3038:
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.
1941:
1712:
39:
136:
348:
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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
754:
1715:
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.
487:(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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2184:
1667:
2061:
600:
1930:
401:
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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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.
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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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1084:
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2324:
839:. 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
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1986:
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932:
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586:
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519:
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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
2195:
835:
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
2072:
3219:
1573:
460:, and artificial spacecraft to a reasonably good approximation, although if a comet passes close to the Sun, then it is also influenced by other
749:{\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).}
1994:
3200:
3175:
413:. Generally when determining trajectories, it may be necessary to account for nonuniform gravitational forces and air resistance (
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1780:
3018:
433:
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103:
3204:
1484:
75:
3042:
388:
122:
2705:
405:. This can be a good approximation for a rock that is thrown for short distances, for example at the surface of the
240:
82:
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2901:
60:
2702:
To find the angle giving the maximum height for a given speed calculate the derivative of the maximum height
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27:
20:
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of the potential, taken at positions along the trajectory. This is the mathematical form of Newton's
436:. In the gravitational field of a point mass or a spherically-symmetrical extended mass (such as the
2617:
1041:
870:
In the analysis that follows, we derive the equation of motion of a projectile as measured from an
366:
1089:
3079:
3074:
1366:
1152:
522:
49:
1253:
2891:{\displaystyle {\mathrm {d} H \over \mathrm {d} \theta }=v^{2}2\cos(\theta )\sin(\theta )/(2g)}
1940:
1711:
1192:
856:
787:
762:
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1951:
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1319:
1218:
1035:
592:
484:
402:
176:
172:
491:, in science as well as technology. It helps to understand and predict an enormous range of
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2302:
234:
3210:
2472:{\displaystyle {\mathrm {d} R \over \mathrm {d} \theta }={2v^{2} \over g}\cos(2\theta )=0}
8:
3245:
3084:
480:
476:
429:
358:
211:
168:
1363:. The co-ordinates of this free-fall frame, with respect to our inertial frame would be
96:
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1971:
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3201:
Parabolic
Projectile Motion: Shooting a Harmless Tranquilizer Dart at a Falling Monkey
1481:
Now translating back to the inertial frame the co-ordinates of the projectile becomes
3119:
139:
Illustration showing the directional trajectory of a bullet fired at an uphill target
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226:
222:
160:
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by
Roberto Castilla-MelĂ©ndez, Roxana RamĂrez-Herrera, and JosĂ© Luis GĂłmez-Muñoz,
3179:
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370:
2264:{\displaystyle \theta ={\frac {1}{2}}\sin ^{-1}\left({\frac {gR}{v^{2}}}\right)}
230:
1318: = 0. The equation of motion of the projectile in this frame (by the
914:
axis is perpendicular to it ( parallel to the gravitational field lines ). Let
821:
813:
568:
represents external forces of a particular kind known as "conservative". Given
414:
218:
3190:
1759:, is the speed at which said object is launched from the point of origin. The
3239:
3114:
3013:
If instead of a uniform downwards gravitational force we consider two bodies
2179:{\displaystyle R=2v^{2}\cos(\theta )\sin(\theta )/g=v^{2}\sin(2\theta )/g\,.}
1854:, is the greatest parabolic height said object reaches within its trajectory
441:
152:
3229:
Java projectile-motion simulation; targeting solutions, parabola of safety.
3022:
1662:{\displaystyle y=-{g\sec ^{2}\theta \over 2v_{0}^{2}}x^{2}+x\tan \theta ,}
472:, which modify the orbit and cause the comet to eject material into space.
418:
1774:
is the respective gravitational pull on the object within a null-medium.
840:
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3129:
2346:
giving the maximum range can be found by considering the derivative or
2189:
This equation can be rearranged to find the angle for a required range
465:
422:
187:
1303:
860:
492:
449:
191:
938:. Relative to the flat terrain, let the initial horizontal speed be
409:. In this simple approximation, the trajectory takes the shape of a
38:
3217:
Java projectile-motion simulation, with first-order air resistance.
3195:
783:
410:
203:
180:
867:, his simplified parabolic trajectory proves essentially correct.
3104:
545:
453:
445:
803:
135:
2056:{\displaystyle v_{h}=v\cos \theta ,\quad v_{v}=v\sin \theta \;}
1745:
848:
844:
488:
457:
199:
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of values calculated by the iterated application of a mapping
3014:
1302:
Assume the motion of the projectile is being measured from a
852:
790:: force equals mass times acceleration, for such situations.
