Liquid Resistance to Motion. 215 in a fluid, as in an equal quantity of solid matter. A pound of water inclosed in a bladder is not more easily thrown to a given height than a pound of ice, or of lead; nor, if falling into the scale of a weighing beam, does it require less as a counterpoise; nor, if made to revolve at the end of a sling, does it render the cord less tight. 361. The force of water moving against an obstacle, or of the resistance of still water to a moving solid, may be deduced from the fact that the pressure of a known height of fluid column produces from an orifice a certain velocity of jet; while conversely, that jet, or any current of equal speed directed against the orifice, supports the column. The impulse given or received, therefore, by the floatboard of a water-wheel, whether of a steam-boat pressing against, or of a corn-mill pressed by, the water, is measured by the weight of a column whose height is according to the velocity, and whose diameter is according to the breadth or extent of the surface concerned. This supposes that the liquid pressure of, or upon, the surface is direct; if it be oblique, there is a diminution according to the rule given under the head of "resolution of forces." One might expect that if a body, as a boat, moving through water at a given rate meet a given resistance, and cost a given expenditure of energy, it should just meet double resistance, and cost double energy when moving twice as fast. But the resistance and energy, with a double rate, are more than four times greater. These facts are but additional examples of a principle already explained (Art. 347). A boat which moves one mile per hour displaces or throws aside a certain quantity of water with a certain velocity. If it move twice as fast, it displaces twice as many particles in the same time, and requires to be moved by twice the force on that account; but it also displaces every particle with a double velocity, for twice as many have to be pushed aside in the same time, and it requires another doubling of the power on this account. In the same manner a ninefold energy is required for a triple speed, three times as many particles being moved, and each particle with three times the velocity. For a speed of four, an energy of sixteen ; for a speed of five, an energy of twenty-five; and so forth. The relation is that which mathematicians indicate by saying that the resistance increases as the square of the speed. Thus, if the resistance at the bow of a vessel were all that had to be considered, one hundred horses would drag the vessel ten times as fast as one horse. But another important element in the calculation 216 Law connecting Liquid Resistance and Velocity. farther increases the disparity between the visible motion and the energy expended, viz., the lessening of the water-pressure on the stern, as the speed of the vessel increases. This pressure, while the vessel is at rest, is just equal to the pressure on the bow; and the energy therefore required to increase the velocity is still considerably greater than in proportion to the square of this velocity. 362. This fact is of very great importance, and explains at what a heavy expense of fuel and machinery high velocities are obtained in steam-boats. If an engine of about 50-horse power would drive a ship seven miles an hour, two engines of more than 50, or one of more than 100, would be required to drive it ten miles, and three such to drive it twelve miles; even supposing the increased resistance at the bow, as already stated, to be the measure of the whole work to be done, which it is not, and that engines worked to the same advantage with a high velocity as with a low, which they do not. For the same reasons, if all the coal which a ship could conveniently carry were just sufficient to drive her 1000 miles at the rate of twelve miles per hour, it would drive her more than 3000 at a rate of seven miles per hour; and more than 6000 at a rate of five miles per hour. The same law shows the error of putting very large sails on a ship. The trifling advantage in point of speed by no means compensates for the additional expense of making and working the sails, and the risk of accidents in bad weather. The ships of the prudent Chinese have not, for the same tonnage, one-third so much sail as those of Europeans, and yet they move with sufficient speed for many purposes. A European ship under jury-masts, or makeshifts after a storm, does not lose nearly so much of her usual speed as one might expect. This explains also why a ship glides through the water one or two miles an hour with a very little wind, although with a strong breeze she would only sail at the rate of six or eight miles. Less than the 100th part of that force of wind which drives her ten miles an hour will drive her one mile an hour, and less than the 400th part will drive her half a mile. Thus, during a calm, a few men pulling in a boat can move a large ship at a sensible rate. 363. These considerations show strikingly of what importance to navigation it might be to have, as a part of a ship's ordinary equipment, one or two water-wheels (or ready means of forming them), to be affixed upon the ship's side when required; so that, by working these in connection with the capstan, the tedium, expense, and ever Resistance of the Air to a moving Body. 217 disastrous results of a long calm at sea might be avoided. A pair of such wheels might also serve other purposes; for, when acted upon by the water as the ship sailed, they would turn with the force of water-wheels on shore, and might be made to move the pumps, to hoist the sails, and to do any work which a steam-engine could perform. Many a vessel has perished because the exhausted crew could no longer labour at the pumps, where such water-wheels would have performed the work required for a much longer time.