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for general service. On the whole, however, small pieces of artillery have been brought into use: thus the battering pieces now approved are the demi-cannon of former times; it being found that their stroke, though less violent than that of a larger piece, is yet sufficiently adapted to the strength of the usual profiles of fortification; and that the facility of their carriage and management, and the ammunition they spare, give them great advantages beyond the whole cannon formerly employed. The method of making a breach, by first cutting off the whole wall as low as possible before its upper part is attempted to be beaten down, seems also to be a considerable modern improvement. . But the most important advance in this artis the method of firing with small quantities of powder, and elevating the piece so that the bullet may just go clear of the parapet of the enemy, and drop into his works. By these means the bullet, coming to the ground at a small angle, and with a small velocity, does not bury itself, but bounds or rolls along in the direction in which it was fired: and therefore, if the piece be placed in a line with the battery it is intended to silence, or the front it is to sweep, each shot rakes the whole length of that battery or front; and has thereby a much greater chance of disabling the defendants, and dismounting their cannon. This method was invented by Vauban, and was by him styled Batterie á Ricochet. It was first practised in 1692 at the siege of Aeth. Something similar was practised by the king of Prussia at the battle of Rosbach. in 1757.
Sect. I.-Of The Effects of GRAvity on Pro-
It has been demonstrated that a body projected in the usual way from the surface of the earth in the atmosphere, must describe a conic section, having the centre of the earth in one focus, and that it will describe round that focus areas proportional to the times: it follows that, if the velocity of projection exceeds 36,700 feet in a second, the body (if not resisted by the air) would describe a hyperbola; if it be just 36,700 it would describe a parabola; and, if it be less, it would describe an ellipsis. If projected directly upwards, in the first case, it would never return, but proceed for ever; its velocity continually diminishing, but never becoming less than an assignable portion of the excess of the initial velocity above 36,700 feet in a second; in the second case it would never return, its velocity would diminish without end, but never be extinguished. In the third case, it would proceed till its velocity was reduced to an assignable portion of the difference between 36700 and its initial velocity; and would then return, regaining its velocity by the same degrees, and in the same places as it lost it. These are necessary consequences of a gravity directed to the centre of the earth, and inversely proportional to the square of the distance. But, in the greatest projections that we are able to make, the gravitations are so nearly equal, and in directions so
nearly parallel, that it would be ridiculous affectation to pay any regard to the deviations from equality and parallelism. A bullet rising a mile above the surface of the earth loses only on of its weight, and a horizontal range of four miles only four of deviation from parallelism. Gravitation may be therefore assumed as equal and parallel. The errors arising from this assumption are quite insensible in all the uses which can be made of this theory; which was the first fruits of mathematical philosophy, and the effort of the genius of the great Galileo. Gravity is a constant or uniform accelerating or retarding force, according as it produces the descent, or retards the ascent, of a body: and, all other forces being ascertained by the accelerations which they produce, they are conveniently measured by comparing their accelerations with the acceleration of gravity. This therefore has been assumed by all the latest and best writers on mechanical philosophy, as the unit by which every other force is measured. It gives a perfectly distinct notion of the force which retains the moon in its orbit, to say it is the 3600th part of the weight of the moon at the surface of the earth: i. e. if a bullet were here weighed by a spring steel-yard, and pulled it out to the mark 3600, if it were then taken to the distance of the moon, it would pull it out only to the mark 1. This assertion is made from observing that a body at the distance of the moon falls from that distance on part of sixteen feet in a second. Forces therefore which are imperceptible are not compared, but the accelerations, which are their indications, effects, and measures. For this reason philosophers have been anxious to determine with precision the fall of heavy bodies, to have an exact value of the accelerating power of terrestrial gravity. This measure may be taken in two ways; by taking the space through which the heavy body falls in a second; or the velocity which it acquires in consequence of gravity having acted on it during a second. The last is the proper measure; for the last is the immediate effect on the body. The action of gravity has changed the state of the body, by giving it a determination to motion downward : this both points out the kind and the degree or intensity of the force of gravity. The space described in a second by falling is not an invariable measure; for, in the successive seconds, the body falls through 16, 48,80, 112, &c., feet, but the changes of the body's state in each second is the same. At the beginning it had no determination to move with any appreciable velocity; at the end of the first second it had a determination by which it would have gone on for ever (had no subsequent force acted on it) at the rate of thirty-two feet per second. At the end of the second second, it had a determination by which it would have moved for ever, at the rate of sixty-four feet per second. At the end of the third second, it had a determination by which it would have moved for ever, at the rate of ninety-six feet per second, &c. &c. The difference of these determinations is a determination to the rate of thirty-two feet per second. This is therefore constant, and the indication and proper measure of the constant or invariable
force of gravity. The space fallen through in the first ...} is of use only as it is one-half of the measure of this determination; and, as halves have the proportion of their wholes, different accelerating forces may be safely affirmed to be in the proportion of the spaces through which they uniformly impel bodies in the same time. But we must always recollect that this is but one-half of the true measure of the accelerating force. Mathematicians of the first rank have committed great mistakes by not attending to this; and it is necessary to notice it here, because cases will occur, in the prosecution of this subject, where we shall be very apt to confound our reasonings by a confusion in the use of those measures.
