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A trajectory or flight path is the path that an
object Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Object (abstract), an object which does not exist at any particular time or place ** Physical object, an identifiable collection of matter * Goal, an ...
with
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different ele ...
in
motion In physics, motion is the phenomenon in which an object changes its position with respect to time. Motion is mathematically described in terms of displacement, distance, velocity, acceleration, speed and frame of reference to an observer and m ...
follows through
space Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consi ...
as a function of time. In
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
, a trajectory is defined by Hamiltonian mechanics via
canonical coordinates In mathematics and classical mechanics, canonical coordinates are sets of coordinates on phase space which can be used to describe a physical system at any given point in time. Canonical coordinates are used in the Hamiltonian formulation of cl ...
; hence, a complete trajectory is defined by position and
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass ...
, simultaneously. The mass might be a
projectile A projectile is an object that is propelled by the application of an external force and then moves freely under the influence of gravity and air resistance. Although any objects in motion through space are projectiles, they are commonly found i ...
or a
satellite A satellite or artificial satellite is an object intentionally placed into orbit in outer space. Except for passive satellites, most satellites have an electricity generation system for equipment on board, such as solar panels or radioiso ...
. For example, it can be an
orbit In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as ...
— the path of a
planet A planet is a large, rounded astronomical body that is neither a star nor its remnant. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud collapses out of a nebula to create a you ...
,
asteroid An asteroid is a minor planet of the inner Solar System. Sizes and shapes of asteroids vary significantly, ranging from 1-meter rocks to a dwarf planet almost 1000 km in diameter; they are rocky, metallic or icy bodies with no atmosphere. ...
, or
comet A comet is an icy, small Solar System body that, when passing close to the Sun, warms and begins to release gases, a process that is called outgassing. This produces a visible atmosphere or coma, and sometimes also a tail. These phenomena ...
as it travels around a central mass. In
control theory Control theory is a field of mathematics that deals with the control system, control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive ...
, a trajectory is a time-ordered set of states of a
dynamical system In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water i ...
(see e.g. Poincaré map). In
discrete mathematics Discrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way analogous to discrete variables, having a bijection with the set of natural numbers) rather than "continuous" (analogously to continuou ...
, 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.


Physics of trajectories

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 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 The Moon is Earth's only natural satellite. It is the fifth largest satellite in the Solar System and the largest and most massive relative to its parent planet, with a diameter about one-quarter that of Earth (comparable to the width of ...
. In this simple approximation, the trajectory takes the shape of a
parabola In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One descri ...
. Generally when determining trajectories, it may be necessary to account for nonuniform gravitational forces and air resistance ( drag and
aerodynamics Aerodynamics, from grc, ἀήρ ''aero'' (air) + grc, δυναμική (dynamics), is the study of the motion of air, particularly when affected by a solid object, such as an airplane wing. It involves topics covered in the field of fluid dy ...
). This is the focus of the discipline of
ballistics Ballistics is the field of mechanics concerned with the launching, flight behaviour and impact effects of projectiles, especially ranged weapon munitions such as bullets, unguided bombs, rockets or the like; the science or art of designing a ...
. One of the remarkable achievements of
Newtonian mechanics Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in motio ...
was the derivation of
Kepler's laws of planetary motion In astronomy, Kepler's laws of planetary motion, published by Johannes Kepler between 1609 and 1619, describe the orbits of planets around the Sun. The laws modified the heliocentric theory of Nicolaus Copernicus, replacing its circular orb ...
. In the gravitational field of a point mass or a spherically-symmetrical extended mass (such as the Sun), the trajectory of a moving object is a
conic section In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a ...
, usually an
ellipse In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in ...
or a
hyperbola In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, ca ...
. This agrees with the observed orbits of planets, comets, and artificial spacecraft to a reasonably good approximation, although if a comet passes close to the Sun, then it is also influenced by other
force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a ...
s 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 Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
known as
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
. It employs the mathematics of
differential calculus In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve ...
(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.
reason Reason is the capacity of consciously applying logic by drawing conclusions from new or existing information, with the aim of seeking the truth. It is closely associated with such characteristically human activities as philosophy, science, lang ...
, in science as well as technology. It helps to understand and predict an enormous range of
phenomena A phenomenon ( : phenomena) is an observable event. The term came into its modern philosophical usage through Immanuel Kant, who contrasted it with the noumenon, which ''cannot'' be directly observed. Kant was heavily influenced by Gottfried ...
; trajectories are but one example. Consider a particle of
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different ele ...
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". 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 second-order
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...
: m \frac = -\nabla V(\vec(t)) \text \vec=(x,y,z). On the right-hand side, the force is given in terms of \nabla V, the
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
of the potential, taken at positions along the trajectory. This is the mathematical form of Newton's second law of motion: force equals mass times acceleration, for such situations.


