; '" ' - - " ''T^^^imi^mrrf^'yf^^vfr^'.«*... AD-A010 816 ON THE DEFLECTION OF PROJECTILES DUE TO ROTATION OF THE EARTH G. Preston Burns Naval Surface Weapons Center Dahlgren Laboratory, Virginia June 1975 DISTRIBUTED BY: Mi] National Technical Information Service U. S. DEPARTMENT OF COMMERCE
SECURITY CLASSIFICATION OF THIS PAGE (Whm» Dalm Enltfd) I REPORT NUMBER TR-3218 REPORT DOCUMENTATION PAGE 4 TITLE («nd Subtitle) CN THE DEFLECTION OF PROJECTILES DUE TO ROTATION OF THE EARTH READ INSTRUCTIONS BEFORE COMPLETING FORM 2. GOVT ACCESSION NO 3. RECIPIENT'S CATALOG NUMBER 5 TYPE OF REPORT 4 PERI OD COVERED 6. PERFORMING ORG. REPORT NUMBER 7. AuTMORft; G. Preston Bums I. CONTRACT OR GRANT NUMBERf») 9 PERFORMING ORGANIZATION NAME AND ADDRESS Naval Surface Weapons Center Dahlgren Laboratory Dahlgren, Va. 22448 10. PROGRAM ELEMENT, PROJECT. TASK AREA ft WORK UNIT NUMBERS ' 1 CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE June 1975 IS. NUMBER OF PAGES 14 MONITORING AGENCY NAME & AODRESSff/ dlttortnt from Controlling Olllc») IS. SECURITY CLASS, (ol Ihli nporl) UNCLASSIFIED IS«. DECLASSlFICATION DOWNGRADING SCHEDULE 16 DISTRIBUTION STATEMENT fo/thli R»por(; Approved for public release; distribution unlimited. 17 DISTRIBUTION STATEMENT (ol Iht»btttmct nlcrad (n Block 20, II dllltfnl Inm Report; IB SUPPLEMENTARY NOTES PRICES SUBJEQ TO CHANGE 19 KEY WORDS (Continue on r»v«raa ajcf«u n«c«««aiy «nd tdmntity by Mock numbmr) Reproduced by NATIONAL TECHNICAL INFORMATION SERVICE US Department of Commorce Spnrglield. VA. J215I 20. ABSTRACT fconilnu«on rmvrt»»ida It n»c»i»mry mnd Idtnllly by block numbtt) An analysis of the effect of the Coriolis acceleration on the flight of projectiles fired fron the surface of the Earth has been extended to include bodies fired or released from any altitude above the surface of the Earth. An equation for Coriolis deflection perpendicular to the line of fire and the general condition for zero deflection are derived. Directions of Coriolis deflections are tabulated for bodies released horizontally at altitude y 0. "*, J N M 7J 1473 EDITION OF I NOV 8» IS OBSOLETE S/N OI02-014-660I ' SECURITY CLASSIFICATION OF THIS»AGE Cirhon Data KntarodJ
r,. - - ;.....-.-- ;,. - NSWC/DL Technical Report rr-321ö June 1975 ON THE DEFLECTION OF PROJECTILES DUE TO ROTATION OF THE EARTH by G. Preston Burns Warfare Analysis Department r 1 * f^v i i! JÜN IS 1975 ^ij^.^ ij. - c; Approved for public release; distribution unlimited.
