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Thus then at length it appears that the first force of the inflamed gunpowder, when occupying only the space at first filled with the powder, is about 1800, that is 1800 times the elasticity of the natural air, or pressure of the atmosphere, in the charges with 8 oz. and 16 oz. of powder, in the two longer guns; but somewhat less in the two shorter, probably owing to the gradual firing of gunpowder in some degree; and also less in the lowest charge 4 oz, in all the guns, which may probably be owing to the less degree of heat in the small charge. But besides the foregoing circumstances that have been noticed, or used in the calculations, there are yet several others that might and ought to be taken into the account, in order to a strict and perfect solution of the blem; such as, the counter pressure of the atmosphere, and the resistance of the air on the fore part of the ball while moving along the bore of the gun; the loss of the elastic fluid by the vent and windage of the gun; the gradual firing of the powder; the unequal density of the elastic fluid in the different parts of the space it occupies between the ball and the bottom of the bore; the difference between pressure and percussion when the ball is not laid close to the powder; and perhaps some others: on all which accounts, it is probable that, instead of 1800, the first force of the elastic fluid is not less than 2000 times the strength of natural air.
Corol. From the theorem last used for the velocity of the
ball and elastic fluid, viz. v = √(
p + w
log.), we may find the velocity of the elas
tic fluid alone, viz. by taking w, or the weight of the ball, 0 in the theorem, by which it becomes barely
log.), for that velocity. And by computing P the several preceding examples by this theorem, supposing the value of n to be 2000, the conclusions come out a little various, being between 4000 and 5000, but most of them nearer to the latter number. So that it may be concluded that the velocity of the flame, or of the fired gun-powder, expands itself at the muzzle of the gun, at the rate of about 5000 feet per second nearly.
ON THE MOTION OF BODIES IN FLUIDS.
To determine the Force of Fluids in Motion; and the Circumstances attending Bodies Moving in Fluids.
1. It is evident that the resistance to a plane, moving perpendicularly through an infinite fluid, at rest, is equal to the pressure or force of the fluid on the plane at rest, and the fluid moving with the same velocity, and in the contrary direction, to that of the plane in the former case. But the force of the fluid in motion, must be equal to the weight or pressure which generates that motion; and which, it is known, is equal to the weight or pressure of a column of the fluid, whose base is equal to the plane, and its altitude equal to the height through which a body must fall, by the force of gravity, to acquire the velocity of the fluid: and that altitude is, for the sake of brevity, called the altitude due to the velocity, So that, if a denote the area of the plane, v the velocity, and n the specific gravity of the fluid; then, the altitude due to the velocity v being the whole
resistance, or motive force m, will be a x n x
g being 16 feet. And hence, cæteris paribus, the resistance is as the square of the velocity.
2. This ratio, of the square of the velocity, may be otherwise derived thus. The force of the fluid in motion, must be as the force of one particle multiplied by the number of them; but the force of a particle is as its velocity; and the number of them striking the plane in a given time, is also as the velocity; therefore the whole force is as v × v or v2, that is, as the square of the velocity.
3. If the direction of motion, instead of being perpendi cular to the plane, as above supposed, be inclined to it in any angle, the sine of that angle being s, to the radius 1: then the resistance to the plane, or the force of the fluid
against the plane, in the direction of the motion, as assigned above, will be diminished in the triplicate ratio of radius to the sine of the angle of inclination, or in the ratio of 1 to s3. For, AB being the direction of the plane, and BD that of the motion, making the angle ABD, whose sine is s; the number of particles, or quantity of the fluid striking the plane, will be diminished in the ratio of 1 to s, or of radius to the f sine of the angle B of inclination; and the force of each particle will also be
diminished in the same ratio of 1 to s: so that, on both these accounts, the whole resistance will be diminished in the ratio of 1 to s2, or in the duplicate ratio of radius to the sine of the said angle. But again, it is to be considered that this whole resitance is exerted in the direction BE perpendicular to the plane; and any force in the direction BE, is to its effect in the direction AE, parallel to BD, as AE to BE, that is as 1 to s. So that finally, on all these accounts, the resistance in the direction of motion, is diminished in the ratio of 1 to s3, or in the triplicate ratio of radius to the sine of inclination. Hence, comparing this with article 1, the whole resistance, or the motive force on the plane, will be anv2ƒ3 4g
4. Also, if w denote the weight of the body, whose plane face a is resisted by the absolute force m; then the
5. And if the body be a cylinder, whose face or end is a, and diameter d, or radius r, moving in the direction of its axis; because then = 1, and a = p31416; the resisting force m will be
and the retarding force ƒ =
6. This is the value of the resistance when the end of the cylinder is a plane perpendicular to its axis, or to the direction of motion. But were its face a conical surface, or an elliptic section, or any other figure every where equally inclined to the axis, the sine of inclination being : then the number of particles of the fluid striking the face being still the same, but the force of each, opposed to the direction
of motion, diminished in the duplicate ratio of radius to the sine of inclination, the resisting force m would be
But if the body were terminated by an end or face of any other form, as a spherical one, or such like, where every part of it has a different inclination to the axis; then a further investigation becomes necessary, such as in the following proposition.
To determine the Resistance of a Fluid to any Body, moving in it, of a Curved End; as a Sphere, or a Cylinder with a Hemispherical End, &c.
1. LET BEAD be a section through the axis ca of the solid, moving in the direction of that axis. To any point of the curve draw the tangent EG, meeting the axis produced in G: also, draw the perpendicular ordinates EF, ef, indefinitely near each other; and draw ae parallel to D CG,
Putting CF, EF =y, BE = %, s = sine 4 G to radius 1, and p = 3·1416: then 2py is the circumference whose radius is EF, or the circumference described by the point E, in revolving about the axis CA; and 2py × Ee or 2pyż is the fluxion of the surface, or it is the surface described by Ee, in the said revolution about CA, and which is the quantity represented by a in art. 3 of the last problem: hence nv253 pnv2s3
× 2pyż or × yż is the resistance on that ring, 4g 2g or the fluxion of the resistance to the body, whatever the figure of it may be. And the fluent of which will be the resistance required,
2. In the case of a spherical form; putting the radius Ca
or CB = r, we have y = √r2 – x2, s
3. But the perpendicular resistance to the circle of the same diameter d or BD, by art. 5 of the preceding problem, is pnv3d2
; which, being double the former, shows that the 16g resistance to the sphere, is just equal to half the direct ́resistance to a great circle of it, or to a cylinder of the same diameter.
4. Since pd is the magnitude of the globe; if N denote its density or specific gravity, its weight w will be
and therefore the retardive force for
m · pn v2 d2
= by art. 8 of the general
theorems in page 342; hence then
xd; which is the space that would be described by the globe, while its whole motion is generated or destroyed by a constant force which is equal to the force of resistance, if no other force acted on the globe to continue its motion. And if the density of the fluid were equal to that of the globe, the resisting force is such, as, acting constantly on the globe without any other force, would generate or destroy its motion in describing the spaced, or of its diameter, by that accelerating'or retarding force.
5. Hence the greatest velocity that a globe will acquire by descending in a fluid, by means of its relative weight in the fluid, will be found by making the resisting force equal to that weight. For, after the velocity is arrived at such a degree, that the resisting force is equal to the weight that urges it, it will increase no longer, and the globe will afterwards continue to descend with that velocity uniformly. Now, N and n being the separate specific gravities of the globe and fluid, N# will be the relative gravity of the globe in the fluid, and therefore wpd3 (N-pd3 (N-1) is the