Projections on./18cents and Descents, fig. 8, 9.
If the mark can be hit only with one di rection A G, the impetus in ascents will be equal to the sum of half the inclined plane and half the perpendicular height, and in descents it will be equal to their difference ; but if the mark can be reach ed with two directions, the impetus will ba greater than that sum or difference. For when A G is the line of direction, the anglegGAbeing..-.MAG=GAs; G g A s, and g z added to or substract ed from both, makes G z half the impetus equal to the sum or difference of A g, a fourth part of the inclined plane, and g z a fourth part of the perpendicular height. In any other direction F P is greater than F o •-= A F ; and F f, added to or subtract ed from both; makesfP half the impetus greater than. the sum or difference of A F, a fourth part of the inclined plane, and F fa fourth part of the perpendicular height. Whence, if in ascents the impe tus be equal to the sum of half the inclin ed plane and half the perpendicular height, or if in descents it be equal to their difference, the mark can be reached only with one direction ; if the impetus is greater than that sum or difference, it May be hit with two directions; and if the im petus is less, the mark can be hit with none at all.
Prob. IL The angles of elevation, the horizontal distance, and perpendicular height, being given, to find the impetus. Fig. 8, 9.
From these data you have the angle of obliquity, and length of the inclined plane ; then as As: AM:: S. angle A M : S. angle A s M : S. angle s A F: S. angle M A F, and AF : A s:: S. angle A S : S. angle M A F ; whence, by the ratio of equality, AF:AM ::S.Angle sA F x S. angle M A : S. angle MAex S. angle M A which gives this rule.
Add the logarithm of A F to twice the logarithmic sine of the angle M A F ; from their sum subtract the logarithmic sines of the angles s A F and M A s, and the remainder will give the logarithm of A M • the impetus.
When the impetus and angles of eleva tion are given, and the length of the in clined plane is required, this is the rule.
Add the logarithm of A al to the logarith mic sines of the angles s A F and M A from their sum subtract twice the loga rithmic sine of angle M A F, and the re mainder will give the logarithm of A F, the fourth part of the length of the inclin ed plane.
It the angle of elevation t A II, and its amplitude A B (fig. 11,) and any other angle of elevation t A H is given ; to find the amplitude A b for that other angle, the impetus A M and angle of obliquity D A H remaining the same.
Describe the circle A G M, take A F a fourth part of A B, and A f a fourth part of A b ; from the points F, f, draw the lines F a, and f p parallel to A M, and cutting the circle in the points 8, p ; then AF:AM:: S. angle sAFx S. angle M A S. angle MAF x S. angle MAP; and A M : A j; : S. angle MAF x S. angle M A F : S. anglep Af x s. angle p A M ; whence by the ratio of equality, A F: Af : : S. angle s A Fx S. angle M A s : S. angle p Af x S. angle p A M, which gives this rule. ' Add the logarithm of A F to the loga rithmic sines of the anglesp At p A M; from their sum subtract the logarithmic sines of the angles s A F, a A M, and the remainder will give the logarithm of A f, a fourth part of the amplitude required.
Prob. III. To find the force or velocity of a ball or projectile at any point of the curve, having the perpendicular height of that point, and the impetus at the point of projection given. From these two data find out the impetus at that point ; then 2 X 16 feet 1 inch is the velocity acquir ed by the descent of a body in a second of the square of which (4 X the square of 16 feet 1 inch) is to the square of the velocity required, as 16 feet 1 inch is to the impetus at the point given; wherefore multiplying that impetus by four times the square of 16 feet 1 inch, and dividing the product by 16 feet 1 inch, the quotient will be the square of the required veloci ty: whence this rule. Multiply the im petus by four times 16 feet 1 inch, or 64 feet I and the square root of the product is the velocity.