A trajectory is the path a moving object follows through space. Types of trajectories include Trajectory of a projectile Trajectory fluid mechanics Trajectory of a point in a moving fluid Trajectory may also refer to Trajectory Hermeneutics , a liberal teaching of Christian Postmodernism Trajectory optimization , the process of minimizing or maximizing performance Trajectories magazine Trajectories magazine , a 1980s tabloid magazine Also in finance the end point from source spending where the money goes In cricket, trajectory is known as flight cricket flight or loop disambig In writing, trajectory is an informative discusstion or map of the various avenues texts can travel as they both affect, and are affected by, the world. de Trajektorie ru ... more details
File Apollo 13 timeline.svg right thumb 350px upright 2 The trajectory followed by Apollo 13 A Circumlunar trajectory , Trans Lunar trajectory or Lunar free return is a type of free return trajectory which takes a spacecraft from Earth, around the far side of the Moon far side of the Moon , and back to Earth. Background The first spacecraft to fly a circumlunar trajectory was Luna 3 . Circumlunar trajectories were also used by Apollo program Apollo missions prior to lunar orbit insertion, ref cite web url http spaceflightnow.com apollo13 retro 09 launch.html title The launch and mission trajectory work Apollo 13 Mission Report publisher Spaceflight Now accessdate 2009 06 30 ref to provide a free return to Earth in the event of a propulsion system malfunction on the way to the Moon. This was used on Apollo 13 , when an oxygen tank rupture necessitated return to Earth without firing the Apollo Service Module Service Module engine, although a number of course corrections using the Apollo Lunar Module Lunar Module were required to maintain this trajectory. ref cite web url http spaceflightnow.com apollo13 retro 14 index3.html title Apollo 13 on Free Return Trajectory last Turnhill first Reginald publisher Spaceflight Now accessdate 2009 06 30 ref A number of manned missions were also proposed to intentionally conduct circumlunar flybys, including the Soviet Soyuz 7K L1 or Zond program Zond programme, and several US proposals, including Gemini program Gemini Centaur rocket stage Centaur and an early Apollo proposal. ref cite web url http www.astronautix.com craftfam manlunar.htm title Manned Circumlunar last Wade first Mark publisher Encyclopedia Astronautica accessdate 2009 06 30 Dead link date October 2010 bot H3llBot ref See also Trans Lunar Injection References reflist Category Exploration of the Moon space stub fa ... more details
to the radial position of orbiting body on the parabolic trajectory math v sqrt 2 cdot v o math ... assumptions, for a body moving along this kind of orbit trajectory an orbital equation becomes ... Under standard assumptions, specific orbital energy math epsilon , math of parabolic trajectory is zero, so the orbital energy conservation equation for this trajectory takes form math epsilon v 2 ... 0 math C 3 0 math Radial parabolic trajectory A radial parabolic trajectory is a non periodic Radial trajectorytrajectory on a straight line where the relative velocity of the two objects is always ... also there is a navbox, such list was redundant orbits DEFAULTSORT Parabolic Trajectory Category Astrodynamics ... more details
Unreferenced date December 2009 When a family of curve s intersects another family at a specific constant angle , the first family is referred to as an isogonal family of the second one, and in this case it is said that every family is an isogonal trajectory of the other. It is assumed that is different from 2 if 2 both families are orthogonal . If the given family is described by the differential equation math frac dy dx f x,y , math then the isogonal family satisfies the differential equation math frac dy dx frac f x,y tan alpha 1 f x,y tan alpha . math DEFAULTSORT Isogonal Trajectory Category Curves ... more details
, math of a hyperbolic trajectory is greater than zero and the orbital energy conservation equation for this kind of trajectory takes form math epsilon v 2 over2 mu over r mu over 2a math where math ... parameter . Radial hyperbolic trajectory A radial hyperbolic trajectory is a non periodic Radial trajectorytrajectory on a straight line where the relative speed of the two objects always exceeds ... iperbolica ja pt Trajet ria hiperb lica simple Hyperbolic trajectory tr Hiperbolik y r nge ... more details
