3049:
proportional to the time since the ball was sent into the air, usually by being struck with a bat. Even when the ball is really descending, near the end of its flight, its angle of elevation seen by the player continues to increase. The player therefore sees it as if it were ascending vertically at constant speed. Finding the place from which the ball appears to rise steadily helps the player to position himself correctly to make the catch. If he is too close to the batsman who has hit the ball, it will appear to rise at an accelerating rate. If he is too far from the batsman, it will appear to slow rapidly, and then to descend.
1952:
1723:
50:
147:
359:
3048:
If a projectile, such as a baseball or cricket ball, travels in a parabolic path, with negligible air resistance, and if a player is positioned so as to catch it as it descends, he sees its angle of elevation increasing continuously throughout its flight. The tangent of the angle of elevation is
765:
1726:
Trajectories of projectiles launched at different elevation angles but the same speed of 10 m/s in a vacuum and uniform downward gravity field of 10 m/s. Points are at 0.05 s intervals and length of their tails is linearly proportional to their speed.
498:(which was also initiated by Newton in his youth). Over the centuries, countless scientists have contributed to the development of these two disciplines. Classical mechanics became a most prominent demonstration of the power of rational thought, i.e.
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2195:
1678:
2072:
611:
1941:
412:
A familiar example of a trajectory is the path of a projectile, such as a thrown ball or rock. In a significantly simplified model, the object moves only under the influence of a uniform gravitational
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303:
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at every relevant position, there is a way to infer the associated force that would act at that position, say from gravity. Not all forces can be expressed in this way, however.
2708:
1372:
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It is theoretically possible for an orbit to be a radial straight line, a circle, or a parabola. These are limiting cases which have zero probability of occurring in reality.
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1095:
1138:
1419:
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1303:
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791:
2803:
2395:
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1977:
1708:
1259:
2808:
2335:
850:. To neglect the action of the atmosphere in shaping a trajectory would have been considered a futile hypothesis by practical-minded investigators all through the
2403:
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1997:
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943:
923:
903:
597:
577:
553:
530:
343:
323:
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at rest with respect to the ground. Associated with the frame is a right-hand coordinate system with its origin at the point of launch of the projectile. The
2206:
846:
The ideal case of motion of a projectile in a uniform gravitational field in the absence of other forces (such as air drag) was first investigated by
2083:
3230:
1584:
471:, and artificial spacecraft to a reasonably good approximation, although if a comet passes close to the Sun, then it is also influenced by other
760:{\displaystyle m{\frac {\mathrm {d} ^{2}{\vec {x}}(t)}{\mathrm {d} t^{2}}}=-\nabla V({\vec {x}}(t)){\text{ with }}{\vec {x}}=(x,y,z).}
2005:
3211:
3186:
424:. Generally when determining trajectories, it may be necessary to account for nonuniform gravitational forces and air resistance (
1871:
1791:
3029:
444:
2959:
114:
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1495:
86:
3053:
399:
133:
2716:
416:. This can be a good approximation for a rock that is thrown for short distances, for example at the surface of the
251:
93:
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2912:
71:
2713:
To find the angle giving the maximum height for a given speed calculate the derivative of the maximum height
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17:
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67:
38:
31:
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of the potential, taken at positions along the trajectory. This is the mathematical form of Newton's
447:. In the gravitational field of a point mass or a spherically-symmetrical extended mass (such as the
2628:
1052:
881:
In the analysis that follows, we derive the equation of motion of a projectile as measured from an
377:
1100:
3090:
3085:
1377:
1163:
533:
60:
1264:
2902:{\displaystyle {\mathrm {d} H \over \mathrm {d} \theta }=v^{2}2\cos(\theta )\sin(\theta )/(2g)}
1951:
1722:
1203:
867:
798:
773:
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1962:
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603:
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413:
187:
183:
502:, in science as well as technology. It helps to understand and predict an enormous range of
3105:
2313:
245:
3221:
2483:{\displaystyle {\mathrm {d} R \over \mathrm {d} \theta }={2v^{2} \over g}\cos(2\theta )=0}
8:
3256:
3095:
491:
487:
440:
369:
222:
179:
1374:. The co-ordinates of this free-fall frame, with respect to our inertial frame would be
107:
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1982:
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3212:
Parabolic
Projectile Motion: Shooting a Harmless Tranquilizer Dart at a Falling Monkey
1492:
Now translating back to the inertial frame the co-ordinates of the projectile becomes
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150:
Illustration showing the directional trajectory of a bullet fired at an uphill target
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237:
233:
171:
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3214:
by
Roberto Castilla-MelĂ©ndez, Roxana RamĂrez-Herrera, and JosĂ© Luis GĂłmez-Muñoz,
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847:
840:
381:
2275:{\displaystyle \theta ={\frac {1}{2}}\sin ^{-1}\left({\frac {gR}{v^{2}}}\right)}
241:
1329: = 0. The equation of motion of the projectile in this frame (by the
925:
axis is perpendicular to it ( parallel to the gravitational field lines ). Let
832:
824:
579:
represents external forces of a particular kind known as "conservative". Given
425:
229:
3201:
1770:, is the speed at which said object is launched from the point of origin. The
3250:
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3024:
If instead of a uniform downwards gravitational force we consider two bodies
2190:{\displaystyle R=2v^{2}\cos(\theta )\sin(\theta )/g=v^{2}\sin(2\theta )/g\,.}
1865:, is the greatest parabolic height said object reaches within its trajectory
452:
163:
3240:
Java projectile-motion simulation; targeting solutions, parabola of safety.
