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the greatest range, therefore, is not made with an elevation of 45o as in tacus, but with an elevation lower, as the first velocity is greater. Thus it is found, that although the larger shells with small velocities range farthest when pro jected at an elevation of about 45°, yet the smaller shells with great velocities range farthest at an elevation not much above 30°.
These instances sufficiently show, that such rules as are deduced from the theory alone, without the aid of experiment, are unfit for directing practice.
Of Projections made on the Horizontal Plane.
194. Prob. 1. To find the velocity of any shot or shell.
Rule. Divide double of the weight of the charge of powder by the weight of the shot, both in lbs. Then if the square root of the quotient be multiplied by 1600, the product will be the velocity, or the number of feet the shot passes over in a second.
Or say, As the square root of the weight of the shot is to the square root of double the weight of powder, so is 1600 feet to the first velocity of the shot.
Er. With what velocity will a 13-inch shell, weighing 196 lbs. be discharged by 9 lbs. of powder?
TX1600=485 feet per second, nearly. Ans.
195. Prob. 2. To find the terminal velocity of a ball or shell; that is, the greatest velocity it can acquire by descending through the air, by its own weight.
Rule. For balls, the terminal velocity is found by multiplying the square root of the diameter of the ball in inches by 1755; and for shells, by multiplying the square root of the diameter of the shell in inches by 147.3.
Er. What is the terminal velocity of a 2 lbs. iron ball, its diameter being 2:45 inches?
175'5/245=175·5 × 15·6524-275 feet. Ans 196. Prob. 3. To find the height from which a body must fall, in vacuo, in order to acquire a given velocity.
Rule. Since the spaces descended by falling bodies are as the square of the velocities, and a fall of 16 feet or 193 inches produces a velocity of 32 feet or 386 inches, therefore 386: the square of the given velocity in inches :: 193: height required in inches. Or, omitting the fractions, if the square of the given velocity in feet be divided by 64, the quotient will be the height required in feet, nearly.
Ex. From what height must a body descend, in order to acquire the velocity of 1670 feet per second?
1670+64 43576 feet. Ans.
197. Prob. 4. To find the greatest range of a ball or shell; and the elevation of the piece to produce that range.
Rule. Divide the given initial velocity by the termina. velocity of the ball or shell, and find the quotient in the first column of the following table; against which, in the second column, will be found the elevation to give the greatest range; and the corresponding number in
the third column multiplied by the height producing the terminal velocity will give the greatest range, nearly.
Elevations giving the greatest range.
Ex. What is the greatest range of a 24 lb. iron ball, when discharged with a velocity of 1640 feet, and the elevation to produce that range, the diameter of the ball being 5:6 inches?
175°5/56=415 the terminal velocity.
4152+64=2691=the height producing that velocity.
The range in the third column against 34° 15′ is 2·9094, and 2'9094 x2691=7829 feet the greatest range, nearly.
198. Prob. 5. The range of any one elevation being given, to find the range of any other elevation, and the converse.
Rule. As the sine of twice the first elevation is to the sine of twice the second, so is the range at the former to that at the latter, the velocity being the same in both cases.
199. Prob. 6. The range for one charge being given, to find the range for another charge, or the charge for another range.
Rule. The ranges at the same elevation are nearly proportional to the charges.
Ex. If, at an elevation of 45°, with a charge of 9 lbs. of powder, a shell range 4000 feet, what charge, at the same elevation, will be required to throw it 3000 feet?
4000 3000:9: 64 lbs. Ans.
200. Prob. 7. The range and elevation being given, to find the time of the flight.
Rule. As radius is to the tangent of elevation so is the range in feet, to the square of 4 times the number of seconds taken up in the flight, nearly.
When the elevation is 45°, then of the square root of the range in feet will be the number of seconds required, nearly.
Er. In what time will a shell range 3000 feet at an elevation of 35°? Rad Tan 35 :: 3000:-2100·6
√2100611'45 seconds nearly. Ans.
201. Prob. 8. The range and elevation being given; to greatest height to which the shell rises above the horizontal p' Rule. As radius is to the tangent of elevation, so is of the to the height required.
Ex. A shell, when discharged at an elevation of 40°, range feet, what is its greatest height during the flight?
Rad: Tan 40 :: 750: 629 feet. Ans.
202. Prob. 9. The range and elevation being given; to find the petus.
Rule. As the sine of twice the elevation is to the radius, so is halt the range to the impetus.
Er. With what impetus must a shell be discharged at an elevation of 35° to strike an object at the distance of 3180 feet?
Sin 70 Rad: 1590: 1692 feet. Ans.
Of Projectiles made on an Inclined Plane.
203 Prob. 10. The inclination of the plane, and the impetus and elevation of the piece being given, to find the range.
Rule. Add together twice the log. secant of the plane's inclination, the log. sine of the elevation above the plane, the log. co-sine of the elevation above the horizon, and the log. of four times the impetus ; then will the sum be the log. of the range.
Er. How far will a shot range on a plane which ascends 10°, and on another which descends 10°, the impetus being 2000 feet, and the elevation of the piece 32°30′?
The elevation above the plane, in the first case, is 22°30′, and in the second 42°30'.
204. Prob. 11. The inclination of the plane, the range and elevation being given, to find the impetus.
Rule. Add together twice the log. co-s'ne of the plane's inclination, the log. co-secant of the elevation above the plane, the log. secant of the elevation above the horizon, and the log of of the range; then will the sum be the log. of the impetus.