Sect. II.-OF THE MEASURE of The AcceleRAtive Power of GRAvity.
The accurate measure of the accelerative power of gravity is the fall 16, feet, if measured by the space, or the velocity of 32 feet per second, if the velocity be taken. It will greatly facilitate ealculation, and will be sufficiently exact for every purpose, to take 16 and 32, supposing that a body falls sixteen feet in a second, and acquires the velocity of thirty-two feet per second. Then, because the heights are as the squares of the times, and as the squares of the acquired velocities, a body will fall one foot in one fourth of a second, and will acquire the velocity of eight feet per second. , Let h express the height in feet, and call it the producing height; v the velocity in feet per second, and call it the produced velocity, the velocity due; and t the time in seconds.-The following formulae, which are of easy recollection, will serve, without tables, to answer all questions relative to projectiles.
I. v = 8 V h, E 8 x 4 t, E 32 t
and therefore its whole velocity will be 138 feet per second. 6. To find how far it will have moved, comound its motion of projection, which will be orty feet in four seconds, with the motion which gravity alone would have given it in that time, which is 256 feet; and the whole motion will be 296 feet. 7. Suppose the body projected as already mentioned, and that it is required to determine the time it will take to go 296 feet downwards, and the velocity it will have acquired. Find the height r, through which it must fall to acquire the velocity of projection, ten feet, and the time y of falling from this height. Then find the time 2 of falling through the height 296 + r, and the velocity v acquired by this fall. The time of describing the 296 feet will be z – y, and v is the velocity required. From such examples it is easy to see the way of answering every question of the kind. Writers on the higher parts of mechanics al.." compute the actions of other accelerating and retarding forces, by comparing them with the acceleration of gravity; and, to render their expressions more general, use a symbol, such as for gravity, leaving the reader to convert it into numbers. Agreeably to this view, the general formulae will stand thus:
Gravity, or its accelerating power, is estimated in all these equations, as it ought to be, by the change of velocity which it generates in a particle of matter in a unit of time. But many mathematicians, in their investigations of curvilineal and other varied motions, measure it by the deflection which it produces in this time from the tangent of the curve, or by the increment by which the space described in a unit of time exceeds the space described in the preceding unit. This is but one-half of the increment which gravity would have produced, had the body moved through the whole moment with the acquired addition of velocity. In this sense of the symbol g, the equations stand thus:
II. t = h v
* - vo - 2 - v
IV.A.-H =se, and v. 2 Vig
It is likewise very common to consider the accelerating force of gravity as the unit of comparison. This renders the expressions much more simple. In this way v expresses not the velocity, but the height necessary for acquiring it, and the velocity itself is expressed by V v. To reduce such an expression of a velocity to numbers, multiply it by V2g, or by 2 Vg. according as g is the generated velocity, or the space fallen through in the unit of time. This will suffice for the perpendićular ascents or descents of heavy bodies; and we proceed to consider their motions when projected obliquely. The circumstance which renders this an interesting subject is, that the flight of cannon shot and shells are instances of such motion, and the art of gunnery must in a great measure depend on Fig. 1.
this doctrine. Let a body
rough A. B. It is required to determine the path of the body, and all the circumstances of its motion in this path?