3033:
1673:{\displaystyle y=-{g\sec ^{2}\theta \over 2v_{0}^{2}}x^{2}+x\tan \theta ,}
483:, which modify the orbit and cause the comet to eject material into space.
429:
1785:
is the respective gravitational pull on the object within a null-medium.
851:
3145:
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giving the maximum range can be found by considering the derivative or
2200:
This equation can be rearranged to find the angle for a required range
476:
433:
198:
1314:
871:
503:
460:
202:
949:. Relative to the flat terrain, let the initial horizontal speed be
420:. In this simple approximation, the trajectory takes the shape of a
49:
3228:
Java projectile-motion simulation, with first-order air resistance.
3206:
794:
421:
214:
191:
878:, his simplified parabolic trajectory proves essentially correct.
3115:
556:
464:
456:
814:
146:
2067:{\displaystyle v_{h}=v\cos \theta ,\quad v_{v}=v\sin \theta \;}
1756:
859:
855:
499:
468:
210:
305:
of values calculated by the iterated application of a mapping
3025:
1313:
Assume the motion of the projectile is being measured from a
863:
801:: force equals mass times acceleration, for such situations.
472:
218:
206:
175:
602:
The motion of the particle is described by the second-order
3036:
and provided much of the motivation for the development of
2287:
875:
510:
417:
167:
1936:{\displaystyle h={v_{i}^{2}\sin ^{2}\theta _{i} \over 2g}}
190:; hence, a complete trajectory is defined by position and
809:
448:
3032:. The derivation of these was one of the major works of
1755:, is the greatest distance the object travels along the
874:. In a near vacuum, as it turns out for instance on the
1308:
770:
On the right-hand side, the force is given in terms of
3016:
is obtained when the projectile is fired straight up.
1847:{\displaystyle R={v_{i}^{2}\sin 2\theta _{i} \over g}}
1743:= highest point of trajectory (indicated with arrows).
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2008:
1985:
1965:
1955:
An example showing how to calculate bullet trajectory
1874:
1794:
1781:, is the angle at which said object is released. The
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1587:
1498:
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1380:
1339:
1267:
1232:
1206:
1166:
1146:
1103:
1055:
1003:
955:
931:
911:
891:
870:, Galileo was able to initiate the future science of
858:. Nevertheless, by anticipating the existence of the
776:
614:
585:
565:
541:
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331:
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254:
3028:
with the mutual gravitation between them, we obtain
3009:{\displaystyle H_{\mathrm {max} }={v^{2} \over 2g}}
1261:, and the maximum altitude at the maximum range is
486:Newton's theory later developed into the branch of
74:. Unsourced material may be challenged and removed.
