Thus suppose the impetus at the point of projection to be 3,000, and the per pendicular height of the other point 100; the impetus at that point will be 2,900. Then 2,900 feet multiplied by 643 feet gives 186,566 feet, the square of 432 nearly, the space which a body would run through in one second, if it moved uni formly.
And to determine the impetus or height, from which a body must descend, so as at the end of the descent it may acquire a given velocity, this is the rule: Divide the square of the given velocity (expressed in feet run through in a se cond) by 643 feet, and the quotient will be the impetus.
The duration of a projection made per pendicularly upwards is to that of a pro jection in any other direction whose im petus is the same, as the sine complement of the inclination of the plane of projec tion (which in horizontal projections is ra dius) is to the sine of the angle contained between the line of direction and that plane.
Draw out A t (fig. 8,) till it meets m B continued in E, the body will reach the mark B in the same time it would have moved uniformly through the line A E; but the time of its fall through M A the impetus, is to the time of its uniform mo tion through A E, as twice the impetus is to A E. And therefore the duration of the perpendicular projection being dou ble the time of its fall, will be to the time of its uniform motion through A E, as four times the impetus is to A E; or as A E is to E 13; that is, as A t is to t which is as the sine of the angle t D A (or M A B its complement to a semicircle) is the sine of the angle t A D.
Hence the time a projection will take to i arrive at any point in the curve may be found from the following data, viz. the im petus, the angle of direction, and the in clination of the plane of projection, which in this case is the angle the horizon make with a line drawn from the point or pro jection to that point, Hence also, in horizontal cases, the du rations of projections in different direc tions with the same impetus are as the sines of the angles of elevation. But in ascents or descents, their durations are as the sines of the angles which the lines of direction make with the inclined plane. Thus, suppose the impetus of any projec tion were 4,500 feet ; then 16 feet 1 inch: 1'r : : 4,500 feet : 275", the square of the time a body will take to fall perpendicu larly through 4,500 feet, the square root of which is 16" nearly, and that doubled gives 32", the duration of the projection made perpendicularly upwards Then, to find the duration of a horizontal pro jection at any elevation, as 20°; say It S.
angle : 32" : duration of a projection at that elevation with the impetus 4,500. Or if with the same impetus a body at the direction of 35° was projected on a plane inclined to the horizon 17°, say as sine 73° : sine : : 32" : duration re quired.
The tables in the net Ieaf, at one view, give all the necessary cases, as well for shooting at objects on the plane of the horizon, with proportions for their solu tions, as for shooting on ascents and de-. scents. We shall in this place mention some or the more important maxims laid down by Mr. Robins, as of use in prac tice. 1. In any piece of artillery, the greater quantity of powder with which it is charged, the greater will be the veloci ty of the bullet. 2. If two pieces of the same bore, but of different lengths, are fired with the same charge of powder, the longer will impel the bullet with a great er celerity than the shorter. 3. The ranges of pieces at a given elevation are no just measures of the velocity of the shot : for the same piece fired successively at an in variable elevation, with the powder, bul let, and every other circumstance, as near ly the same as possible, will yet range to very different distances. 5. The greatest part of the uncertainty in the ranges of pieces arises from the resistance of the air. 6. The resistance of the air acts up on projectiles by opposing their motion, and diminishing celerity; and it also di verts them from the regular track which they would otherwise follow. 7. If the piece of cannon be successively fired at an invariable elevation, but with va rious charges of powder, the greatest charge being the whole weight of the ball in powder, and the least not less than the fifth part Of that weight ; then, if the ele vation be not loss than eight or ten de gress, it will be found that some ranges, with the least charge, will exceed some of those with the greatest. 8. if two pieces of cannon with the same bore, but of different lengths, are successively fired at the same elevation, with the same charge of powder, then it will frequently happen that some of the ranges with the shorter piece will exceed some of those With the longer. 9. Whatever operations are performed with artillery, the least charges of powder with which they can be effected are always to be preferred. 10. No field-piece ought at any time to be loaded with more than one-sixth, or at most one-fifth, of the weight of its bullet in powder, nor should the charge of any battering piece exceed one-third of the weight of its bullet.