.iili.!jij^ip,i)ipijjii^i.^!i 'wijwmwbiil..u»t*^^, ', «r^?^www^i^w!^^wr!?^v7-^rpt FOREWORD This report, referred to as Reference 7 in NWL Technical Report TR-3061 published in April, 1974, is one of several written on the effect of Coriolis acceleration on the flight of a projectile. It contains a derivation of the expression for DOMEGA, the Coriolis deflection perpendicular to tii line of fire. This work was performed in the Aeroballistics Division of the Warfare Analysis Department under ORDTASK 551028090. Released by: Q^k oyuju^ RALPH A. NIEMANN Head, Warfare Analysis Department
^M'MH" *!.1 ^W. WiiiLL!^i..!lJ.Ui:w1j..y..n i.n. T--,T7r.i-wri-r. " '-.TTWiTWf! ABSTRACT An analysis of the effect of the Corlolis acceleration on the flight of projectiles fired from the surface of the Earth has been extended to include bodies fired or released from any altitude above the surface of the Earth. An equation for Coriolis deflection perpendicular to the line of fire and the general condition for zero deflection are derived. Directions of Coriolis deflections are tabulated for bodies released horizontally at altitude y. ii
w ""^'""" nvw>'"!»<ik,i,t i^y!iipj^!ij<*pipwpwi5ipi CONTENTS Page FOREWORD i ABSTRACT ti I. INTRODUCTION 1 II. ANALYSIS AND RESULTS 2 III. CONCLUSIONS 4 References 8 iii
mßr. ity." r V^- T "~^r)j",»w. -' I. LNiRODUCiION An analysis was made*- of the effect of the Corlolls acceleration on the flight of projectiles fired from the surface of the Earth and the condition for zero deflection was derived. It was assumed that the acceleration of gravity is constant; velocities due to Coriolis accelerations may be neglected in computation of Coriolis accelerations; air resistance is negligible; and the impact posicion is in the plane tangent to the Earth at the gun. The analysis has been extended, using the same assumptions, to include bodies fired or released from any altitude (y 0 ) above the surface of the Earth.
r^^c^-'-t """" II. ANALYSIS AND RESULTS Consider a projectile fired from altitude y 0 above a point on the Earth at latitude 9 in the northern hemisphere with initial velocity Vo having azimuth 0 (measured clockwise from north) and elevation angle t. The time of flight is given by T = (V 0 sin «+ (vs sins e + Zg'y V^/g' (1) where g' = g - 2V 0 U) cos e sin 0 cos 0. The second term in the expression for g' is the vertical component (ZV^io cos 9) of the Cortolis acceleration. Vp is the magnitude of the horizontal east component of projectile velocity and (JU is the angular rate of rotation of the Earth. The horizontal Corlolis displacements are, in the east (x) direction^, x = VQUJ T 2 sin 9 cos e cos 0 -» resulting from the horizontal north component (V N )of projectile velocity; and, in the south (z) directionl, z = VQIUT 2 sin 9 cos e sin 0 resulting from the horizontal east component (V,,). In the west (-x) direction d^x/dts = -2V v a) sin (90-9) from which, using Vy = V sin e - g't and integrating from t = 0 to t = r, ' 0 x = -VQUJT 2 sin e cos 9 + (Ug'l^Ccos 9)/3. Vy is the magnitude ol the vertical component oi projectile velocity. The net horizontal displacement perpendicular to the line of fire is DT = V ^T 2 (sin 9 cos cos p ti + sin 9 cos e sin :, 0 - sin cos 9 cos 0 + g^t cos 9 cos 0) 3V 0 = V 0 U)'L' 2 (sin 9 cos e - sin c cos 9 cos 0 + g' T cos 0 cos 0). (2) 3Vn For no deflection and tan 9 = (tan e DT = 0 ^x 3V 0 cos ) cos 0. (3)
pwyin <i ;* r."*7'?ji^>'.y»m i^.inwhipm^t'.'.ij. - - -w^i ivt^.'^^w.^j.alyi ^..^lymipi» ^inj.. IIHHUIM^MMWIH ' " H.-T- ^^,,. -.. This is the general condition for zero detlection. If y 0 = 0, T = 2V 0 (sin e )/g,, and Eq. (3) reduces to tan e = 3(tan 9)/cos 0. (4) Conclusions for this case are given in the previous analysis. If = 0, T = (Zyo/g') 2, and we have Irom Eq. (3) V 0 = -(2g, y 0 )^ (cos 0)/(3 tan 9). (5) therefore, neglecting air resistance, the initial velocity for zero deflection of a body projected horizontally at azimuth 0 from altitude y 0 above the. surface of the Earth at latitude 9 varies directly as the square root of the altitude.