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2369:
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2290:function is such that there are two solutions for
2274:
2189:
2066:
1991:
1971:
1935:
1846:
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989:
937:
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759:
591:
571:
547:
524:
337:
317:
297:
1571:{\displaystyle y=x\tan(\theta )-g(x/v_{h})^{2}/2}
818:Trajectories of a mass thrown at an angle of 70°:
37:"Flightpath" redirects here. For other uses, see
3248:
3054:Trajectory of a projectile § Catching balls
2588:
3207:Projectile Lab, JavaScript trajectory simulator
3202:An interactive simulation on projectile motion
2778:{\displaystyle H=v^{2}\sin ^{2}(\theta )/(2g)}
1140:. The maximum range for a given initial speed
298:{\displaystyle (f^{k}(x))_{k\in \mathbb {N} }}
380:. There might be a discussion about this on
2536:{\displaystyle 2\theta =\pi /2=90^{\circ }}
2949:{\displaystyle \theta =\pi /2=90^{\circ }}
2063:
463:. This agrees with the observed orbits of
451:), the trajectory of a moving object is a
432:). This is the focus of the discipline of
348:
2614:
2282:(Equation II: angle of projectile launch)
2183:
400:Learn how and when to remove this message
289:
134:Learn how and when to remove this message
3052:For a proof of the above statement, see
1950:
1721:
905:-axis is tangent to the ground, and the
813:
232:, a trajectory is a time-ordered set of
145:
555:. Physically speaking, mass represents
14:
3249:
810:Uniform gravity, neither drag nor wind
439:One of the remarkable achievements of
1946:
1714:is the acceleration due to gravity).
2671:, so the maximum height obtained is
1309:Derivation of the equation of motion
1038:{\displaystyle v_{v}=v\sin(\theta )}
990:{\displaystyle v_{h}=v\cos(\theta )}
506:; trajectories are but one example.
352:
72:adding citations to reliable sources
43:
3019:
2618:{\displaystyle R_{\max }=v^{2}/g\,}
2569:{\displaystyle \theta =45^{\circ }}
2493:which has a nontrivial solution at
1717:
1482:{\displaystyle y=-g(x/v_{h})^{2}/2}
24:
3216:The Wolfram Demonstrations Project
2975:
2972:
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997:and the initial vertical speed be
777:
680:
657:
623:
25:
3273:
3177:
3164:
3043:
3030:Kepler's laws of planetary motion
1045:. It will also be shown that the
445:Kepler's laws of planetary motion
2703:{\displaystyle {v^{2} \over 4g}}
1367:{\displaystyle y=x\tan(\theta )}
494:. It employs the mathematics of
357:
48:
2037:
1959:In terms of angle of elevation
1710:is the angle of elevation, and
59:needs additional citations for
3184:Projectile Motion Flash Applet
3158:
3066:
2896:
2887:
2879:
2873:
2864:
2858:
2772:
2763:
2755:
2749:
2664:{\displaystyle \sin(\pi /2)=1}
2652:
2638:
2471:
2462:
2172:
2163:
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2127:
2118:
2112:
1551:
1529:
1520:
1514:
1462:
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1361:
1355:
1317:frame which happens to be at (
1292:
1283:
1200:, i.e. the initial angle is 45
1097:, and the maximum altitude is
1032:
1026:
984:
978:
862:, later to be demonstrated on
751:
733:
724:
710:
707:
701:
695:
686:
651:
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274:
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255:
13:
1:
3151:
1090:{\displaystyle 2v_{h}v_{v}/g}
248:, a trajectory is a sequence
182:, a trajectory is defined by
2576:. The maximum range is then
1325:) = (0,0) at
1133:{\displaystyle v_{v}^{2}/2g}
205:. For example, it can be an
7:
3079:
1414:{\displaystyle y=-gt^{2}/2}
1193:{\displaystyle v_{h}=v_{v}}
804:
39:Flightpath (disambiguation)
32:Trajectory (disambiguation)
10:
3278:
3051:
1298:{\displaystyle v^{2}/(4g)}
178:as a function of time. In
36:
29:
3189:14 September 2008 at the
3167:The Principles of Physics
1690:is the initial velocity,
1219:{\displaystyle ^{\circ }}
3059:
2956:. So the maximum height
2397:and setting it to zero.
786:{\displaystyle \nabla V}
3091:Displacement (geometry)
3086:Aft-crossing trajectory
2798:{\displaystyle \theta }
2390:{\displaystyle \theta }
2350:{\displaystyle \theta }
2303:{\displaystyle \theta }
1972:{\displaystyle \theta }
1703:{\displaystyle \theta }
1254:{\displaystyle v^{2}/g}
947:acceleration of gravity
509:Consider a particle of
349:Physics of trajectories
221:as it travels around a
27:Path of a moving object
3010:
2950:
2903:
2799:
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2391:
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2351:
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2191:
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1993:
1973:
1956:
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1704:
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1483:
1415:
1368:
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1220:
1194:
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1134:
1091:
1039:
991:
939:
919:
899:
868:Evangelista Torricelli
843:
787:
761:
593:
573:
549:
526:
443:was the derivation of
339:
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151:
3197:Trajectory calculator
3136:Range of a projectile
3038:differential calculus
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2780:
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2330:{\displaystyle d_{h}}
2305:
2277:
2192:
2069:
1994:
1974:
1954:
1938:
1849:
1759:in the I sector. The
1725:
1705:
1675:
1573:
1484:
1416:
1369:
1331:equivalence principle
1300:
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604:differential equation
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496:differential calculus
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300:
188:canonical coordinates
184:Hamiltonian mechanics
149:
3165:Metha, Rohit. "11".