1. By the continual action of gravity, the body will be continually deflected from the line BC, and will describe a curve line BVG, concave towards the earth.
2. This curve line is a parabola, of which the vertical line A BE is diameter, B the vertex of this diameter, and BC a tangent in B. Through any two points, VG, of the curve draw VC, GH, parallel to AB, meeting BC in C and H, and draw V.E, G K,
arallel to BC, meeting AB in E, K. It fol
|. from the composition of motions, that the body would arrive at the points V, G, of the curve in the same time that it would have uniformly described BC, BH, with the velocity of projection; or that it would have fallen through B.E, BK, with a motion uniformly accelerated by gravity; therefore the times of describing BC, BH, uniformly, are the same with the time of falling through BE, BK. But, because the motion along B H is uniform, BC is to BH as the time of describing BC to the time of describing BH, which we may express thus, BC : BH = T, BC : T, BH, - T, BE : T, BK. But, because the motion along BK is uniformly accelerated, we have BE: B K = To, BE: To, BK, = BC* : BH", = EV* : K.G”; therefore the curve BVG is such, that the abscissae BE, BK, are as the squares of the corresponding ordinates EV, KG; that is, the curve BVG is a parabola, and BC, parallel to the ordinates, is a tangent in the point B.
3. If the horizontal line AD d be drawn through the point A, it is the directrix of the parabola. Let B E be taken equal to AB. The time of falling through B E is equal to the time of falling through AB; but BC is described with the velocity acquired by falling through AB; and therefore by number 4 of perpendicular descents, BC is double of AB, and E V is double of BE; therefore, EV* = 4 BE", = 4 B E x AB, - BE x 4 AB, and 4 AB is the parameter or latus rectum of the parabola BVG, and, A B being one fourth of the parameter, AD is the directrix.
4. The times of describing the different arches BV, VG, of the parabola are as the portions BC, BH, of the tangent, or as the portions AD, Ad, of the directrix, intercepted by the same vertical lines AB, CV, HG; for the times of describing BV, BVG, are the same with those of describing
the corresponding parts BC, BH, of the tangent,
and are proportional to these parts, because the motion along BH is uniform; and BC, BH, are proportional to AD, A d. Therefore the motion estimated horizontally is uniform. 5. The velocity in any point G of the curve is the same with that which a heavy body would acquire by falling from the directrix along d G. Draw the tangent GT, cutting the vertical AB in T; take the points a, f, equidistant from A and a, and extremely near them, and draw the verticles a b,fi. let the points a, f, continually approach A and d, and ultimately coincide with them. B b will therefore ultimately be to g G in the ratio of the velocity at B to the velocity at G (for the portions of the tangent ultimately coincide with the portions of the curve; and are described in equal times); but B b is to Gg as BH to TG: therefore the velocity at B is to that at G as BH to TG. But, by the properties of the parabola, BH” is to TG” as AB to d6; and A B is to d G as the square of the velocity acuired by falling through A B to the square of the velocity acquired by falling through d G; and the velocity in BH, or in the point B of the parabola, is the velocity acquired by falling along AB; therefore the velocity in TG, or in the point G of the parabola, is the velocity acquired by falling along d G. The preceding propositions contain all the theory of the motion of projectiles in vacuo, or independent on the resistance of the air; and being a very easy and neat exhibition of mathematical philosophy, and connected with a very interesting practice, they have been much commented on, and have furnished matter for many splendid volumes. But the resistance of the air occasions such a prodigious diminution of motion in the great velocities of military projectiles, that this parabolic theory, as it is called, is of little practical use. A musketball, discharged with the ordinary allotment of powder, issues from the piece with the velocity of 1670 feet per second: this velocity would be acquired by falling from the height of eight miles. If the piece be elevated to an angle of 45°, the parabola should be of such extent that it would reach sixteen miles on the horizontal plain; whereas it does not reach much above half a mile. Similar deficiencies are observed in the ranges of cannon shot. It is unnecessary, therefore, to enlarge upon this theory. Facts prove, beyond all doubt, how deficient the parabolic theory is, and how unfit for directing the practice of the artillerist. A very simple consideration is sufficient for rendering this obvious to the most uninstructed. The resistance of the air to a very light body may greatly exceed its weight. Any one will feel this in trying to move a fan very rapidly through the air; therefore this resistance would occasion a greater deviation from uniform motion than gravity would in that body. Its path, therefore, through the air may differ more from a parabola than a parabola itself deviates from the straight line. For these reasons, we affirm that the voluminous treatises which have been published on this subject are nothing but ingenious amusements for young mathematicians. All that seems possible to do for the practical artillerist is, to multiply judicious experiments on real pieces of ordnance, with the charges that are .P. actual service, and to furnish him with tables calculated from such experiments.