With what impetus must a shell be discharged to strike an object at the distance of 2662 feet on an inclined plane which ascends 10o the elevation of the mortar being 32°30′ ?
Plane's inclination........ 10°...... 2 cos.... 19.986703
Elevation above the horizon 32°30′
of the range....................... 6655....log.......
Impetus =2000 feet, neark = 3 300982
nitions.-1. HYDROSTATICS treat of the nature, gravity, and motion, of fluids in general, and of the methods of ang solids in them. And its mechanical practice, called ulics, relates particularly to the motion of water through , &c.
2. A fluid is a body, the parts of which yield to any impression, and are easily moved among each other.
Fluids are either non-elastic and incompressible, as water, oil, mercury, &c. or elastic and compressible, as air, steam, and the different gases.
Obs —1. Heat, or motion, is supposed to be the cause of fluidity: for xample; ice, without heat, is a solid; with heat, it becomes a fluid, in ater; and with more heat, an elastic fluid, in steam. In the first state, ae atoms are fixed in crystals; in the second, are thrown into intestine motion; and, in the third state, are forced asunder with an amazing expansive force.
2. Philosophers have usually assumed, that the particles of fluids are round and smooth, since they are so easily moved among one another. This supposition will account for some circumstances belonging to them. If the particles are round, there must be vacant spaces between them, in the saine manner as there are vacuities between cannon-balls that are piled together; between these balls smaller shot may be placed, and between these, others still smaller, or gravel, or sand, may be diffused. In a similar manner, a certain quantity of particles of sugar can be taken up in water without increasing the bulk, and when the water has dissolved the sugar, salt may be dissolved in it, and, yet the bulk remain the same; and, admitting that the particles of water are round, this is easily accounted for.
3. Others have supposed that the cause of fluidity is the mere want of cohesion of the particles of water, oil, &c. and from this imperfect cohesion, fluids in small quantities, and under pecuiiar circumstances, arrange themselves in a spherical manner and form drops.
Fluids are subject to the same laws of gravity with solids; but their want of cohesion occasions some peculiarities. The parts of a solid are so connected as to form a whole, and their weight is concentrated in a single point, called the centre of gravity: but the atoms of a fluid gravitate independently of each other.
PROP. 1.-If one part of a fluid be higher than another, the higher parts will continually descend to the lower places, and will not be at rest till the surface of it is quite level.
For the parts of a fluid being movable every way, if any part is above the rest, it will descend by its own gravity as low as it can get. And af terwards other parts, that are now become higher, will descend as the other did, till at last they will all be reduced to a level or horizontal plane.
Co 1. Hence water that communicates by means of a channel or pipe 4ther water, will settle at the same height in both places.
Cor. 2. For the same reason, if a fluid gravitates towards a centre; it will dispose itself into a spherical figure, whose centre is the centre of force; as the sca in respect of the earth.
PROP. 2.-If a fluid be at rest in a vessel whose base is parallel to the horizon, equal parts of the base are equally pressed by the fluid.
For upon every part of the base there is an equal column of the fluid supported by it. And, as all these columns are of equal weight, they must press the base equally; or equal parts of the base will sustain an equal pressure.
Cor. 1. All parts of the fluid press equally at the same depth.
For, imagine a plane drawn through the fluid parallel to the horizon. Then the pressure will be the same in any part of that plane, and therefore the parts of the fluid at the same depth sustain the same pressure.
Cor. 2. The pressure of a fluid, at any depth, is as the depth of the fluid. For the pressure is as the weight, and the weight is as the height, of a column of the fluid.
PROP. 3.-If a fluid is compressed by its weight or otherwise; at any point it presses equally, in all manner of directions.
This arises from the nature of fluidity; which is, to yield to any force in any direction. If it cannot give way to any force applied, it will press against other parts of the fluid in direction of that force. And the pressure in all directions will be the same. For, if any one was less, the fluid would move that way, till the pressure be equal every way.
Cor. In any vessel containing a fluid, the pressure is the same against the bottom as against the sides, or even upwards, at the same depth.
PROP. 4-The pressure of a fluid upon the base of the contain ing vessel, is as the base and perpendicular altitude, whatever be the figure of the vessel that contains it.
Let ABIC, EGKH, figs. 1 and 2, be two vessels. Then (Prop. 2. Cor. 2.) the pressure upon an inch on the base AB height CDX 1 inch; and the pressure upon an inch on the base HK is height FH x 1 inch. But (Prop. 2.) equal parts of the bases are equally pressed, therefore the pressure on the base AB is CD x number of inches in AB; and pressure on the base HK is FH x number of inches in HK. That is, the pressure on AB is to the pressure on IIK, as base AB × height CD, to the base HK X height FH.
Obs. From this proposition may be calculated, the pressure upon, and the strength required for, dams, cisterns, pipes, &c.
Cor. 1. Hence, if the heights be equal, the pressures are as the bases. And, if both the heights and bases be equal, the pressures are equal in both; though their contents be ever so different.
For, the reason that the wider vessel EK has no greater pressure at the bottom, is, because the oblique sides EH, GK, take off part of the weight. And in the narrower vessel CB, the sides CA, IB, re-act against the pressure of the water, which is all alike at the same depth, and by this re-action the pressure is increased at the bottom, so as to become the same every where.
Cor. 2. The pressure against the base of any vessel, is the same as of a cylinder of an equal base and height.
Cor. 3. If there be a recurve tube ABF, fig. 3, in which are two different fluids CD, EF; their heights in the two legs CD, EF, will be reciprocally as their specific gravities, when they are at rest.