3106:Orbit (group theory)
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2580:
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2084:
2077:giving the range as
2006:
1983:
1963:
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1792:
1731:= time from launch,
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1585:
1496:
1425:
1378:
1337:
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1230:
1204:
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1144:
1101:
1053:
1001:
953:
929:
909:
889:
866:by his collaborator
799:second law of motion
774:
612:
583:
563:
539:
516:
370:confusing or unclear
329:
309:
252:
246:discrete mathematics
197:The mass might be a
162:is the path that an
68:improve this article
30:For other uses, see
3233:3 July 2012 at the
3096:Galilean invariance
2909:which is zero when
1898:
1818:
1638:
1118:
492:classical mechanics
488:theoretical physics
441:Newtonian mechanics
378:clarify the article
180:classical mechanics
3111:Orbital trajectory
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2775:
2700:
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2615:
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2387:
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2347:
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2310:for a given range
2300:
2272:
2187:
2064:
1989:
1979:and initial speed
1969:
1957:
1947:Angle of elevation
1933:
1884:
1844:
1804:
1745:
1735:= time of flight,
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1670:
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783:
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589:
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481:radiation pressure
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194:, simultaneously.
152:
3131:Projectile motion
3004:
2834:
2698:
2454:
2429:
2370:{\displaystyle R}
2266:
2224:
1992:{\displaystyle v}
1931:
1842:
1640:
1160:is obtained when
1153:{\displaystyle v}
938:{\displaystyle g}
918:{\displaystyle y}
898:{\displaystyle x}
727:
716:
698:
672:
642:
592:{\displaystyle V}
572:{\displaystyle V}
548:{\displaystyle V}
525:{\displaystyle m}
410:
409:
402:
338:{\displaystyle x}
318:{\displaystyle f}
144:
143:
136:
118:
16:(Redirected from
3269:
3171:
3170:
3162:
3101:Orbit (dynamics)
3073:
3070:
3020:Orbiting objects
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3013:
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2785:with respect to
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2735:
2734:
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2706:
2701:
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2648:
2625:. At this angle
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2377:with respect to
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1761:initial velocity
1718:Range and height
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1226:. This range is
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838:
830:
822:
792:
790:
789:
784:
766:
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763:
758:
729:
728:
720:
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715: with
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238:dynamical system
209:— the path of a
174:follows through
139:
132:
128:
125:
119:
117:
76:
52:
44:
21:
3277:
3276:
3272:
3271:
3270:
3268:
3267:
3266:
3247:
3246:
3235:Wayback Machine
3224:, ScienceWorld.
3191:Wayback Machine
3180:
3175:
3174:
3163:
3159:
3154:
3121:Planetary orbit
3082:
3077:
3076:
3071:
3067:
3062:
3057:
3046:
3022:
2996:
2990:
2986:
2984:
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2940:
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2600:
2596:
2587:
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2560:
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2523:
2512:
2498:
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2401:
2382:
2379:
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2362:
2359:
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2342:
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2317:
2315:
2312:
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2295:
2292:
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2256:
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2226:
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2175:
2151:
2147:
2136:
2100:
2096:
2085:
2082:
2081:
2042:
2038:
2013:
2009:
2007:
2004:
2003:
1984:
1981:
1980:
1964:
1961:
1960:
1949:
1923:
1916:
1912:
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848:Galileo Galilei
836:
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613:
610:
609:
584:
581:
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564:
561:
560:
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537:
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534:potential field
517:
514:
513:
406:
395:
389:
386:
375:
362:
358:
351:
345:of its source.
330:
327:
326:
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249:
140:
129:
123:
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77:
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65:
53:
42:
35:
28:
23:
22:
15:
12:
11:
5:
3275:
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3264:
3259:
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3209:
3204:
3199:
3194:
3179:
3178:External links
3176:
3173:
3172:
3169:. p. 378.