SECT. III.-OF THE CAUSEs of THE DEFIcIENcy of THE PARAbolic Theory.
It is, however, the business of the philosopher to enquire into the causes of such a prodigious deviation from a well founded theory; and, having discovered them, to ascertain precisely the deviations they occasion. Thus we shall obtain another theory, either in the form of the parabolic theory corrected, or as a subject of independent discussion. The motion of projectiles being performed in the atmosphere, the air is displaced, or put in motion. Whatever motion it acquires must be taken from the bullet. The motion communicated to the air must be in the proportion of the quantity of air put in motion, and of the velocity communicated to it. If, therefore, the displaced air be always similarly displaced, whatever be the velocity of the bullet, the motion communicated to it, and lost by the bullet, must be proportional to the square of the velocity of the bullet and to the density of the air jointly. Therefore the diminution of its motion must be greater when the motion itself is greater; and in the very great velocity of shot and shells it must be prodigious. From Mr. Robins's experiments it is plain that a globe of four inches and a half in diameter, moving with the velocity of twentyfive feet in a second, sustained a resistance of 315 grains, nearly three-quarters of an ounce. Suppose this ball to move 800 feet in a second, that is, thirty-two times faster, its resistance would be 32 x 32 times three-quarters of an ounce, or 768 ounces, or forty-eight pounds. This is four times the weight of a ball of cast iron of this diameter; and, if the initial velocity had been 1600 feet per second, the resistance would be at least sixteen times the weight of the ball. It is indeed much greater. So great a resistance, operating constantly and uniformly on the ball, must take away four times as much from its velocity as its gravity would do in the same time. In one second gravity would reduce the velocity 800 to 768, if the ball were projected straight upwards. This resistance of the air would therefore reduce it in one second to 672, if it operated uniformly; but as the velocity diminishes continually by the resistance, and the resistance diminishes along with the velocity, the real diminution will be somewhat less than 128 feet. We shall, however, find, that in one second its velocity will be reduced from 800 to 687. From this instance it is clear that the resistance of the air must occasion great deviation from parabolic motion. To judge accurately of its effect, we must consider it as a retarding force, as we consider
view, by adding the differential mark to the letter R or r, which expresses the aggregate resistance. If we thus consider resistance as a retarding force, we can compare it with any other such force by means of the retardation which it produces in similar circumstances. We would compare it with gravity by comparing the diminution of velocity which its uniform action produces in a given time with the diminution produced in the same time by gravity. But we have no opportunity of doing this directly; for, when the resistance of the air diminishes the velocity of a body, it diminishes it gradually, which occasions a gradual diminution of its own intensity. This is not the case with gravity, which has the same action on a body in motion or at rest. We cannot, therefore, observe the uniform action of the resistance of the air as a retarding force. We must make the comparison in some other way. We can state them both as dead pressures. A ball may be fitted to the rod of a spring steelyard, and exposed to the impulse of the wind. This will compress the steelyard to the mark 3, for instance. Perhaps the weight of this ball will compress it to the mark 6. Half this weight would compress it to 3. We reckon this equal to the pressure of the air, because they balance the same elasticity of the spring. In this way we can estimate the resistance by weights whose pressures are equal to its pressure; and we can thus compare it with other resistances, weights, or any other pressures. In fact, we are measuring them by all the elasticity of the spring. This elasticity in its different positions is supposed to have the proportions of the weights which keep it in these positions. Thus we reason from the nature of gravity, no longer considered as a dead pressure, but as a retarding force; and we apply our conclusions to resistances which exhibit the same pressures, but which we cannot make to act uniformly. This sense of the words must be remembered whenever we speak of resistances in pounds and Ounces. The most convenient and direct way of