Orphan date February 2009 unreferenced date November 2007 Technology trajectory refers to a single branch in the evolution of a technological design of a product service, with nodes representing separate designs. Movement along the technology trajectory is associated with research and development. Due to institutionalisation of ideas, markets and professions, a technology development can get stuck within one trajectory, and firms and engineers unable to adopt to ideas and innovation from outside. See also Innovation Thomas Samuel Kuhn Social shaping of technology References cite book last Kuhn first Thomas title The structure of scientific revolutions location Chicago publisher University of Chicago Press year 1962 authorlink Thomas Kuhn Giovanni Dosi 1982 Technological Paradigms and Technological Trajectories a suggested interpretation of the determinants of technological change , Research Policy, Vol. 11, pp. 147 62. Category Innovation Category Technological change Category Science and technology studies marketing stub sociology stub technology stub ... more details
Trajectory optimization is the process of designing a trajectory that minimizes or maximizes some figure ..., the goal of solving a trajectory optimization problem is essentially the same as solving an optimal ... for a Multiple Pulse Missile , AIAA Paper 88 0334, Jan., 1988 ref History Trajectory optimization ... of optimal control. Early application of trajectory optimization had to do with the optimization ... drag region and the low thrust of early jet aircraft, trajectory optimization was the key ... based on optimal control solutions. In the early phase of trajectory optimization many of the solutions ... of time math t 1 le t le t 2 math is called the singular control case and the optimal trajectory ... to a trajectory that is usually the optimal singular control . ref H.J. Kelley, R.E. Kopp ... performance. Many of the early triumphs of trajectory optimization have moved into the background ... dimensional problem. Trajectory optimization shares the same optimization algorithms as other optimization ... in the first derivative. The smoothness requirement imposes a burden on flight trajectory analysts in that most highly detailed trajectory simulations do not exhibit smoothness. This restriction was a problem in the early days of trajectory optimization when computer computation speed was an issue. Often, special approximate trajectory models had to be used to work with non linear programming ... in the number of functional samples but exhibit robustness to non smoothness in the trajectory ... 2357 2363 ref ref A hybrid multiagent approach for global trajectory optimization M. Vasile and M .... ref Survey of Numerical Methods for Trajectory Optimization John T. Betts Journal of Guidance ... for the optimization of the performance of a missile. ref Trajectory Optimization for a Missile Using ..., September October 2000 ref Software Examples of trajectory optimization programs include Copernicus Trajectory Design and Optimization System http research.jsc.nasa.gov PDF Eng 12.pdf OTIS Optimal Trajectories ... more details
In astrodynamics and celestial mechanics a radial trajectory is a Kepler orbit with zero angular momentum . Two objects in a radial trajectory move directly towards or away from each other in a straight line. Classification There are three types of radial trajectories orbits ref William Tyrrell Thomson 1986 , Introduction to Space Dynamics, Dover ref . Radial elliptic trajectory a nonperiodic orbit corresponding to the part of a degenerate ellipse from the moment the bodies touch each other and move away from each other until they touch each other again. The relative speed of the two objects is less than the escape velocity . This is an elliptic orbit with semi minor axis 0 and eccentricity 1. Although the eccentricity is 1 this is not a parabolic orbit. Radial parabolic trajectory , a nonperiodic orbit where the relative speed of the two objects is always equal to the escape velocity. There are two cases the bodies move away from each other or towards each other. Radial hyperbolic trajectory a nonperiodic orbit where the relative speed of the two objects always exceeds the escape velocity. There are two cases the bodies move away from each other or towards each other. This is a hyperbolic ... velocity at any time. Parabolic trajectory math t x sqrt frac 2 x 3 9 mu math where t is the time ... trajectories see Barker s Equation . Elliptic trajectory math t x,w frac arcsin sqrt w , x sqrt ... . Hyperbolic trajectory math t x,w frac sqrt w x 2 w x ln sqrt w x sqrt 1 w x sqrt 2 mu , w 3 2 math ... coincide, and x is the separation. Universal form any trajectory The radial Kepler equation can be made ... and velocity at any time. Parabolic trajectory math x t left frac 9 2 mu t 2 right frac 1 3 ... orbit . Universal form any trajectory Two intermediate quantities are used w, and the separation at time t the bodies would have if they were on a parabolic trajectory, p. math w frac 1 x 0 frac ... DEFAULTSORT Radial Trajectory Category Orbits Category Celestial mechanics Category Johannes Kepler ... more details