461:
207:
195:
164:
591:
The motion of the particle is described by the second-order
3025:
and provided much of the motivation for the development of
2276:
864:
499:
406:
156:
1925:{\displaystyle h={v_{i}^{2}\sin ^{2}\theta _{i} \over 2g}}
179:; hence, a complete trajectory is defined by position and
798:
437:
3021:. The derivation of these was one of the major works of
1744:, is the greatest distance the object travels along the
863:. In a near vacuum, as it turns out for instance on the
1297:
759:
On the right-hand side, the force is given in terms of
3005:
is obtained when the projectile is fired straight up.
1836:{\displaystyle R={v_{i}^{2}\sin 2\theta _{i} \over g}}
1732:= highest point of trajectory (indicated with arrows).
2951:
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1997:
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1944:
An example showing how to calculate bullet trajectory
1863:
1783:
1770:, is the angle at which said object is released. The
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1576:
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1328:
1256:
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1092:
1044:
992:
944:
920:
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859:, Galileo was able to initiate the future science of
847:. Nevertheless, by anticipating the existence of the
765:
603:
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320:
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243:
3017:
with the mutual gravitation between them, we obtain
2998:{\displaystyle H_{\mathrm {max} }={v^{2} \over 2g}}
1250:, and the maximum altitude at the maximum range is
475:Newton's theory later developed into the branch of
63:. Unsourced material may be challenged and removed.
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2279:function is such that there are two solutions for
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1960:
1924:
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978:
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774:
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560:
536:
513:
326:
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286:
1560:{\displaystyle y=x\tan(\theta )-g(x/v_{h})^{2}/2}
807:Trajectories of a mass thrown at an angle of 70°:
26:"Flightpath" redirects here. For other uses, see
3237:
3043:Trajectory of a projectile § Catching balls
2577:
3196:Projectile Lab, JavaScript trajectory simulator
3191:An interactive simulation on projectile motion
2767:{\displaystyle H=v^{2}\sin ^{2}(\theta )/(2g)}
1129:. The maximum range for a given initial speed
287:{\displaystyle (f^{k}(x))_{k\in \mathbb {N} }}
369:. There might be a discussion about this on
2525:{\displaystyle 2\theta =\pi /2=90^{\circ }}
2938:{\displaystyle \theta =\pi /2=90^{\circ }}
2052:
452:. This agrees with the observed orbits of
440:), the trajectory of a moving object is a
421:). This is the focus of the discipline of
337:
2603:
2271:(Equation II: angle of projectile launch)
2172:
389:Learn how and when to remove this message
278:
123:Learn how and when to remove this message
3041:For a proof of the above statement, see
1939:
1710:
894:-axis is tangent to the ground, and the
802:
221:, a trajectory is a time-ordered set of
134:
544:. Physically speaking, mass represents
3238:
799:Uniform gravity, neither drag nor wind
428:One of the remarkable achievements of
1935:
1703:is the acceleration due to gravity).
2660:, so the maximum height obtained is
1298:Derivation of the equation of motion
1027:{\displaystyle v_{v}=v\sin(\theta )}
979:{\displaystyle v_{h}=v\cos(\theta )}
495:; trajectories are but one example.
341:
61:adding citations to reliable sources
32:
3008:
2607:{\displaystyle R_{\max }=v^{2}/g\,}
2558:{\displaystyle \theta =45^{\circ }}
2482:which has a nontrivial solution at
1706:
1471:{\displaystyle y=-g(x/v_{h})^{2}/2}
13:
3205:The Wolfram Demonstrations Project
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986:and the initial vertical speed be
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612:
14:
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3019:Kepler's laws of planetary motion
1034:. It will also be shown that the
434:Kepler's laws of planetary motion
2692:{\displaystyle {v^{2} \over 4g}}
1356:{\displaystyle y=x\tan(\theta )}
483:. It employs the mathematics of
346:
37:
2026:
1948:In terms of angle of elevation
1699:is the angle of elevation, and
48:needs additional citations for
3173:Projectile Motion Flash Applet
3147:
3055:
2885:
2876:
2868:
2862:
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2847:
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2744:
2738:
2653:{\displaystyle \sin(\pi /2)=1}
2641:
2627:
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2152:
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2116:
2107:
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1540:
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1306:frame which happens to be at (
1281:
1272:
1189:, i.e. the initial angle is 45
1086:, and the maximum altitude is
1021:
1015:
973:
967:
851:, later to be demonstrated on
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244:
1:
3140:
1079:{\displaystyle 2v_{h}v_{v}/g}
237:, a trajectory is a sequence
171:, a trajectory is defined by
2565:. The maximum range is then
1314:) = (0,0) at
1122:{\displaystyle v_{v}^{2}/2g}
194:. For example, it can be an
7:
3068:
1403:{\displaystyle y=-gt^{2}/2}
1182:{\displaystyle v_{h}=v_{v}}
793:
28:Flightpath (disambiguation)
21:Trajectory (disambiguation)
10:
3267:
3040:
1287:{\displaystyle v^{2}/(4g)}
167:as a function of time. In
25:
18:
3178:14 September 2008 at the
3156:The Principles of Physics
1679:is the initial velocity,
1208:{\displaystyle ^{\circ }}
3048:
2945:. So the maximum height
2386:and setting it to zero.