stating the comparison between the resistance of the air and the accelerating force of gravity, is to take a case in which we know that they are equal. Since the resistance is here assumed as proportional to the square of the velocity, it is evident that the velocity may be so increased that the resistance shall equal or exceed the weight of the body. If a body be already moving downwards with this velocity, it cannot accelerate; because the accelerating force of gravity is balanced by an equal retarding force of resistance. It follows from this remark that this velocity is the greatest that a body can acquire by the force of gravity only. Nay, we shall see that it never can completely attain it; because, as it aproaches to this velocity, the remaining accelerating force decreases faster than the velocity increases. It may therefore be called the limiting or terminal velocity by gravity. Let a be the height through which a heavy oody must fall, in vacuo, to acquire its terminal velocity in air. If projected directly upwards with this velocity, it will rise again to this height, and the height is half the space which it would describe uniformly, with this velocity, in the time of its ascent. Therefore, the resistance to this velocity being equal to the weight of the body, it would extinguish this velocity, by its uniform action, in the same time, and after the same distance, that gravity would. Now let g be the velocity which gravity generates or extinguishes during an unit of time, and let u be the terminal velocity of any particular body. The theorems
- tu for perpendicular ascents give us g = 2 o' zz
small a quantity that it would be pedantic affectation to attend to it. It is much more than o: when we make g = 32 feet, instead of 32h, which it should be. Let e be the time of this ascent in opposition to gravity. The same theorems give use u = 2 a; and, since the resistance competent to this terminal velocity is equal to gravity, e will also be the time in which it would be extinguished by the uniform action of the resistance; for which reason we may call it the extinguishing time for this velocity. Let R and E mark the resistance and extinguishing time for the same body moving with the velocity 1. As the resistances are as the squares of the ve
and 2 a is equal to E, the time in which the velocity 1 will be extinguished by the uniform action of the resistance competent to this velocity. The velocity 1 would in this case be extinguished after a motion uniformly retarded, in which the space described is one-half of what would be uniformly described during the same time with the constant velocity 1. Therefore the space thus described by a motion which begins with the velocity 1, and is uniformly retarded by the resistance competent to this velocity, is equal to the height through which this body must fall in vacuo in order to acquire its terminal velocity lil alt. The following description may render all these circumstances more easily conceived by some readers. The terminal velocity is that where the resistance of the air balances and is equal to the weight of the body. The resistance of the air to any particular body is as the square of the velocity; therefore let R be the whole resistance to the body moving with the velocity 1, and r the resistance to its motion with the terminal velocity u; we must have r = R x u”, and this must be = W, the weight. Therefore, to obtain the terminal velocity, divide the weight by the resistance to the velocity 1, and the quotient is the - - W square of the terminal velocity, or R = u”: and this is a very expeditious method of determining it, if R be previously known. Then the common theorems give a, the fall necessary for pro2
of the fill = t = e, and eu = 2*, + the space uniformly described with the velocity u during the time of the fall, or its equal, the time of the extinction by the uniform action of the resistance r ; and, since r extinguishes it in the time e, R which is u” times smaller will extinguish it in the time u’e, and R will extinguish the velocity 1, which is u times less than u, in the time u e, that is, in the time 2 a ; and the body moving uniformly during the time 2 a = E, with the velocity 1, will describe the space 2 a ; and if the body begin to move with the velocity 1, and be uniformly opposed by the resistance R, it will be brought to rest when it has described the space a ; and the space in which the resistance to the velocity 1 will extinguish that velocity by its uniform action is equal to the height through which that body must fall in vacuo in order to acquire its terminal velocity in air. And thus every thing is regulated by the