* 364. The law, that resistance to a body moving in a fluid increases in a greater proportion than the speed of the body, applies where the fluid is aëriform, as well as where it is liquid. A bullet shot through the air with a double velocity, for the reasons given above, experiences four times as much resistance in front as with a single velocity; the motion being retarded also by the loss or diminution on the posterior surface of the usual atmospheric pressure of 15 lbs. per square inch. It is true, further, that when the velocities of bodies moving in air are very great, the resistance in front increases in a still quicker ratio than in liquids; and possibly because the compressibility of air allows it to be much condensed, or heaped up, before the quick moving body. This will be again referred to. 365. The rule of action between a solid and fluid now explained is reciprocal, and holds the same when the fluid is in motion against the solid, as when the solid moves through the fluid. If a ship be anchored in a tide's way, where the current is four miles an hour, the strain on her cable is not one-fourth part so great as if the current were eight miles. A wind moving three miles an hour is scarcely felt; if moving six miles, it is a pleasant breeze; if twenty or thirty miles, it is a brisk gale; if sixty, it is a storm; and beyond eighty, it is a frightful hurricane, tearing up trees by the roots, and generally destructive. Supposing the wind to move one hundred miles per hour, there are one hundred times as many particles of matter striking any body exposed to it as when it moves only one mile per hour, and * The suggestion here made was acted upon a year after the publication of the first edition of this work. 218 Resistance depending on the Shape of the Body. each particle strikes, moreover, with one hundred times the velocity or force, so that the whole increase of force is a hundred times a hundred, or ten thousand. This explains how the soft invisible air may, by motion, acquire force sufficient to unroof houses, to level oaks, the roots of which have been spreading wide and gathering strength for centuries, and, in some forms of hurricane, absolutely to brush away everything projecting from the surface of the earth. The explosive force of ignited gunpowder illustrates the same principle. 366. This law of rapidly increasing resistance assigns a limit to many velocities, both natural and artificial. It limits the velocities of bodies falling through the air. By the law of gravity a body would fall with a constantly accelerated speed; but as the resistance of the air increases still more quickly than the speed, at a certain point, this resistance and the gravity balance each other, and the motion becomes uniform. No ship under canvas or with steam power sails faster than about twenty miles an hour; and it is because the frictional resistance to be overcome by steam-carriages on railways does not increase with their velocity, like the water resistance to ships, that the speed of the former may so much exceed that of the latter. No fish swims with a velocity much exceeding twenty miles an hour; not the dolphin, when shooting ahead of our swiftest frigates; nor the salmon, when darting forward with the speed that lifts it over a waterfall. 367. The resistance between a meeting fluid and solid depends greatly on the shape of the solid. Experiment shows that a round mass of wood floating in water can be drawn along at a certain rate by about half the force required to draw a cubical block of the same material and of the same diameter and weight. As a plough opens and penetrates the ground with ease proportioned to the sharpness of its wedge-like form, so does the wedge-formed ship plough easily through the water. In the case of the plough, the furrow left behind remains open, and the form of the hind part of the plough is immaterial; but in the case of the ship, the water has to close in behind, and by its pressure to counterbalance in a degree the resistance of the water in front. If the stern part of a ship were to be abrupt like the end of a packing case, the water would fall in but slowly Illustrations from Nature. 219 to fill up the furrow called the ship's wake, and hence would arise an important cause of retardation. To favour, therefore, the motion of a ship through the water, the wedge-shape or tapering is required behind as well as before; a gradual tapering of the hind part, or a fine run, as it is technically called, allows the water to apply itself readily to it as it passes along, and is essential for quick sailing. 368. Nature herself furnishes us with the models of our sailing vessels. Fishes are wedge-like both before and behind, their form being modified, however, in relation to other purposes of equal importance to them as mere speed of motion. Of birds the same is true, and in flying they are observed to stretch out their necks, so as to make their form perfect for dividing the air. There are boats used in China called snake-boats, seen on festive occasions, which are only about two feet broad, while perhaps a hundred feet long, and when they are moved by a multitude of rowers, their swiftness is extreme. The problem which has for its object to assign for a ship's hull or bottom the best possible form to give speed of sailing with capacity for cargo, is of much importance, but, being of very great complexity, is not yet satisfactorily solved ; so that a kind of empiricism prevails in the matter, and unexpected results often arise. The flight of birds through the thin air has a limited celerity. The crow, when flying homewards against the storm, does not face the wind in the open sky, but skims along near the surface of the earth in the deep valleys, or wherever the swiftness of the wind is retarded by terrestrial obstructions. The great albatross of the South Sea, stemming upon the wing the current of a gale so as to remain in company with a driving ship, where the air is passing at the rate of eighty miles an hour, often takes short shelter on the lee side of a lofty billow. The bird called the stormy petrel abides chiefly in the midst of the Atlantic Ocean, but the violence of the wind can sweep it from the waves and cause its appearance on the solid shores. Vessels from the high sea, approaching a coast from which strong wind blows, often become resting-places to exhausted land-birds, driven off the shore by wind which they have not strength of wing to stem ;--sad evidence of the myriads which are constantly perishing where no resting-place is found, and where no eye notes their fate. 369. The following instances exhibit the mutual influence of mecting solids and fluids, where the surface of the solid is plane or |