Angle of Elevation.
PRO ( 888 )
If then the Celerity of two 'Pro-
PRO
jetftles be the fame, the Parameter is the fame. Wherefore, fince the Semiparameter of the Path, in the one Cafe, is to the Amplitude, as the whole Sine, to the Sine of double the Angle of Elevation ; and the Semiparameter of the Path in the other Cafe is to the Amplitude as the whole Sine to the Sine of double the Angle of Elevation : We may fay farther, as the Amplitude is to the Sine of the Angle of double the Elevation in the one Cafe, fo is the Amplttude to the Sine of the Angle of double the Elevation in the other Cafe. The Amplitudes, therefore, or Magnitudes of the Paths, are as the Sines of double the Angles of Ele- vation ; the Velocity of the ProjeBile remaining the fame.
6?. The Celerity of the Projetlile being the fame, the Amplitude AB is greateft, i. e. the Range of the 'Pro. jetlile is greateft, at an Angle of Elevation of 45 ; and the Amplitudes or Ranges, at Angles of Elevation equally diftant from 45 , are equal.
This is found by Experiment ; and is likewile demon- ftrable thus : Since the Ratio of the Sine of double the Angle of Elevation to the Amplitude is always the fame, while the Celerity of the Projectile remains the fame ; as the Sine of double the Angle of Elevation increafes, the Amplitude will increafe. Wherefore, fince the Sine of double the Angle of Elevation of 45° is Radius, or the largeft Sine ; the Amplitude, or Range in that Elevation mutt be the greateft. Again, fince the Sines of Angles equi- diftant from right Angles ; e.gr. 8c« and ioo" are the fame ; and the double Angles mutt be equi-diflant from a righr Angle, if the fimple ones be fo : The Amplitudes or° Ranges at Elevations equi-diftant from 45°, muft be equal.
Hence, fince as the whole Sine is to the Sine of double the Angle of Elevation ; fo is the Semi-parameter to the Amplitude ; and the whole Sine is equal to double the Sine of the Angle of Elevation, if that be 45' : Under the Angle of Elevation 45°, the Amplitude is equal to the Semi-parameter.
7'. 'The greateft Range or Amplitude being given ; to determine the Amplitude or Range under any other given Angle of Elevation ; the Celerity remaining the fame. Say thus ; As the whole Sine is to the Sine of double the Angle of any other Elevation ; fo is the greateft Amplitude or Range, to the Amplitude required.
Thus, fuppofe the greateft Range of a Mortar at 45', to be 6000 Paces, and the length of the Range at 30°, requi- red ; it will be found 5195 Faces.
S°. The Velocity of a Projectile being given, to find its greateft Range or Amplitude. Since the Celerity of the 'Projehile is given in the Space it will pafs over by the im- preffed Force ; e.gr. in one Second; there is nothing re- ciuired but to find the Parameter of the Path, (by Corel. 2. o'fthe 3d Law.) for half of this is the Amplitude or Range required.
Suppofe, e.gr. the Celerity of the Proyttile fuch as that in one Second it will run over 1000 Feet, or nooolnches : If then 144000000 be divided by 1S1, the Quotient will give the Parameter of the Path 79558a Inches, or 0^298 Feet. The Range or Amplitude required, therefore, ■'■
3149.
AnyObieft, therefore, found within this Extent fcribe the Path A m B.
1 2. The Altitude of the Range t m is to the eighth part of the Parameter, as the verfed Sine of double the Angle of Elevation to the whole Sine.
Hence, 1. fince, as the whole Sine is to the verfed Sine of double the Angle of Elevation in one Cafe ; fo is the eighth part of the Parameter to the Altitude of the Ra„g e . And as the whole Sine is to the verfed Sine of double the Angle of Elevation in any other Cafe ; fo is the eighth part of the Parameter to the Altitude : but the Velocity remaining the fame, the Parameter, in different Angles of Elevation, will likewife be the fame : The Altitudes of the Ranges under different Angles of Elevations are a s the verfed Sines of double their Angles, a. Hence, alfo, the Velocities remaining the fame, the Altitudes of the Ranges are in a duplicate Ratio of the Sines of double the Angles of Elevation.
13. The Horizontal Diftance of any Mark or ObjeB, ro . gethcr ivith its Height above, or Depth beneath the Hori- zon, being given ; to find the Angle of Elevation required to hit the faid ObjeB.
Wolfius gives us the following Theorem, the Rcfult of a regular Inveftigation : Suppofe the Parameter = a ; ln = b, AI=c, the whole Sine=r. Then, as c is t„ i a -^-y'Q a ' — ab~c'-.) So is the whole Sine 1 5 to the Angle of Elevation required R A B.
Dr. Halley gives the following eafy, and compendious Geometrical Conftruclion of the Problem ; which he like. wife deduces from an analytical Inveftigation.
