- Trajectory
**Trajectory**is the path a moving object follows through space. The object might be aprojectile or asatellite , for example. It thus includes the meaning oforbit - the path of aplanet , anasteroid or acomet 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 time-ordered set of states of adynamical system (see e.g.Poincaré map ). Indiscrete mathematics , a**trajectory**is a sequence$(f^k(x))\_\{k\; in\; mathbb\{N$ 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

metaphor ically, 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 themoon . In this simple approximation the trajectory takes the shape of aparabola . Generally, when determining trajectories it may be necessary to account for nonuniform gravitational forces, air resistance (drag andaerodynamics ). This is the focus of the discipline ofballistics .One of the remarkable achievements of

Newtonian mechanics was the derivation of thelaws of Kepler , in the case of the gravitational field of a single point mass (representing theSun ). The trajectory is aconic section , like anellipse or aparabola . This agrees with the observed orbits ofplanets andcomets , to a reasonably good approximation. Although if a comet passes close to the Sun, then it is also influenced by otherforce s, such as thesolar wind andradiation 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 asclassical mechanics . It employs the mathematics ofdifferential 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 ofphenomena . Trajectories are but one example.Consider a particle of

mass $m$, moving in apotential field $V$. Physically speaking, mass representsinertia , 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 second-order

differential equation :$m\; frac\{mathrm\{d\}^2\; vec\{x\}(t)\}\{mathrm\{d\}t^2\}\; =\; -\; abla\; V(vec\{x\}(t))$ with $vec\{x\}\; =\; (x,\; y,\; z)$

On the right-hand side, the force is given in terms of $abla\; V$, 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 aparabola . To model this, one chooses $V\; =\; m\; g\; z$, where $g$ is the acceleration of gravity. This gives theequations of motion :$frac\{mathrm\{d\}^2\; x\}\{mathrm\{d\}t^2\}\; =\; frac\{mathrm\{d\}^2\; y\}\{mathrm\{d\}t^2\}\; =\; 0$ :$frac\{mathrm\{d\}^2\; z\}\{mathrm\{d\}t^2\}\; =\; -\; g$

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 theMiddle Ages inEurope . Nevertheless, by anticipating the existence of thevacuum , later to be demonstrated on Earth by his collaboratorEvangelista Torricelli , Galileo was able to initiate the future science ofmechanics . And in a near vacuum, as it turns out for instance on theMoon , 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 $\{v\_v^2\}/2g,$. 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.The path of the projectile is known to be a parabola so:$p(t)\; =\; (\; A\; t,\; 0\; ,\; a\; t^2\; +\; b\; t\; +\; c\; ),$where $A,,a,,b,,c$ are parameters to be found. The first and second derivatives of $p$ are::$p\text{\'}(t)\; =\; (\; A\; ,\; 0\; ,\; 2\; a\; t\; +\; b\; ),quad\; p"(t)\; =\; (\; 0\; ,\; 0\; ,\; 2\; a\; ).$At $t=0$ :$p(0)=\; (0,\; 0,\; 0)\; p\text{\'}(0)=(v\_h,0,v\_v),\; p"(0)=(0,0,-g)$so:$A\; =\; v\_h,\; a\; =\; -g/2,\; b\; =\; v\_v,\; c\; =\; 0$. This yields the formula for a parabolic trajectory::$p(t)\; =\; (v\_h\; t,0,v\_v\; t\; -\; g\; t^2/2),qquad$ (Equation I: trajectory of parabola).

