Dictionary Definition
trajectory n : the path followed by an object
moving through space [syn: flight]
User Contributed Dictionary
English
Noun
 The path of a body as it travels through space.
 The ordered set of intermediate states assumed by a dynamical system as a result of time evolution.
 Metaphorically, a course of development, such as that of a war or career.
Related terms
 sense cybernetics run
Translations
path of a body
 Czech: dráha , trajektorie
 Finnish: rata, lentorata
 Norwegian: bane
 Portuguese: trajetória
 Russian: траектория (trajektórija)
ordered set of intermediate states
 Finnish: kehittyminen
 Portuguese: trajetória
 Russian: траектория (trajektórija)
course of development
 Finnish: kulku, kehittyminen
 Portuguese: trajetória
Extensive Definition
Trajectory is the path a moving object follows
through space. The object might be a projectile or a satellite, for example. It
thus includes the meaning of orbit  the path of a planet, an asteroid or a comet as it travels around a
central mass. A trajectory can be described mathematically either
by the geometry of the path, or as the position of the object over
time.
In control
theory a trajectory is a timeordered set of states
of a dynamical
system (see e.g. Poincaré
map). In discrete
mathematics, a trajectory is a sequence (f^k(x))_ of values
calculated by the iterated application of a mapping f to an element
x of its source.
The word trajectory is also often used metaphorically, for instance,
to describe an individual's career.
Physics of trajectories
A familiar example of a trajectory is the path of a projectile such as a thrown ball or rock. In a greatly simplified model the object moves only under the influence of a uniform homogenous gravitational force field. This can be a good approximation for a rock that is thrown for short distances for example, at the surface of the moon. In this simple approximation the trajectory takes the shape of a parabola. Generally, when determining trajectories it may be necessary to account for nonuniform gravitational forces, air resistance (drag and aerodynamics). This is the focus of the discipline of ballistics.One of the remarkable achievements of Newtonian
mechanics was the derivation of the laws of
Kepler, in the case of the gravitational field of a single
point mass (representing the Sun). The trajectory is
a conic
section, like an ellipse or a parabola. This agrees with the
observed orbits of planets and comets, to a reasonably good
approximation. Although if a comet passes close to the Sun, then it
is also influenced by other forces, such as the solar wind and
radiation
pressure, which modify the orbit, and cause the comet to eject
material into space.
Newton's theory later developed into the branch
of theoretical
physics known as classical
mechanics. It employs the mathematics of differential
calculus (which was, in fact, also initiated by Newton, in his
youth). Over the centuries, countless scientists contributed to the
development of these two disciplines. Classical mechanics became a
most prominent demonstration of the power of rational thought, i.e.
reason, in science as
well as technology. It helps to understand and predict an enormous
range of phenomena.
Trajectories are but one example.
Consider a particle of mass m, moving in a potential
field V. Physically speaking, mass represents inertia, and the field V
represents external forces, of a particular kind known as
"conservative". That is, given V 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.
The motion of the particle is described by the
secondorder differential
equation
 m \frac = \nabla V(\vec(t)) with \vec = (x, y, z)
On the righthand side, the force is given in
terms of \nabla V, the gradient of the potential,
taken at positions along the trajectory. This is the mathematical
form of Newton's second law of motion: mass times acceleration
equals force, for such situations.
Examples
Uniform gravity, no drag or wind
The case of uniform gravity, disregarding drag and wind, yields a trajectory which is a parabola. To model this, one chooses V = m g z, where g is the acceleration of gravity. This gives the equations of motion \frac = \frac = 0
 \frac =  g
Simplifications are made for the sake of studying
the basics. The actual situation, at least on the surface of
Earth, is
considerably more complicated than this example would suggest, when
it comes to computing actual trajectories. By deliberately
introducing such simplifications, into the study of the given
situation, one does, in fact, approach the problem in a way that
has proved exceedingly useful in physics.
The present example is one of those originally
investigated by Galileo
Galilei. To neglect the action of the atmosphere, in shaping a
trajectory, would (at best) have been considered a futile
hypothesis by practical minded investigators, all through the
Middle
Ages in Europe.