In physics , the ballistic trajectory of a projectile is the path that a thrown or launched projectile will take under the action of gravity , neglecting all other forces, such as friction from air resistance, without Vehicle propulsion propulsion . The United States Department of Defense and NATO define a Ballistics ballistic trajectory as a trajectory traced after the propulsive force is terminated and the body is acted upon only by gravity and aerodynamic drag. ref Cite web url http www.dtic.mil doctrine jel doddict data b 00622.html title Ballistic trajectory accessdate 2008 06 10 publisher Defense Technical Information Center ref The following applies for ranges which are small compared to the size of the Earth. For longer ranges see sub orbital spaceflight . Notation In the equations on this page, the following variables will be used var g var the gravitational acceleration &mdash usually taken to be 9.81 m s sup 2 sup near the Earth s surface var var the angle at which the projectile is launched var v var the velocity at which the projectile is launched var y sub 0 sub var the initial height of the projectile var d var the total horizontal distance traveled by the projectile Conditions at the final position of the projectile Distance travelled The total horizontal distance var d var travelled. math d frac v cos theta g left v sin theta sqrt v sin theta 2 2gy 0 right math When the surface the object is launched from and is flying over is flat the initial height is zero ... of flight var t var is the time it takes for the projectile to finish its trajectory. math t frac ... required to hit coordinate var x var , var y var Image Trajectory for changing launch angle.gif right thumb 320px Vacuum trajectory of a projectile for different launch angles To hit a target at range ... pm sqrt v 4 g gr 2 cos 2 phi 2v 2r sin phi gr cos phi right math Trajectory of a projectile with air ... air resistance. See also Ballistic coefficient Range of a projectile Trajectory References ... more details
A free return trajectory is one of a very small sub class of trajectories in which the trajectory of a satellite traveling away from a primary body for example, the Earth is modified by the presence of a secondary body for example, the Moon causing the satellite to return to the primary body. ref cite book last Schwaninger first Arthur J. title Trajectories in the Earth Moon Space with Symmetrical Free Return Properties publisher NASA Marshall Space Flight Center location Huntsville, Alabama date 1963 series Technical Note D 1833 url http ntrs.nasa.gov archive nasa casi.ntrs.nasa.gov 19630007117 1963007117.pdf ref This method has been used by several spacecraft , most notably the Apollo 8 , Apollo 10 , and Apollo 11 lunar missions. Those spacecraft were launched into a free return trajectory to allow their safe return in the event of a systems failure after launch. They all successfully inserted into orbit upon arriving at the Moon, and so did not take advantage of the free return. Due to the landing site restrictions that resulted from constraining the launch to a free return that flew by the Moon, subsequent Apollo missions, starting with Apollo 12 and including the ill fated Apollo 13 , used a hybrid trajectory that launched to a highly elliptical Earth orbit that fell short of the Moon with effectively a free return to the atmospheric entry corridor. They then performed a mid course maneuver to change to a trans Lunar trajectory that was not a free return. This retained the safety characteristics of being on a free return upon launch, and only departed from free return once ... used the lunar module to maneuver from its planned lunar orbit insertion trajectory to a free return trajectory. Apollo 13 was the only Apollo mission to actually complete a free return trajectory. A true free return trajectory is completed without the assistance of any mid course corrections or maneuvers. The free return trajectory allows a great distance to be covered relatively directly without ... more details