**Range and height**The

**range**$R$ of the projectile is found when the $z$-component of $p$ is zero, that is when:$0\; =\; v\_v\; t\; -\; g\; t^2/2\; =\; t\; left(\; v\_v\; -\; g\; t/2\; ight),$which has solutions at $t=0$ and$t\; =\; 2\; v\_v\; /g$(the**hang-time of the projectile**).The range is then$R\; =\; 2\; v\_h\; v\_v/g.,$From the symmetry of the parabola the

**maximum height**occurs at the halfway point $t=v\_v/g$ at position:$p(v\_v/g)=(v\_h\; v\_v/g,0,v\_v^2/(2g)),$This can also be derived by finding when the $z$-component of $p\text{'}$ is zero.**Angle of elevation**In terms of angle of elevation $heta$ and initial speed $v$::$v\_h=v\; cos\; heta,quad\; v\_v=v\; sin\; heta\; ;$giving the range as:$R=\; 2\; v^2\; cos(\; heta)\; sin(\; heta)\; /\; g\; =\; v^2\; sin(2\; heta)\; /\; g,.$This equation can be rearranged to find the angle for a required range:$\{\; heta\; \}\; =\; frac\; 1\; 2\; sin^\{-1\}\; left(\; \{\; \{g\; R\}\; over\; \{\; v^2\; \}\; \}\; ight)$ (Equation II: angle of projectile launch)Note that the

sine function is such that there are two solutions for $heta$ for a given range $d\_h$. Physically, this corresponds to a direct shot versus a mortar shot up and over obstacles to the target.The angle $heta$ giving the maximum range can be found by considering the derivative or $R$ with respect to $heta$ and setting it to zero.:$\{mathrm\{d\}Rover\; mathrm\{d\}\; heta\}=\{2v^2over\; g\}\; cos(2\; heta)=0$which has a non trivial solutions at $2\; heta=pi/2=90^circ$.The maximum range is then $R\_\{max\}\; =\; v^2/g,$. At this angle $sin(pi/2)=1$ so the maximum height obtained is $\{v^2\; over\; 4g\}$.To find the angle giving the maximum height for a given speed calculate the derivative of the maximum height $H=v^2\; sin(\; heta)\; /(2g)$ with respect to $heta$, that is $\{mathrm\{d\}Hover\; mathrm\{d\}\; heta\}=v^2\; cos(\; heta)\; /(2g)$which is zero when $heta=pi=180^circ$. So the maximum height $H\_\{max\}=\{v^2over\; 2g\}$ 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

$y=-\{gsec^2\; hetaover\; 2v\_0^2\}x^2+x\; an\; heta+h$,

where "v"

_{0}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 $heta$ 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::$frac\{R\_s\}\; \{R\}=(1-cot\; heta\; an\; alpha)sec\; alpha$ (Equation 11)

In this equation, downhill occurs when $alpha$ is between 0 and -90 degrees. For this range of $alpha$ we know: $an(-alpha)=-\; an\; alpha$ and $sec\; (\; -\; alpha\; )\; =\; sec\; alpha$. Thus for this range of $alpha$,$R\_s/R=(1+\; an\; heta\; an\; alpha)sec\; alpha$. Thus $R\_s/R$ is a positive value meaning the range downhill is always further than along level terrain. The lower level of terrain causes the projectile to remain in the air longer, allowing it to travel further horizontally before hitting the ground.

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" $heta\_\{cr\}$::$1=(1-\; an\; heta\; an\; alpha)sec\; alpha\; quad\; ;$ :$heta\_\{cr\}=arctan((1-csc\; alpha)cot\; alpha)\; quad\; ;$

Equation 11 may also be used to develop the "

rifleman's rule " for small values of $alpha$ and $heta$ (i.e. close to horizontal firing, which is the case for many firearm situations). For small values, both $an\; alpha$ and $an\; heta$ 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::$frac\{R\_s\}\; \{R\}=(1-0)sec\; alpha$And solving for level terrain range, $R$:$R=R\_s\; cos\; alpha$ "Rifleman's rule"Thus if the shooter attempts to hit the level distance R, s/he will actually hit the slant target. "In other words, pretend that the inclined target is at a horizontal distance equal to the slant range distance multiplied by the cosine of the inclination angle, and aim as if the target were really at that horizontal position." [*http://www.snipertools.com/article4.htm*]**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::$m\; x=-frac\{g\}\{2v^2\{cos\}^2\; heta\}x^2\; +\; frac\{sin\; heta\}\{cos\; heta\}\; x$:$x=frac\{2v^2cos^2\; heta\}\{g\}left(frac\{sin\; heta\}\{cos\; heta\}-m\; ight)$ (Solving above x)This value of x may be substituted back into the linear equation 12 to get the corresponding y coordinate at the intercept::$y=mx=m\; frac\{2v^2cos^2\; heta\}\{g\}\; left(frac\{sin\; heta\}\{cos\; heta\}-m\; ight)$Now the slant range $R\_s$ is the distance of the intercept from the origin, which is just the

