is to one hour. So is the distance of the two places, to the distance the wind, will pass over in one hour. Note, By a similar experiment, the velocity of running waters may be computed. PROBLEM VIII. Heights or depths may be estimated from the velocities acquired by fal, ling bodies, and the spaces fallen through in given times, or from the time of falling a In successive equal parts of time, such as 1, 2, 3, 4, &c., the spaces passed over, are in the series of the odd numbers, 1, 3, 5, 7, 9, 11, &c., and the acquired velocities, as 1, 2, 3, 4, &c. Hence, it is plain, that the velocities are as the times, and the spaces passed over, are as the square of the times of falling. Thus, in a quarter of a second, from the instant of beginning to fall, a body will fall 1 foot ; in half a second, it will have fallen 4 feet, in three quarters, 9 feet, and in one second, 16 feet. In the next second, it will fall through 16X3=48, which added to the velocity at the end of the former second, will give 64, the whole space fallen through in two seconds. In the third second, the body will fall through 5*16=80, which being added to the last sum, 64, will give 144, the space passed over in 3 seconds, and so on continually. For the continued addition of the odd numbers, gives the squares of all numbers from unity and upwards. 2 Thus, In 1 second, a body will fall 16 feet, which is 1' *16. In 2 seconds, !+3=4=2? *16=64. 1 EXAMPLE I. How far will a body fall in 6 seconds ? 6 36 the square of the time, 216 36 576 feet. EXAMPLE II. In what time will a body descend throuị h11 664 feet? 16)11.664(729(27 seconds. 4 112 46 47)329 144 EXAMPLE III. seconds of time. 2 16 twice the time. 15*16=240, the last acquired velocity. PROBLEM IX. f co To meafure heights and diftances by the geometrical square. When the plane is horizontal, the instrument is to be supa ported and placed horizontally at any point A, and it is to be turned till the remote point F, whose distance is to be measured, is seen through the fixed lights; then turn the index, till through the fights upon it, you see any accessible object B; then place the instrument at the point B, directing the fixed lights to the first station A, and the moveable ones to the point F; and if the index cut the reclined side of the square, as in the point E, then, from similar triangles, ES : SB :: as BA : AG; but if the index cut the right side of the square K, it will be BR : RK :: BA : AF. In either of these cases, the distance required may be found by the rule of three *. Perpendicular heights, when accessible, may be obtained by the quadrant only. For example: If you wanted the height of a house, tree, &t. approach towards or retire from the object, till it subtends an angle of 45° ; then shall the height of the object be equal to its horizontal distance. Euclid, I. 6. A similar obfervation may be made of the other instruments used for heights and distances; but this, and many more, will daily occur in practice. : : * The fide DE is called the right fide, E the reclined side. Regu lifecom TABLES The vclocity acquired at the end of any given time may be found thus. Suppose a body begins to move with a celerity constantly encreasing in such a manner as would carry it through 16 feet in one second, at the end of this space it will have acquired such a degree of velocity as would carry it 32 feet in the next second, though it should then receive no new impulfo from the cause by which its motion had been accelerated. But as the same accelerating cause continues constantly to act, it will move 16 feet farther the next second, consequently it will have run 64 feet, and acquire such velocity as would, in the same time, carry it over double the space. And so on. In what time will a body descend through 11664 feet? 16)1.664(729(27 seconds. Required the last acquired velocity, when a body has fallen 8 seconds of time. 32 the additional velocity per second. 256 the last acquired velocity is 256 feet per second. EXAMPLE EXAMPLE IV. If a body move at the rate of 1376 feet per second, How far muft it fall to acquire that velocity? In the following Table, the column titled T denotes the le. conds of time from i" to 60"; S the spaces passed over in any second of time. The third column gives the heights from which a body would fall at the end of any given time, from }" to 60"; and column 4th denotes the last acquired velocity at the end of any given time. Thus, at the end of 22 seconds, the body has fallen from the height of 7744 feet, and moves with a velocity of 704 feet per second. |