MT 



GUNNERY. 



CCNM-UV. 



and substituting hi (2) the value of ( derived from (1), 



v sin a . * 2* 



9 



xtono ~ - 



rcosa " v : co*a 2vooV (8) 



Let A represent the height which a body must fall to obtain the 

 Telocity (T), when acted upon by the force of gravity only ; then 



V = 2.;*, 

 and substituting this value in equation (3) 



which is the equation to the parabola, the diameters of the curve being 

 parallel to A v. From this equation all the properties relating to 

 projectiles moving in a vacuum may be derived. 



To fnd Ike Time of Plight. To find the time of flight T we have 



GX = AO sin o= vr sin a 



.'.4?T > =vt sin a .'. and T = 



2 v sin o (5) 



a ' 



The time of flight varies, therefore, with the sine of the angle of 

 elevation. 



The same result might also have been obtained by putting y= o in 

 equation (2). 



To Jitid the Horizontal Range. To find the horizontal range we 

 have, 



A x = A o cos a 



AQ = VT 



2 vein a 

 T= - by (5) 



or, AX 



2 v 8 sin a cos a 



9 



= 2 A sin 2 a, 



(c) 



The same result would be obtained by putting y=o in equation (3). 



It appears, then, that when h, or the initial velocity, is constant, 

 A X, or the horizontal range, varies as sine 2 a ; and the value of A X is 

 greatest when 2a = 90, or a = 45, in which case AX = 2A. 



Again, with the same initial velocity there are generally two 

 angles which will give the same horizontal range. For the sine 

 2o = sin (108 2 a) and the ranges will be equal at equal distances 

 from 45*. 



Lastly, with the same angle of projection the initial velocities are as 

 the square root of the ranges. 



To .find the greatett Iteiyht to which the Shot will Rite. In order to 

 find the greatest height to which the shot will rise, bisect A x in L, 

 then A L = L sin 2 o-2 A sin a cog a , and putting ML = Z, x = A n = 2/i 

 sin a cos a z, substituting this value (8), 



(2 h sin o cos a z) 1 

 up=y=(2Aain a cos az) tan a 



4 h cos* a 



f z* 1 



= 2 A sin 1 a z tan a < h sin* o 2 tan o + 4 h COB t } 



=h sin* a - 



4 A cos' a 



(7) 



11 p or .y has, therefore, its greatest possible value when z = o. 



LK is therefore this greatest height, and is equal to h sin 1 a. 

 Equation (7) shows that the curve is symmetrical with respect to the 

 line LK. 



To find the Time of Flight, Range, d-c., when the Plane it not Ifori- 

 :o*tal. Should the line joining the object and the piece not lie in a 

 horizontal plane, then , as the direction of the force of gravity is not 

 perpendicular to the plane in which the gun and object are situated, 

 the foregoing foniiuhc will not be applicable. 



Figure 2. 



> 



\. : \\ represent a section of the horizontal plane, and A B that of 

 the plane containing the gun and object B, 



T a time of describing the curve A F B. 

 OB:A.o::sinoAB:sinABO 



^QAB = (O fl); 



Z A Bo = 180- / ABX ; 

 sin ABu=sin/ ABX=COS/S; 



By substituting these values hi the above proportion we have 



{'IT 3 : VT : : sin (a-/3) : cos P 

 {</ T 1 cos (3 = v T sin (a /3) 

 2 v sin (a /8) 





0) 



Substituting this value of T in the equator A o= 

 2 sin (o /5) * 



v T, 



-I/. 



j/oos/S 

 an (a 13) 



Oaf d 



Also, sin AOB = COS GA x = cos a 

 sin A Bd = sin ABX=cos/3. 



Substituting these values in the proportion 



AB :AO : : Bin AUB : sin ABO 

 we have, 



, sin (o fl) 

 A B : 4 A co, n : : cos a : cos $ 



sin (a J3). cos a 

 ::AB= cos/3 (2)* 



If the plane be a descending one, the angle 3 must be considered 

 negative. 



On the Application of the Equation to the Curve deterilted by a Body 

 in a Vacuum to Mortar Practice. Although the foregoing theory will 

 give results very different from those found in practice with shot 

 projected with great velocities, still, for the ordinary practice with 

 mortars, it will not be far from correct, as may be seen by an inspection 

 of the following tables, extracted from a work entitled ' Traitd de 

 Blastique,' by M. Didion ; the mortars corresponded to our 8-inch and 

 10-inch. 



Table of Ranges calculated without taking into account the resistance 

 of the air, and the experimental ranges : 



* Equations (1) and (2) may also be determined from the equations to the 

 cutve In tbe following manner : 



AB COS /S = X \ COS OT 



AB in /8=y = v sin at i 0T* 

 tan 0= - = tan o 



m 2 v cos o 



2 v (tan o tan f) cos a 



(1) 



Again, 



:% -in (,:-,-') 



I cci- li 



y.z tan a 



f*' 



2 v cos 1 a 



n sin /S=RCO/! Una 



roo'0 



iv'coa'a" 60 ** tano 

 _ ain(a-l) 



ic> a 

 2 v' ln (o t ) co a 



*" "~ g cos* t 

 4 A sin (a -0) coin 



f R COS* 