Nevertheless, by anticipating the existence of the vacuum, later to be demonstrated
on Earth by his collaborator Evangelista
Torricelli, Galileo was able to initiate the future science of
mechanics. And in a
near vacuum, as it turns out for instance on the Moon, his simplified
parabolic trajectory proves essentially correct.
Relative to a flat terrain, let the initial
horizontal speed be v_h\,, and the initial vertical speed be v_v\,.
It will be shown that, the range
is 2v_h v_v/g\,, and the maximum altitude is /2g\,. The maximum
range, for a given total initial speed v, is obtained when
v_h=v_v\,, i.e. the initial angle is 45 degrees. This range is
v^2/g\,, and the maximum altitude at the maximum range is a quarter
of that.
Derivation
The equations of motion may be used to calculate
the characteristics of the trajectory.
Let
 p(t)\; be the position of the projectile, expressed as a vector
 t\; be the time into the flight of the projectile,
 v_h \; be the initial horizontal velocity (which is constant)
 v_v \; be the initial vertical velocity upwards.
 p(t) = ( A t, 0 , a t^2 + b t + c )\,
 p'(t) = ( A , 0 , 2 a t + b ),\quad p(t) = ( 0 , 0 , 2 a ).
 p(0)= (0, 0, 0)\ p'(0)=(v_h,0,v_v),\ p(0)=(0,0,g)
 A = v_h,\ a = g/2,\ b = v_v,\ c = 0.
 p(t) = (v_h t,0,v_v t  g t^2/2)\,\qquad (Equation I: trajectory of parabola).
Range and height
The range R of the projectile is found when the zcomponent of p is zero, that is when 0 = v_v t  g t^2/2 = t \left( v_v  g t/2\right)\,
From the symmetry of the parabola the maximum
height occurs at the halfway point t=v_v/g at position
 p(v_v/g)=(v_h v_v/g,0,v_v^2/(2g))\,
Angle of elevation
In terms of angle of elevation \theta and initial speed v: v_h=v \cos \theta,\quad v_v=v \sin \theta \;
 R= 2 v^2 \cos(\theta) \sin(\theta) / g = v^2 \sin(2\theta) / g\,.
 = \frac 1 2 \sin^ \left( \right) (Equation II: angle of projectile launch)
 = \cos(2\theta)=0
To find the angle giving the maximum height for a
given speed calculate the derivative of the maximum height H=v^2
sin(\theta) /(2g) with respect to \theta, that is =v^2 \cos(\theta)
/(2g) which is zero when \theta=\pi=180^\circ. So the maximum
height H_= is obtain when the projectile is fired straight up. The
equation of the trajectory of a projectile fired in uniform gravity
in a vacuum on Earth in Cartesian coordinates is
y=x^2+x\tan\theta+h,
where v0 is the initial speed, h is the height
the projectile is fired from, and g is the acceleration due to
gravity).
Uphill/downhill in uniform gravity in a vacuum
Given a hill angle \alpha and launch angle \theta as before, it can be shown that the range along the hill R_s forms a ratio with the original range R along the imaginary horizontal, such that: \frac =(1\cot \theta \tan \alpha)\sec \alpha (Equation 11)
In this equation, downhill occurs when \alpha is
between 0 and 90 degrees. For this range of \alpha we know:
\tan(\alpha)=\tan \alpha and \sec (  \alpha ) = \sec \alpha.
Thus for this range of \alpha, R_s/R=(1+\tan \theta \tan
\alpha)\sec \alpha . Thus R_s/R is a positive value meaning the
range downhill is always further than along level terrain. This
makes perfect sense as it is expected that gravity will assist the
projectile, giving it greater range.
While the same equation applies to projectiles
fired uphill, the interpretation is more complex as sometimes the
uphill range may be shorter or longer than the equivalent range
along level terrain. Equation 11 may be set to R_s/R=1 (i.e. the
slant range is equal to the level terrain range) and solving for
the "critical angle" \theta_:
 1=(1\tan \theta \tan \alpha)\sec \alpha \quad \;
 \theta_=\arctan((1\csc \alpha)\cot \alpha) \quad \;
Equation 11 may also be used to develop the
"rifleman's
rule" for small values of \alpha and \theta (i.e. close to
horizontal firing, which is the case for many firearm situations).
