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104. THE two foregoing propositions contain the whole theory of projectiles, with theorems for all the cases, regularly arranged for use, both for oblique and horizontal planes. But, before they can be applied to use in resolving the several cases in the practice of gunnery, it is necessary that some more data be laid down, as derived from good experiments made with balls or shells discharged from cannon or mortars, by gunpowder, under different circumstances. For, without such experiments and data, those theorems can be of very little use in real practice, on account of the imperfections and irregularities in the firing of gunpowder, and the expulsion of balls from guns, but more especially on account of the enormous resistance of the air to all projectiles made with any velocities that are considerable. As to the cases in which projectiles are made with small velocities, or such as do not exceed 200, or 300, or 400 feet per second of time, they may be resolved tolerably near the truth, especially for the larger shells, by the parabolic theory, laid down above. But, in cases of great projectile velocities, that theory is quite înadequate, without the aid of several data drawn from many and good experiments. For so great is the effect of the resistance of the air to projectiles of considerable velocity, that some of those which in the air range only between 2 and 3 miles at the most, would in vacuo range about ten times as far, or between 20 and 30 miles.
The effects of this resistance are also various, according to the velocity, the diameter, and the weight of the projectile. So that the experiments made with one size of ball or shell, will not serve for another size, though the velocity should be the same; neither will the experiments made with one vefocity, serve for other velocities, though the ball be the same. And therefore it is plain that, to form proper rules for practical gunnery, we ought to have good experiments made with each size of mortar, and with every variety of charge, from the least to the greatest. And not only so, but these ought also to be repeated at many different angles of elevation, namely, for every single degree between 30° and 60° elevation, and at intervals of 5° above 60° and below 30°, from the vèrtical direction to point blank. By such a course of experiments it will be found, that the greatest range, instead of being constantly that at an elevation of 45°, as in the parabolic theory, will be at all intermediate degrees between 45 and 30,
being more or less, both according to the velocity and the weight of the projectile; the smaller velocities and larger shells ranging farthest when projected almost at an elevation of 45°; while the greatest velocities, especially with the smaller shells, range farthest with an elevation of about 30°.
105. There have, at different times, been made certain small parts of such a course of experiments as is hinted at above. Such as the experiments or practice carried on in the year 1773, on Woolwich Common; in which all the sizes of mortars were used, and a variety of small charges of powder. But they were all at the elevation of 45°; consequently these are defective in the higher charges, and in all the other angles of elevation.
Other experiments were also carried on in the same place in the years 1784 and 1786, with various angles of elevation indeed, but with only one size of mortar, and only one charge of powder, and that but a small one too: so that all those nearly agree with the parabolic theory. Other experiments have also been carried on with the ballistic pendulum, at different times; from which have been obtained some of the laws for the quantity of powder, the weight and velocity of the ball, the length of the gun, &c. Namely, that the velocity of the ball varies as the square root of the charge directly, and as the square root of the weight of ball reciprocally; and that, some rounds being fired with a medium, length of one-pounder gun, at 15° and 45° elevation, and with 2, 4, 8, and 12 ounces of powder, gave nearly the velocities, ranges, and times of flight, as they are here set down in the following Table.
106. But as we are not yet provided with a sufficient number and variety of experiments, on which to establish true rules for practical gunnery, independent of the parabolic theory, we must at present content ourselves with the data of VOL. II.
some one certain experimented range and time of flight, at a given angle of elevation; and then by help of these, and the rules in the parabolic theory, determine the like circumstances for other elevations that are not greatly different from the former, assisted by the following practical rules.
SOME PRACTICAL RULES IN GUNNERY.
I. To find the Velocity of any Shot or Shell.
RULE. Divide double the weight of the charge of powder by the weight of the shot, both in lbs. Extract the square root of the quotient. Multiply that root by 1600, and the product will be the velocity in feet, or the number of feet the
passes over per
Or say-As the root of the weight of the shot, is to the root of double the weight of the powder, so is 1600 feet, to the velocity.
II. Given the range at One Elevation; to find the Range at
RULE. As the sine of double the first elevation, is to its range; so is the sine of double another elevation, to its range.
