force employed in moving
the portion of the fluid in contact with it, which force is the sole
source of resistance in the discontinued fluid. Also, the condensation
of the air in front of the body will influence considerably the relation
between the resistances and the velocities of an oblique surface: and it
is highly probable that although the resistances to a globe may for slow
motions be nearly proportional to the squares of the velocities, they
will for great velocities increase in a much higher ratio.
ON THE VELOCITY WITH WHICH AIR WILL RUSH INTO A VACUUM.
~The velocity of the rush of air into a vacuum.~
When considering the resistance of the air to a body in motion, it is
important that the velocity with which air will rush into a vacuum
should be determined; and this will depend upon its pressure or
elasticity.
~Result.~
It has been calculated, that air will rush into a vacuum at the rate of
about 1,344 feet per second when the barometer stands at 30 inches, so
that should a projectile be moving through the atmosphere at a greater
velocity than this, say 1,600 feet per second, then would there be a
vacuum formed behind the ball, and instead of having merely the
resistance due to the inertia of the particles of the air, it would, in
addition, suffer that from the whole pressure of a column of the medium,
equal to that indicated by the barometer.
UPON THE RESISTANCE OF THE AIR TO BODIES OF DIFFERENT FORMS.
~Difficulties of the question.~
The influence of the form of a body upon the resistance offered to it by
a fluid, is a problem of the greatest difficulty; and although the most
celebrated mathematicians have turned their attention to the subject,
still, even for slow motions, they have only been able to frame strictly
empirical formula, founded upon the data derived from practice; while
with regard to the resistance at very high velocities, such as we have
to deal with, very little light has hitherto been thrown upon the
subject.
~Compressed fluid.~
When a body moves in the atmosphere, the particles which are set in
motion by the projectile, act upon those in proximity to them, and these
again upon others; and also from the elasticity of the fluid, it would
be compressed before the body in a degree dependant upon the motion and
form of the body. Moreover, the atmosphere itself partakes so much of
the nature of an infinitely compressed fluid, as to constantly follow
the body without loss of density when the motion is slow, but not when
the velocity is great, so that the same law will not hold good for both.
In an infinitely compressed fluid (that is, one which would fill up the
space left behind the body instantaneously) the parts of the fluid which
the body presses against in its motion would instantaneously communicate
the pressure received by them throughout the whole mass, so that the
density of the fluid would not undergo any change, either in front of
the body or behind it, consequently the resistance to the body would be
much less than in a fluid partially compressed like the atmosphere; and
the form of the body would not have the same effect in diminishing or
increasing the amount of resistance.
~When a vacuum is formed behind the ball.~
When the velocity of a body moving in the atmosphere is so great that a
vacuum is formed behind it, the action of the fluid approaches to that
of the discontinued fluid.
RESULTS OF EXPERIMENTS WITH SLOW MOTIONS.
~Resistance in proportion to surface.~
1st. It appears from the various experiments that have been made upon
bodies moving in the atmosphere, that the resistance is nearly as the
surface, increasing a very little above that proportion in the greater
surfaces.
~Resistance as squares of velocity.~
2nd. That the resistance to the same surface with _different_
velocities, is in _slow_ motions nearly as the squares of the velocity,
but gradually increasing more and more in proportion as the velocities
increase.
~Rounded and pointed ends suffer less resistance.~
3rd. The round ends, and sharp ends of solids, suffer less resistance
than the flat or plane ends of the same diameter. Hence the flat end of
the cylinder and of a hemisphere, or of a cone, suffer more resistance
than the round or sharp ends of the same.
~Sharp ends not always least resistance.~
4th. The sharper ends have not always the smaller resistances; for
instance, the round end of a hemisphere has less resistance than the
pointed end of a cone, whose angle with the axis is 25° 42′.
~Form of base affects resistance.~
5th. When the hinder parts of bodies are of different forms, the
resistances are different, though the fore parts are the same. Hence the
resistance to the fore part of a cylinder is less than that on the
equally flat surface of the cone or hemisphere, owing to the shape of
the _base_ of the cylinder. The base of the hemisphere has less
resistance than the cone, and the round side of the hemisphere less than
that of the whole sphere.
~Only proved for slow motions.~
The above refers only to _slow_ motions, and the results given, from
experiments with very small velocities; and it is to be expected, that
with very rapid motions the form of the fore, as well as the hind part,
of the projectile, will influence the amount of resistance in a much
higher degree.
~Form of hind part.~
That form for the hind part will be best which has the greatest pressure
upon it, when moving with a certain velocity.
~Best shape for fore and hind part.~
The ogivale form seems, from experiment, to fulfil the former condition.
The best form for the _hind_ part, for _rapid_ motions, has not been
determined; it may, however, be considered to be of much less importance
than the shape of the fore part.
~Form determined by extent of range.~
Of course the best form can be determined by extent of range, but
deductions from this will depend upon such a variety of circumstances,
the effects of some of which must be entirely hypothetical, that the
correctness of any formulæ obtained in this manner must be very
uncertain.
~Form suggested by Sir I. Newton.~
Sir Isaac Newton, in his “Principia,” has given an indication of that
form of body, which, in passing through a fluid, would experience less
resistance than a solid body of equal magnitude of any other form. It is
elongated.
~Axis of elongated bodies must be fixed.~
It is plain, however, that the minimum of resistance would not be
obtained with a shot of an elongated form, unless the axis can be kept
in the direction of the trajectory; as not only will the axis
perpetually deviate from the true direction, but the projectile will
turn over and rotate round its shorter axis, that is, if fired out of a
smooth bore.
~Advantages of conical bullets.~
Conical bullets have an advantage, from their pointed end, which enables
them to pass through the air with greater facility; and for the same
reason they are better calculated to penetrate into any matter than
spherical ones.
~Disadvantages of conical bullets.~
A _solid_ bullet cannot be pointed without sending backward the centre
of gravity. The sharper the point, the more it is liable to injury, and
if the apex of the cone does not lie true, in the axis of the
projectile, then such an imperfection of figure is calculated to cause
greater deflections in the flight than any injury which a round surface
is likely to sustain. In penetrating into solid bodies, it is also
important that the centre of gravity should be near its work.
RESISTANCE OF THE AIR, AS AFFECTED BY THE WEIGHT OF PROJECTILES.
~Resistance overcome by weight.~
Bodies of similar volume and figure overcome the resistance of the air
in proportion to their densities. The amount of the air’s resistance is
in proportion to the magnitude of the surface.
~Contents of circles.~
The superficial contents of circles are as the _squares_ of their
diameters. Hence if the ball A. (plate 23, fig. 3) be 2in. in diameter,
and the ball B. 4in., the amount of resistance experienced would be as
four to sixteen.
~Contents of spheres.~
The cubical contents, or weights of spheres, are in proportion to the
_cubes_ of their diameters. Hence the power to overcome resistance in
the balls A and B would be as _eight_ to _sixty-four_. Thus the power to
overcome resistance increases in much greater proportion than the
resistance elicited by increasing the surface.
~Advantages of elongated bullets.~