Examples


Uniform gravity, neither drag nor wind

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
Galileo Galilei Galileo di Vincenzo Bonaiuti de' Galilei (15 February 1564 – 8 January 1642) was an Italian astronomer, physicist and engineer, sometimes described as a polymath. Commonly referred to as Galileo, his name was pronounced (, ). He ...
. 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
Middle Ages In the history of Europe, the Middle Ages or medieval period lasted approximately from the late 5th to the late 15th centuries, similar to the post-classical period of global history. It began with the fall of the Western Roman Empire ...
in
Europe Europe is a large peninsula conventionally considered a continent in its own right because of its great physical size and the weight of its history and traditions. Europe is also considered a Continent#Subcontinents, subcontinent of Eurasia ...
. Nevertheless, by anticipating the existence of the
vacuum A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or " void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often ...
, later to be demonstrated on
Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's sur ...
by his collaborator
Evangelista Torricelli Evangelista Torricelli ( , also , ; 15 October 160825 October 1647) was an Italian physicist and mathematician, and a student of Galileo. He is best known for his invention of the barometer, but is also known for his advances in optics and wo ...
, Galileo was able to initiate the future science of
mechanics Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objec ...
. In a near vacuum, as it turns out for instance on the
Moon The Moon is Earth's only natural satellite. It is the fifth largest satellite in the Solar System and the largest and most massive relative to its parent planet, with a diameter about one-quarter that of Earth (comparable to the width of ...
, his simplified parabolic trajectory proves essentially correct. In the analysis that follows, we derive the equation of motion of a projectile as measured from an
inertial frame In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleration ...
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 x-axis is tangent to the ground, and the yaxis is perpendicular to it ( parallel to the gravitational field lines ). Let g be the acceleration of gravity. Relative to the flat terrain, let the initial horizontal speed be v_h = v \cos(\theta) and the initial vertical speed be v_v = v \sin(\theta). It will also be shown that the range is 2v_h v_v/g, and the maximum altitude is v_v^2/2g. The maximum range for a given initial speed v is obtained when v_h=v_v, i.e. the initial angle is 45^\circ. This range is v^2/g, and the maximum altitude at the maximum range is v^2/(4g).


Derivation of the equation of motion

Assume the motion of the projectile is being measured from a free fall frame which happens to be at (''x'',''y'') = (0,0) at ''t'' = 0. The equation of motion of the projectile in this frame (by the
equivalence principle In the theory of general relativity, the equivalence principle is the equivalence of gravitational and inertial mass, and Albert Einstein's observation that the gravitational "force" as experienced locally while standing on a massive body (su ...
) would be y = x \tan(\theta). The co-ordinates of this free-fall frame, with respect to our inertial frame would be y = - gt^2/2. That is, y = - g(x/v_h)^2/2. Now translating back to the inertial frame the co-ordinates of the projectile becomes y = x \tan(\theta)- g(x/v_h)^2/2 That is: : y=-x^2+x\tan\theta, (where ''v''0 is the initial velocity, \theta is the angle of elevation, and ''g'' is the acceleration due to gravity).


Range and height

The range, ''R'', is the greatest distance the object travels along the x-axis in the I sector. The initial velocity, ''vi'', is the speed at which said object is launched from the point of origin. The initial angle, ''θi'', is the angle at which said object is released. The ''g'' is the respective gravitational pull on the object within a null-medium. : R= The height, ''h'', is the greatest parabolic height said object reaches within its trajectory : h=


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 \; giving the range as :R= 2 v^2 \cos(\theta) \sin(\theta) / g = v^2 \sin(2\theta) / g\,. This equation can be rearranged to find the angle for a required range : \theta = \frac 1 2 \sin^ \left( \frac \right) (Equation II: angle of projectile launch) Note that the
sine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opp ...
function is such that there are two solutions for \theta for a given range d_h. The angle \theta giving the maximum range can be found by considering the derivative or R with respect to \theta and setting it to zero. := \cos(2\theta)=0 which has a nontrivial solution at 2\theta=\pi/2=90^\circ, or \theta=45^\circ. The maximum range is then R_ = v^2/g\,. At this angle \sin(\pi/2)=1, so the maximum height obtained is . To find the angle giving the maximum height for a given speed calculate the derivative of the maximum height H=v^2 \sin^2(\theta) /(2g) with respect to \theta, that is =v^2 2\cos(\theta)\sin(\theta) /(2g) which is zero when \theta=\pi/2=90^\circ. So the maximum height H_\mathrm= is obtained when the projectile is fired straight up.


Orbiting objects

If instead of a uniform downwards gravitational force we consider two bodies
orbit In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as ...
ing with the mutual gravitation between them, we obtain
Kepler's laws of planetary motion In astronomy, Kepler's laws of planetary motion, published by Johannes Kepler between 1609 and 1619, describe the orbits of planets around the Sun. The laws modified the heliocentric theory of Nicolaus Copernicus, replacing its circular orb ...
. The derivation of these was one of the major works of
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, Theology, theologian, and author (described in his time as a "natural philosophy, natural philosopher"), widely ...
and provided much of the motivation for the development of
differential calculus In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve ...
.


Catching balls

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 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.


Notes


See also

* Aft-crossing trajectory *
Displacement (geometry) In geometry and mechanics, a displacement is a vector whose length is the shortest distance from the initial to the final position of a point P undergoing motion. It quantifies both the distance and direction of the net or total motion along a ...
* Galilean invariance * Orbit (dynamics) * Orbit (group theory) * Orbital trajectory * Planetary orbit * Porkchop plot * Projectile motion * Range of a projectile * Rigid body *
World line The world line (or worldline) of an object is the path that an object traces in 4-dimensional spacetime. It is an important concept in modern physics, and particularly theoretical physics. The concept of a "world line" is distinguished from c ...


References


External links


Projectile Motion Flash Applet
:)




Projectile Lab, JavaScript trajectory simulator

Parabolic Projectile Motion: Shooting a Harmless Tranquilizer Dart at a Falling Monkey
by Roberto Castilla-Meléndez, Roxana Ramírez-Herrera, and José Luis Gómez-Muñoz, The Wolfram Demonstrations Project.
Trajectory
ScienceWorld.

* ttp://www.geogebra.org/en/upload/files/nikenuke/projTARGET01.html Java projectile-motion simulation; targeting solutions, parabola of safety. {{Authority control Ballistics Mechanics