III. Given the Range for One Charge; to find the Range for Another Charge, or the Charge for Another Range.
RULE. The ranges have the same proportion as the charges; that is, as one range is to its charge, so is any other range to its charge; the elevation of the piece being the same in both cases.
107.. Example 1. If a ball of 1 lb. acquire a velocity of 1600 feet per second, when fired with 8 ounces of powder; it is required to find with what velocity each of the several kinds of shells will be discharged by the full charges of powder, viz.
108. Exam. 2. If a shell be found to range 1000 yards, when discharged at an elevation of 45°; how far will it
range when the elevation is 30° 16', the charge of powder being the same? Ans. 2612 feet, or 871 yards.
109. Exam. 3. The range of a shell, at 45° elevation, being found to be 3750 feet; at what elevation must the piece be set, to strike an object at the distance of 2810 feet, with the same charge of powder?
Ans. at 24° 16′, or at 65° 44′.
110. Exam. 4. With what impetus, velocity, and charge of powder, must a 13-inch shell be fired, at an elevation of 32° 12', to strike an object at the distance of 3250 feet?
Ans. impetus 1802, veloc. 340, charge 4lb. 71oz.
111. Exam. 5. A shell being found to range 3500 feet, when discharged at an elevation of 25° 12'; how far then will it range at an elevation of 36° 15′ with the same charge of powder ? Ans. 4332 feet.
112. Exam. 6. If, with a charge of 9lb. of powder, a shell range 4000 feet; what charge will suffice to throw it 3000 feet, the elevation being 45° in both cases?
Ans. 64lb. of powder.
113. Exam. 7. What will be the time of flight for any given range, at the elevation of 45° ?
Ans. the time in secs. is 4 the sq. root of the range in feet. 114. Exam. 8. În what time will a shell range 3250 feet, at an elevation of 32° ? Ans. 114 sec. nearly.
115. Exam. 9. How far will a shot range on a plane which ascends 8° 15′, and another which descends 8° 15′'; the impetus being 3000 feet, and the elevation of the piece 32° 30'?
Ans. 4244 feet on the ascent,
and 6745 feet on the descent.
116. Exam. 10. How much powder will throw a 13-inch shell 4244 feet on an inclined plane, which ascends 8° 15′, the elevation of the mortar being 32o 30'?
Ans. 7.3765lb. or 7lb. 6oz.
117. Exam. 11. At what elevation must a 13-inch mortar
be pointed, to range 6745 feet, on a plane which descends 8° 15'; the charge 73lb. of powder?
Ans. 32° 28'.
118. Exam. 12. In what time will a 13-inch shell strike a plane which rises 6o 30', when elevated 45°, and discharged with an impetus of 2304 feet? Ans. 14 seconds.
THE DESCENT OF BODIES ON INCLINED PLANES AND CURVE SURFACES.-THE MOTION OF PENDULUMS.
119. If a weight w be Sustained on an Inclined Plane AB, by ☛ Power P, acting in a Direction WP, Parallel to the Plane. Then
The Weight of the Body, w
FOR, draw CD perpendicular to the plane. Now here are three forces, keeping one another in equilibrio; namely, the weight, or force of gravity, acting perpendicular to AC, or parallel to BC; the power acting parallel to DB; and the pressure
perpendicular to AB, or parallel to DC: but when three forces keep one another in equilibrio, they are proportional to the sides of the triangle CBD, made by lines in the direction of those forces, by prop. 8; therefore those forces are to one another as BC, BD, CD. But the two triangles ABC, CBD, are equiangular, and have their like sides proportional; therefore the three BC, BD, CD, are to one another respectively as the three AB, BC, AC; which therefore are as the three forces w, P, p.
120. Corol. 1. Hence the weight w, power P2 and pressure p, are respectively as radius, sine, and cosine, of the plane's elevation BAC above the horizon.
For, since the sides of triangles are as the sines of their opposite angles, therefore the three AB, BC, AC,
are respectively as
radius, sine, cosine,
sin. C, sin. A, sin. B,
of the angle A of elevation.
Or, the three forces are as AC, CD, AD; perpendicular to their directions.
121. Corol, 2. The power or relative weight that urges a