775:{\displaystyle \nabla V}
3080:Displacement (geometry)
3075:Aft-crossing trajectory
2787:{\displaystyle \theta }
2379:{\displaystyle \theta }
2339:{\displaystyle \theta }
2292:{\displaystyle \theta }
1961:{\displaystyle \theta }
1692:{\displaystyle \theta }
1243:{\displaystyle v^{2}/g}
936:acceleration of gravity
498:Consider a particle of
338:Physics of trajectories
210:as it travels around a
16:Path of a moving object
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2360:
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2180:
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1962:
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1404:
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1183:
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1080:
1028:
980:
928:
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857:Evangelista Torricelli
832:
776:
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582:
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432:was the derivation of
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140:
3186:Trajectory calculator
3125:Range of a projectile
3027:differential calculus
3000:
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2319:{\displaystyle d_{h}}
2294:
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2181:
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1983:
1963:
1943:
1927:
1838:
1748:in the I sector. The
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1473:
1405:
1358:
1320:equivalence principle
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593:differential equation
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485:differential calculus
329:
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177:canonical coordinates
173:Hamiltonian mechanics
138:
3154:Metha, Rohit. "11".
3095:Orbit (group theory)
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2196:
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2066:giving the range as
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1972:
1952:
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1781:
1720:= time from launch,
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1574:
1485:
1414:
1367:
1326:
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1193:
1153:
1133:
1090:
1042:
990:
942:
918:
898:
878:
855:by his collaborator
788:second law of motion
763:
601:
572:
552:
528:
505:
359:confusing or unclear
318:
298:
241:
235:discrete mathematics
186:The mass might be a
151:is the path that an
57:improve this article
19:For other uses, see
3222:3 July 2012 at the
3085:Galilean invariance
2898:which is zero when
1887:
1807:
1627:
1107:
481:classical mechanics
477:theoretical physics
430:Newtonian mechanics
367:clarify the article
169:classical mechanics
3100:Orbital trajectory
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2784:
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2336:
2316:
2299:for a given range
2289:
2261:
2176:
2053:
1978:
1968:and initial speed
1958:
1946:
1936:Angle of elevation
1922:
1873:
1833:
1793:
1734:
1724:= time of flight,
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1659:
1613:
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1400:
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904:
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772:
746:
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470:radiation pressure
324:
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284:
183:, simultaneously.
141:
3120:Projectile motion
2993:
2823:
2687:
2443:
2418:
2359:{\displaystyle R}
2255:
2213:
1981:{\displaystyle v}
1920:
1831:
1629:
1149:is obtained when
1142:{\displaystyle v}
927:{\displaystyle g}
907:{\displaystyle y}
887:{\displaystyle x}
716:
705:
687:
661:
631:
581:{\displaystyle V}
561:{\displaystyle V}
537:{\displaystyle V}
514:{\displaystyle m}
399:
398:
391:
327:{\displaystyle x}
307:{\displaystyle f}
133:
132:
125:
107:
3258:
3160:
3159:
3151:
3090:Orbit (dynamics)
3062:
3059:
3009:Orbiting objects
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2793:
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2785:
2774:with respect to
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2637:
2614:. At this angle
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2366:with respect to
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1750:initial velocity
1707:Range and height
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1215:. This range is
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1165:
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885:
827:
819:
811:
781:
779:
778:
773:
755:
753:
752:
747:
718:
717:
709:
706:
704: with
703:
689:
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662:
660:
659:
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227:dynamical system
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65:
41:
33:
3266:
3265:
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3260:
3259:
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3236:
3235:
3224:Wayback Machine
3213:, ScienceWorld.
3180:Wayback Machine
3169:
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3163:
3152:
3148:
3143:
3110:Planetary orbit
3071:
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523:potential field
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334:of its source.
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3167:External links
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3158:. p. 378.