Having the right Angle LD A (j%.480 make DA, DF, the greateft Range, DG the horizontal Diftance, and DB, D C, the perpendicular Height of the Objefl ; and draw GB, and make DE equal thereto, Then with the Ra- dius A C, and Centre E, fweep an Arch, which, if the thing be poffible, will interfect the Line A D in H ; and the Line D H being laid both ways from F, will give the Points K and L j to which draw the Lines G L, GK.
Here, the Angles L G D, K G D, are the Elevations re- quired for hitting the Object B.
But note, that if B be below the Horizon, its Dcfcent D C = D B, muft be laid upon A, fo as to have A C = to AD+DC. Note, likewife, that if in Defcents, DH be greater than F D, and fo K fall below D ; the Angle K G D, fhall be the Depreffion below the Horizon.
It may be here obferved, that the Elevation fought con- ftantly biSects the Angle between the Perpendicular and the Objeft. This, the Author was not aware of, when he gave the firft Solution of the Problem 5 but upon difcover- ing it, obferves, that nothing can be more compendious; or bid fairer for the perfection of the Art of Gunnery ; fince 'tis here, as eafy to fhoot with a Mortar at any Objecl in any fituation, as if it were on the Level ; nothing more be- ingrequired but to lay the Piece, fo as to pafs in the middle Line between the Zenith and the Objeft, and giving it the due Charge. See Mortar.
14. The Times of the Projections, or Cafls under different Angles of Elevation, the Velocity remaining the fame, are as the Sines of the Angles of Elevation.
15. TheVelocity o/« Projectile, together ivith the Angle of Elevation R A B being given (fig-470 5 to find the Range, or Amplitude A B, and the Altitude of the Range t m, and de-
Tothe horizontal Line A B erect
may be ftruck by thtprojetfite.
9°. The greateft Range or Amplitude being given ; to find the Velocity of the Projeaile, or the Horizontal Space it will pafs over in a Second. Since double the greateft Am- plitude is the Parameter of the Path ; between double the greateft Amplitude and the Space pafs'd over in a Second by a Body falling perpendicularly, viz. 181 Paris In- ches, find a mean Proportional; for this will be the Space defctibed by the ProjeSilc in the given Second.
Thus, if the greateft Amplitude be 1000 Feet, or 12000 Inches, the Space required will be =/ ("12000.18 1)= 120 Feet and 4 Inches.
10. To determine the greateft Altitude to which aSody obliquely fYo)n&ti willrife. The Rule is; Biffefl the Am- plitude A B in t, and from the Point t erect a perpendicular tm; this tm will be the greateft Altitude to which the Body projected, according to the Direction A R, will arife.
11. The Range or Amplitude A B, and the Angle of E- levation B A R, being given ; to determine the greateft Al- titude of the Projectile. If A R be taken for the whole Sine, B R will be the Sine, and A B the Co-fine of the Angle of Elevation BAR: Wherefore, fay, as the Co- iine of the Angle of Elevation is to the Sine of the fame ; fo is the Amplitude A B to a fourth Number ; which will be B R ; the fourth part whereof is the greateft Altitude required.
Hence, fince from the given Velocity of a Projetlile, its greateft Range or Amplitude, and thence its Range under any other Angle, is found ; the Velocity being given, the greateft Altitude of the Projecfile is likewife found.
a Perpendicular A D, which is to be the Altitude whence the ProjeQile falling, might acquire the given Velocity : On A D defcribe a Semi-circle A Q_D, cutting the Line of Direction A R in Q_: Thro' Q_draw C m parallel to A B, and make C Q_= Qffz- From the Point m let fall a Per- pendicular mt to AH : Laftly, thro' the Vertex M de- fcribe the Parabola A m B.
Here, Am B is the Path fought, 4 C QJts Amplitude or Range, tm the Altitude of the Range, and 4 CD the Parameter.
Hence, 1. The Velocity of a 'ProjeEiile being given, the Amplitudes and Altitudes of all the poffible Ranges are given at the fame time. For, drawing E A, we have under the Angle of Elevation EAB, the Altitude A I, and the Amplitude 4 I E : Under the Angle of Elevation FAB, the Altitude A H, the Amplitude 4 H F. 2. Since A Bis perpendicular to A D, it is a Tangent to the Circle in A : Hence the Angle A D Q_is equal to the Angle of Elevation R A B ; confequently A I M is double the Angle of Ele- vation, and therefore C Q; the fourth part of the Amplitude. is the right Sine ; A C the Altitude of the Range, the ver- fed Sine of double the Angle of Elevation.
i«. The Altitude tm of a Caft>w, or its Amplitude AB, together with the Angle of Elevation R A B, being gi ven i to find the Velocity wherewith the Projetlile firft moved, that is, the Altitude A D, in falling from whence it «<»»<} acquire the like Velocity : Since A C = t m is the verier! Sine, CO=i AB, the right Sine of double the Angle ot Elevation A I Qj To the verfed Sine of double the Angle of Elevation, find the whole Sine, and the height or tns