3156:
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3093:
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3081:
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3061:
3058:
3045:
3044:Catching balls
3042:
3021:
3018:
3002:
2999:
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2989:
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2324:
2320:
2299:
2286:Note that the
2284:
2283:
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2236:
2233:
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2059:
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2053:
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2041:
2036:
2033:
2030:
2027:
2024:
2021:
2016:
2012:
1988:
1968:
1948:
1945:
1944:
1943:
1929:
1926:
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1915:
1911:
1906:
1902:
1896:
1891:
1887:
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1877:
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1531:
1528:
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1522:
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1478:
1474:
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1439:
1436:
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1396:
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1007:
986:
983:
980:
977:
974:
971:
968:
963:
959:
934:
914:
894:
883:inertial frame
811:
808:
806:
803:
782:
779:
768:
767:
756:
753:
750:
747:
744:
741:
738:
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723:
712:
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694:
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685:
682:
679:
676:
668:
664:
659:
653:
650:
647:
641:
638:
630:
625:
617:
588:
568:
544:
532:, moving in a
521:
408:
407:
365:
363:
356:
350:
347:
334:
325:to an element
314:
291:
287:
284:
280:
276:
273:
270:
265:
261:
257:
230:control theory
142:
141:
56:
54:
47:
26:
9:
6:
4:
3:
2:
3274:
3263:
3260:
3258:
3255:
3254:
3252:
3245:
3241:
3238:
3236:
3232:
3229:
3226:
3223:
3220:
3217:
3213:
3210:
3208:
3205:
3203:
3200:
3198:
3195:
3192:
3188:
3185:
3182:
3181:
3168:
3161:
3157:
3147:
3144:
3142:
3139:
3137:
3134:
3132:
3129:
3127:
3126:Porkchop plot
3124:
3122:
3119:
3117:
3114:
3112:
3109:
3107:
3104:
3102:
3099:
3097:
3094:
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3065:
3055:
3050:
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3035:
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3027:
3017:
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2991:
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2964:
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2926:
2922:
2919:
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2893:
2890:
2883:
2876:
2870:
2867:
2861:
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2840:
2836:
2830:
2820:
2792:
2769:
2766:
2759:
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2737:
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2727:
2723:
2720:
2711:
2694:
2691:
2685:
2681:
2658:
2655:
2649:
2645:
2641:
2635:
2632:
2611:
2607:
2601:
2597:
2593:
2584:
2561:
2557:
2553:
2550:
2528:
2524:
2520:
2517:
2513:
2509:
2506:
2503:
2500:
2477:
2474:
2468:
2465:
2459:
2456:
2451:
2445:
2441:
2437:
2431:
2425:
2415:
2400:
2399:
2398:
2384:
2364:
2344:
2337:. The angle
2322:
2318:
2297:
2289:
2268:
2261:
2257:
2252:
2249:
2243:
2239:
2234:
2231:
2227:
2221:
2218:
2213:
2210:
2203:
2202:
2201:
2184:
2180:
2176:
2169:
2166:
2160:
2157:
2152:
2148:
2144:
2141:
2137:
2130:
2124:
2121:
2115:
2109:
2106:
2101:
2097:
2093:
2090:
2087:
2080:
2079:
2078:
2060:
2057:
2054:
2051:
2048:
2043:
2039:
2034:
2031:
2028:
2025:
2022:
2019:
2014:
2010:
2002:
2001:
2000:
1986:
1966:
1953:
1927:
1924:
1917:
1913:
1909:
1904:
1900:
1894:
1889:
1885:
1878:
1875:
1868:
1867:
1866:
1864:
1860:
1839:
1833:
1829:
1825:
1822:
1819:
1814:
1809:
1805:
1798:
1795:
1788:
1787:
1786:
1784:
1780:
1773:
1772:initial angle
1769:
1762:
1758:
1754:
1750:
1742:
1738:
1734:
1730:
1724:
1715:
1713:
1697:
1686:
1667:
1664:
1661:
1658:
1655:
1652:
1647:
1643:
1634:
1629:
1625:
1621:
1616:
1613:
1608:
1604:
1600:
1594:
1591:
1588:
1581:
1580:
1579:
1565:
1561:
1555:
1545:
1541:
1536:
1532:
1526:
1523:
1517:
1511:
1508:
1505:
1502:
1499:
1490:
1476:
1472:
1466:
1456:
1452:
1447:
1443:
1437:
1434:
1431:
1428:
1408:
1404:
1398:
1394:
1390:
1387:
1384:
1381:
1358:
1352:
1349:
1346:
1343:
1340:
1332:
1328:
1324:
1320:
1316:
1306:
1289:
1286:
1279:
1273:
1269:
1248:
1244:
1238:
1234:
1211:
1208:
1185:
1181:
1177:
1172:
1168:
1147:
1127:
1124:
1120:
1114:
1109:
1105:
1084:
1080:
1074:
1070:
1064:
1060:
1056:
1048:
1029:
1023:
1020:
1017:
1014:
1009:
1005:
981:
975:
972:
969:
966:
961:
957:
948:
932:
912:
892:
884:
879:
877:
873:
869:
865:
861:
857:
853:
849:
842:
834:
826:
816:
802:
800:
796:
780:
754:
748:
745:
742:
739:
736:
730:
721:
704:
692:
683:
677:
674:
666:
662:
648:
636:
628:
615:
608:
607:
606:
605:
600:
586:
566:
558:
542:
535:
519:
512:
507:
505:
501:
497:
493:
489:
484:
482:
478:
474:
470:
466:
462:
458:
455:, usually an
454:
453:conic section
450:
446:
442:
437:
435:
431:
427:
423:
419:
415:
404:
401:
393:
390:November 2011
383:
382:the talk page
379:
373:
371:
366:This article
364:
355:
354:
346:
332:
312:
285:
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271:
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138:
135:
127:
116:
113:
109:
106:
102:
99:
95:
92:
88:
85: –
84:
80:
79:Find sources:
73:
69:
63:
62:
57:This article
55:
51:
46:
45:
40:
33:
19:
3244:
3166:
3160:
3068:
3047:
3034:Isaac Newton
3023:
2712:
2492:
2285:
2199:
2076:
1958:
1862:
1858:
1856:
1782:
1775:
1771:
1764:
1760:
1752:
1748:
1746:
1740:
1739:= range and
1736:
1732:
1728:
1711:
1684:
1682:
1491:
1326:
1322:
1318:
1312:
880:
845:
769:
601:
508:
485:
475:such as the
438:
430:aerodynamics
411:
396:
387:
376:Please help
367:
242:Poincaré map
227:
223:central mass
196:
159:
155:
153:
130:
121:
111:
104:
97:
90:
83:"Trajectory"
78:
66:Please help
61:verification
58:
18:Trajectories
1421:. That is,
1333:) would be
852:Middle Ages
841:Newton drag
833:Stokes drag
414:force field
160:flight path
3257:Ballistics
3251:Categories
3222:Trajectory
3152:References
3146:World line
3141:Rigid body
2805:, that is
477:solar wind
434:ballistics
372:to readers
240:(see e.g.
199:projectile
156:trajectory
94:newspapers
3262:Mechanics
2942:∘
2923:π
2917:θ
2877:θ
2871:
2862:θ
2856:
2831:θ
2793:θ
2753:θ
2747:
2642:π
2636:
2562:∘
2551:θ
2529:∘
2510:π
2504:θ
2469:θ
2460:
2426:θ
2385:θ
2345:θ
2298:θ
2240:
2232:−
2211:θ
2170:θ
2161:
2131:θ
2125:
2116:θ
2110:
2061:θ
2058:
2032:θ
2029:
1967:θ
1914:θ
1910:
1830:θ
1823:
1698:θ
1665:θ
1662:
1617:θ
1614:
1595:−
1578:That is:
1524:−
1518:θ
1512:
1435:−
1388:−
1359:θ
1353:
1315:free fall
1212:∘
1030:θ
1024:
982:θ
976:
872:mechanics
778:∇
725:→
696:→
681:∇
678:−
640:→
504:phenomena
490:known as
461:hyperbola
286:∈
203:satellite
124:July 2022
3231:Archived
3187:Archived
3080:See also
3026:orbiting
823:without
805:Examples
795:gradient
422:parabola
215:asteroid
192:momentum
3116:Phugoid
1683:(where
945:be the
557:inertia
465:planets
457:ellipse
368:may be
108:scholar
1859:height
1757:x-axis
860:vacuum
856:Europe
837:
829:
821:
793:, the
500:reason
473:forces
469:comets
244:). In
234:states
211:planet
172:motion
164:object
110:
103:
96:
89:
81:
3060:Notes
2543:, or
1749:range
1047:range
864:Earth
839:with
831:with
459:or a
236:of a
219:comet
217:, or
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