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3033:Catching balls
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872:inertial frame
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577:
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521:, moving in a
510:
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314:to an element
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280:
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219:control theory
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45:
43:
36:
15:
9:
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3116:
3115:Porkchop plot
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2326:. The angle
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2037:
2032:
2028:
2023:
2020:
2017:
2014:
2011:
2008:
2003:
1999:
1991:
1990:
1989:
1975:
1955:
1942:
1916:
1913:
1906:
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1808:
1803:
1798:
1794:
1787:
1784:
1777:
1776:
1775:
1773:
1769:
1762:
1761:initial angle
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1423:
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1417:
1397:
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1387:
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1347:
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831:
823:
815:
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791:
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769:
743:
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731:
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719:
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681:
672:
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663:
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625:
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555:
547:
531:
524:
508:
501:
496:
494:
490:
486:
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478:
473:
471:
467:
463:
459:
455:
451:
447:
444:, usually an
443:
442:conic section
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435:
431:
426:
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420:
416:
412:
408:
404:
393:
390:
382:
379:November 2011
372:
371:the talk page
368:
362:
360:
355:This article
353:
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98:
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91:
88:
84:
81:
77:
74: –
73:
69:
68:Find sources:
62:
58:
52:
51:
46:This article
44:
40:
35:
34:
29:
22:
3233:
3155:
3149:
3057:
3036:
3023:Isaac Newton
3012:
2701:
2481:
2274:
2188:
2065:
1947:
1851:
1847:
1845:
1771:
1764:
1760:
1753:
1749:
1741:
1737:
1735:
1729:
1728:= range and
1725:
1721:
1717:
1700:
1673:
1671:
1480:
1315:
1311:
1307:
1301:
869:
834:
758:
590:
497:
474:
464:such as the
427:
419:aerodynamics
400:
385:
376:
365:Please help
356:
231:Poincaré map
216:
212:central mass
185:
148:
144:
142:
119:
110:
100:
93:
86:
79:
72:"Trajectory"
67:
55:Please help
50:verification
47:
1410:. That is,
1322:) would be
841:Middle Ages
830:Newton drag
822:Stokes drag
403:force field
149:flight path
3246:Ballistics
3240:Categories
3211:Trajectory
3141:References
3135:World line
3130:Rigid body
2794:, that is
466:solar wind
423:ballistics
361:to readers
229:(see e.g.
188:projectile
145:trajectory
83:newspapers
3251:Mechanics
2931:∘
2912:π
2906:θ
2866:θ
2860:
2851:θ
2845:
2820:θ
2782:θ
2742:θ
2736:
2631:π
2625:
2551:∘
2540:θ
2518:∘
2499:π
2493:θ
2458:θ
2449:
2415:θ
2374:θ
2334:θ
2287:θ
2229:
2221:−
2200:θ
2159:θ
2150:
2120:θ
2114:
2105:θ
2099:
2050:θ
2047:
2021:θ
2018:
1956:θ
1903:θ
1899:
1819:θ
1812:
1687:θ
1654:θ
1651:
1606:θ
1603:
1584:−
1567:That is:
1513:−
1507:θ
1501:
1424:−
1377:−
1348:θ
1342:
1304:free fall
1201:∘
1019:θ
1013:
971:θ
965:
861:mechanics
767:∇
714:→
685:→
670:∇
667:−
629:→
493:phenomena
479:known as
450:hyperbola
275:∈
192:satellite
113:July 2022
3220:Archived
3176:Archived
3069:See also
3015:orbiting
812:without
794:Examples
784:gradient
411:parabola
204:asteroid
181:momentum
3105:Phugoid
1672:(where
934:be the
546:inertia
454:planets
446:ellipse
357:may be
97:scholar
1848:height
1746:x-axis
849:vacuum
845:Europe
826:
818:
810:
782:, the
489:reason
462:forces
458:comets
233:). In
223:states
200:planet
161:motion
153:object
99:
92:
85:
78:
70:
3049:Notes
2532:, or
1738:range
1036:range
853:Earth
828:with
820:with
448:or a
225:of a
208:comet
206:, or
196:orbit
190:or a
165:space
155:with
104:JSTOR
90:books
2277:sine
1846:The
1736:The
865:Moon
814:drag
500:mass
468:and
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2857:sin
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1038:is
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3242::
3182::)
3029:.
2927:90
2699:.
2547:45
2514:90
1988::
1850:,
1763:,
1752:,
1740:,
1478:.
1294:.
456:,
425:.
214:.
202:,
143:A
3207:.
3045:.
2990:g
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2981:2
2977:v
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2923:=
2920:2
2916:/
2909:=
2886:)
2883:g
2880:2
2877:(
2873:/
2869:)
2863:(
2854:)
2848:(
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2834:2
2830:v
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2759:g
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2753:(
2749:/
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2739:(
2731:2
2721:2
2717:v
2713:=
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2684:g
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2648:1
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2639:2
2635:/
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2467:0
2464:=
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2208:1
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2138:v
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2127:/
2123:)
2117:(
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2102:(
2091:2
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2080:=
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1917:g
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1526:/
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1234:/
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1114:2
1110:/
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1070:/
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1016:(
1007:v
1004:=
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956:=
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744:.
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735:,
732:y
729:,
726:x
723:(
720:=
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691:(
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676:(
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664:=
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576:V
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532:V
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373:.
363:.
322:x
302:f
279:N
272:k
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120:(
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111:(
101:·
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87:·
80:·
53:.
30:.
23:.
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