hypotenuse of x and y::$R\_s=sqrt\{x^2+y^2\}=sqrt\{left(frac\{2v^2cos^2\; heta\}\{g\}left(frac\{sin\; heta\}\{cos\; heta\}-m\; ight)\; ight)^2+left(m\; frac\{2v^2cos^2\; heta\}\{g\}\; left(frac\{sin\; heta\}\{cos\; heta\}-m\; ight)\; ight)^2\}$::$=frac\{2v^2cos^2\; heta\}\{g\}\; sqrt\{left(frac\{sin\; heta\}\{cos\; heta\}-m\; ight)^2+m^2\; left(frac\{sin\; heta\}\{cos\; heta\}-m\; ight)^2\}$::$=frac\{2v^2cos^2\; heta\}\{g\}\; left(frac\{sin\; heta\}\{cos\; heta\}-m\; ight)\; sqrt\{1+m^2\}$Now $alpha$ is defined as the angle of the hill, so by definition of tangent, $m=\; an\; alpha$. This can be substituted into the equation for $R\_s$::$R\_s=frac\{2v^2cos^2\; heta\}\{g\}\; left(frac\{sin\; heta\}\{cos\; heta\}-\; an\; alpha\; ight)\; sqrt\{1+\; an^2\; alpha\}$Now this can be refactored and the

trigonometric identity for $sec\; alpha\; =\; sqrt\; \{1\; +\; an^2\; alpha\}$ may be used::$R\_s=frac\{2v^2cos\; hetasin\; heta\}\{g\}left(1-frac\{sin\; heta\}\{cos\; heta\}\; analpha\; ight)secalpha$Now the flat range $R=v^2sin\; 2\; heta\; /\; g\; =\; 2v^2sin\; hetacos\; heta\; /\; g$ by the previously usedtrigonometric identity and $sin\; heta/cos\; heta=tan\; heta$ so::$R\_s=R(1-\; an\; heta\; analpha)secalpha\; ;$:$frac\{R\_s\}\{R\}=(1-\; an\; heta\; analpha)secalpha$**Orbiting objects**If instead of a uniform downwards gravitational force we considertwo 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 ofIsaac Newton and provided much of the motivation for the development ofdifferential calculus .**ee also***

Aft-crossing trajectory

*Orbit (dynamics)

*Orbit (group theory)

*Planetary orbit

*Porkchop plot

*Rigid body

*Trajectory of a projectile **External links*** [

*http://www.physics-lab.net/applets/projectile-motion Projectile Motion Flash Applet*]

* [*http://hyperphysics.phy-astr.gsu.edu/hbase/traj.html Trajectory calculator*]

* [*http://www.phy.hk/wiki/englishhtm/ThrowABall.htm An interactive simulation on projectile motion*]

* [*http://publicliterature.org/tools/projectile_motion/ Projectile Motion Simulator, java applet*]

* [*http://www.thewritingpot.com/projectilelab/ Projectile Lab, JavaScript trajectory simulator*]

* [*http://www.excelcalcs.com/content/view/74/109/ 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.

* [*http://demonstrations.wolfram.com/ParabolicProjectileMotionShootingAHarmlessTranquilizerDartAt/ 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 .

* [*http://scienceworld.wolfram.com/physics/Trajectory.html Trajectory*] , ScienceWorld.

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