For small values, both \tan \alpha and \tan \theta have a small
value and thus when multiplied together (as in equation 11), the
result is almost zero. Thus equation 11 may be approximated as:
 \frac =(10)\sec \alpha
 R=R_s \cos \alpha \ "Rifleman's rule"
Derivation based on equations of a parabola
The intersect of the projectile trajectory with a hill may most easily be derived using the trajectory in parabolic form in Cartesian coordinates (Equation 10) intersecting the hill of slope m in standard linear form at coordinates (x,y): y=mx+b \; (Equation 12) where in this case, y=d_v, x=d_h and b=0
Substituting the value of d_v=m d_h into Equation
10:
 m x=\fracx^2 + \frac x
 x=\frac\left(\fracm\right) (Solving above x)
 y=mx=m \frac \left(\fracm\right)
 R_s=\sqrt=\sqrt

 =\frac \sqrt
 =\frac \left(\fracm\right) \sqrt
Now \alpha is defined as the angle of the hill,
so by definition of
tangent, m=\tan \alpha. This can be substituted into the
equation for R_s:
 R_s=\frac \left(\frac\tan \alpha\right) \sqrt
 R_s=\frac\left(1\frac\tan\alpha\right)\sec\alpha
 R_s=R(1\tan\theta\tan\alpha)\sec\alpha \;
 \frac=(1\tan\theta\tan\alpha)\sec\alpha
Orbiting objects
If instead of a uniform downwards gravitational force we consider two bodies orbiting with the mutual gravitation between them, we obtain Kepler's laws of planetary motion. The derivation of these was one of the major works of Isaac Newton and provided much of the motivation for the development of differential calculus.See also
External links
 Projectile Motion Flash Applet
 Trajectory calculator
 An interactive simulation on projectile motion
 Projectile Motion Simulator, java applet
 Projectile Lab, JavaScript trajectory simulator
 Projectile calculation in MS Excel – calculation of the projectile position after a given time, the maximum height reached and the range of the projectile. The projectile path is plotted on an Excel chart and all cell formulae are shown in mathematical notation.
 Parabolic Projectile Motion: Shooting a Harmless Tranquilizer Dart at a Falling Monkey by Roberto CastillaMeléndez, Roxana RamírezHerrera, and José Luis GómezMuñoz, The Wolfram Demonstrations Project.
 Trajectory, ScienceWorld.
trajectory in Catalan: Trajectòria
trajectory in German: Trajektorie (Physik)
trajectory in Estonian: Trajektoor
trajectory in Spanish: Trayectoria
trajectory in French: Trajectoire
trajectory in Italian: Traiettoria
trajectory in Lithuanian: Trajektorija
trajectory in Japanese: 弾道
trajectory in Polish: Trajektoria
trajectory in Portuguese: Trajetória
trajectory in Russian: Траектория материальной
точки
trajectory in Slovak: Trajektória
trajectory in Ukrainian: Траекторія
Synonyms, Antonyms and Related Words
air lane, altitude peak, aphelion, apogee, astronomical longitude,
automatic control, autumnal equinox, beat, blastoff, burn, burnout, ceiling, celestial equator,
celestial longitude, celestial meridian, circle, circuit, colures, course, descent, ecliptic, end of burning,
equator, equinoctial, equinoctial
circle, equinoctial colure, equinox, flight, flight path, galactic
longitude, geocentric longitude, geodetic longitude, great circle,
heliocentric longitude, ignition, impact, itinerary, launch, liftoff, line, longitude, meridian, orbit, path, perigee, perihelion, period, primrose path, road, rocket launching, round, route, run, sea lane, shoot, shortcut, shot, small circle, solstitial
colure, tour, track, trade route, traject, trajet, velocity peak, vernal
equinox, walk, zodiac, zone