Summary Information Description Simulation of the motion of a sand grain following a saltation trajectory Source self made Date 4 30 1999 Location N A Author User Leaflet Leaflet User talk Leaflet talk 05 15, 7 March 2008 UTC Licensing self cc by sa 3.0 GFDL ... more details
In 2005, a new trajectory that an air launched rocket could take to put satellite s into orbit was tested. Until this time, launch vehicle s such as the Pegasus rocket , or rocket planes such as the Bell X 1 X 1 , X 15 , or SpaceShipOne , which were carried under an aircraft pointing in the same direction as the fuselage , would have their engines ignited either just before being air dropped or a few seconds afterward. They would then be expected to accelerate and climb in front of the carrier aircraft, crossing its flight path. This was considered dangerous due to the potential for a crashes between the rocket and the launch vehicle. The aft crossing trajectory is an alternate flight path for a rocket. The rocket s rotation induced by the deployment from the aircraft is slowed by a small parachute attached to its tail, then ignited once the carrier aircraft has passed it. It is ignited before it is pointing fully vertically, however it will turn to do so, and accelerates to pass behind the carrier aircraft. The principal advantage of this method is its safety for the crew of the carrier aircraft. See also AirLaunch LLC t Space References Aviation Week & Space Technology June 27, 2005, page 32. cite web url http www.transformspace.com document library media AIAA 2006 1040.pdf title Flight Testing of a New Air Launch Method for Safely Launching... publisher t Space date 2006 Category Spaceflight ... more details
In fluid mechanics , meteorology and oceanography , a trajectory traces the motion of a single point, often called a parcel, in the flow. Trajectories are useful for tracking atmospheric contaminants, such as smoke plumes, and as constituents to Lagrangian coordinates Lagrangian simulations, such as contour advection or semi Lagrangian scheme s. Suppose we have a time varying flow field, math vec v vec x, t math . The motion of a fluid parcel, or trajectory, is given by the following system of ordinary differential equations math frac d vec x dt vec v vec x, t math While the equation looks simple, there are at least three concerns when attempting to solve it numerically . The first is the numerical ordinary differential equations integration scheme . This is typically a Runge Kutta , although others can be useful as well, such as a leapfrog integration leapfrog . The second is the method of determining the velocity vector, math vec v math at a given position, math vec x math , and time, t . Normally, it is not known at all positions and times, therefore some method of interpolation is required. If the velocities are gridded in space and time, then bilinear interpolation bilinear , trilinear interpolation trilinear or higher dimensional linear interpolation is appropriate. Bicubic interpolation Bicubic , Tricubic interpolation tricubic , etc., interpolation is used as well, but is probably not worth the extra computational overhead. Velocity fields can be determined by measurement, e.g. from weather balloons , from numerical models or especially from a combination of the two, e.g. data assimilation assimilation models . The final concern is metric corrections. These are necessary for geophysical fluid flows on a spherical Earth. The differential equations for tracing a two dimensional, atmospheric trajectory in longitude latitute coordinates are as follows math frac d theta ... and buoys in the ocean . External links http ctraj.sourceforge.net ctraj A trajectory integrator ... more details
Orphan date February 2009 Symbolic trajectory evaluation STE is a lattice group lattice based model checking technology that uses a form of symbolic simulation . STE is essentially used for computer hardware , that is circuit verification . The technique uses abstraction , meaning that details of the circuit behaviour are removed from the circuit model. It was first developed by Carl Seger and Randy Bryant in 1995 as an alternative to classical symbolic model checking, References C. J. H. Seger, and R. E. Bryant, Formal Verification by Symbolic Evaluation of Partially Ordered Trajectories, Formal Methods in System Design, Vol. 6, No. 2 March, 1995 , pp. 147&ndash 190 Category Model checking Comp sci stub ... more details