miauiuiw^ri^iiiji.ipij;!«!«iii. CONCLUSIONS The following conclusions are drawn from a study of liqs. (3) and (5) and Figure 1 for a body projected horizontally or dropped vertically downward from a point above the surface of the Earth: (1) If the body is released horizontally in a northward direction (270 < 0 < 90 ) in the northern hemisphere (Figure 1), the deflection is to the right for all velocities. If it is released horizontally in a southward direction (90 < 0 < 270 ) in the northern hemisphere, the deflection is to the right or left depending on whether the horizontal velocity is greater or less, respectively, than that given by Eq. (5) for zero deflection. Horizontal velocities required for zero deflection are given in lable I for different latitudes for azimuth 180 degrees. These were calculated using g = 32 ft/sec^, the approximate average value for altitudes between zero and 100,000 feet. To obtain values for other azimuth angles, multiply the table values by the fibsolute value of cos 0. (.2; If the body is released horizontally in a southward direction in the southern hemisphere (Figure 1), the deflection is to the left for all velocities. If it is released horizontally in a northward direction in the southern hemisphere, the deflection is to the left or right depending on whether the horizontal velocity is greater or less than that given by Eq. (5) for zero deflection. (3) A body projected or released directly east (0 = 90 ) or vest (0 = 270 ) from a point above the surface of the Earth will deflect to the right in the northern hemisphere and to the left in the southern hemisphere; if released directly east or west horizontally from a point directly above the equator, the deflection is zero. (4) A body projected horizontally from a point directly above the north pole will deflect to the right; a body projected horizontally from a point directly above the south pole will deflect to the left, (5) A body dropped vertically down from a point above the surface of the Earth, except at the poles, will deflect toward east; if dropped vertically down from a point above either pole, the deflection is zero. A summary of ^oriolis deflections of bodies released horizontally or vertically downward at altitude y 0 above the surface of the Earth is given in Table II.
Jli i,l«^«hi)»ijwwi^jij,«l«ul*itk^ 7»-u..«.i»i.!!1«;J-ti!.i..,»»- ui.»wt i' FIGURE 1 EARm AND BODY VELOCIiT COMPONENTS FOR HORIZONiAL, BODY PROJECTION
- i n.l^bljnp TABLE I Horizontal Velocities (fc/sec) Required tor Zero Cortolis Deflection (0 = 180 )* Allitude (It) 15 Latitude {den) 30 hi 60 15 0 0 0 0 0 0 500 223 103 60 34 16 1000 315 146 84 49 23 1500 385 179 103 60 28 2000 445 207 119 69 32 2500 498 231 133 77 3 b 3000 545 253 146 84 39 4000 629 292 169 97 45 5000 704 327 189 109 51 booo 771 358 207 119 55 7000 833 386 223 129 60 8000 890 413 239 138 64 9000 944 438 253 14b 68 10000 995 462 267 134 71 20000 1408 653 377 216 101 30000 1724 800 462 26 7 124 40000 1991 924 533 308 143 50000 2226 1033 596 344 160 60000 2438 1131 653 377 175 70000 2633 1222 706 407 189 'To obtain values for other azimuth angles, multiply l\\a table valut-: the absolute value of cos 0. by
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.11 Jji^il.ll i.i w^ni^w^^^^wrk^w"^^ '^ ' ' mi<" '<W'-f.Wm''jmiim liv^- EKT^"«" 1«",!. - REFERENCES» 1. Burns. G. Preston. "Deflection of Projectiles due J 0 J* 0^1 f 1 the Earth." wrlcm Journal of Physics. November 1971, p. 1329.