The Ballistic Trajectory Extended Range Munition BTERM is a precision guided rocket assisted 5 inch artillery shell under development by Alliant Techsystems for the U.S. Navy . The program is an extension of the earlier Autonomous Naval Support Round ANSR . ref name GLOBSEC http www.globalsecurity.org military systems munitions bterm.htm Ballistic Trajectory Extended Range Munition BTERM Global Security ref Specification Caliber 127 mm 5 in Length Weight Speed Range Guidance GPS Inertial guidance system INS Propulsion Solid propellant rocket motor Warhead Blast Fragmentation weaponry fragmentation References reflist See also Extended Range Guided Munition Long Range Land Attack Projectile M712 Copperhead M982 Excalibur External links http www.atk.com Customer Solutions MissionSystems cs ms w gp bterm.asp Ballistic Trajectory Extended Range Munition BTERM ATK Category Artillery shells US mil stub ... more details
Move to Commons Summary Information Description Vacuum trajectory of a projectile for different launch angles Source I User Skorkmaz Skorkmaz User talk Skorkmaz talk created this work entirely by myself. Date 16 03, 29 October 2009 UTC Author User Skorkmaz Skorkmaz User talk Skorkmaz talk other versions Licensing PD self date October 2009 ... more details
In mathematics , orthogonal trajectories are a family of curves in the plane that intersect a given family of curves at right angle s. The problem is classical, but is now understood by means of complex analysis see for example harmonic conjugate . For a family of level curve s described by math g x, y C math , where math C math is a constant, the orthogonal trajectories may be found as the level curves of a new function math f x, y math by solving the partial differential equation math nabla f cdot nabla g 0 math for math f x, y math . This is literally a statement that the gradient s of the functions which are perpendicular to the curves are orthogonal. Note that if math f math and math g math are functions of three variables instead of two, the equation above will be nonlinear and will specify orthogonal surface s. The partial differential equation may be avoided by instead equating the tangent of a parametric curve math vec r t math with the gradient of math g x, y math math frac d d t left vec r t right nabla g math which will result in two possibly coupled ordinary differential equations, whose solutions are the orthogonal trajectories. Note that with this formula, if math g math is a function of three variables its level set s are surfaces, and the family of curves math vec r t math are orthogonal to the surfaces. Example circle In polar coordinates , the family of circles centered about the origin is the level curves of math r R 0 math where math R math is the radius of the circle. Then the orthogonal trajectories are the level curves of math f math defined by math nabla f cdot nabla r R 0 math math left frac partial f partial r , frac 1 r frac partial f partial theta right cdot left frac partial partial r r R , frac 1 r frac partial partial theta r R right 0 math math frac partial f partial r 0 quad Rightarrow quad f f theta math The lack of complete boundary data prevents determining math f theta math . However, we want our orthogonal trajectories to span ... more details
Cycler may refer to Lunar cycler , a spacecraft trajectory that encounters the Earth and the Moon on a regular basis, or a spacecraft on such a trajectory Mars cycler or Earth Mars cycler , a spacecraft trajectory that encounters Earth and Mars on a regular basis, or a spacecraft on such a trajectory Thermal cycler , a laboratory apparatus used to amplify segments of DNA See also Cyclist , a person who rides a bicycle disambig fr Cycler ... more details
BadJPEG Summary Event Segment and State Trajectory for an SP DEVS model for a crosswalk light controller Licensing GFDL self migration relicense ... more details
Summary Black line the running trajectory of the rat. Red dots every red dot correspond to one spike recorded from a single neuron. C Torkel Hafting Licensing GFDL self with disclaimers migration relicense ... more details
Summary Information Description ion s trajectory penning trap Source self made Date 24 03 08 Location Author jrzs other versions Licensing PD self date March 2008 PNG version available Penningtrajec.png ... more details
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