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| {{For|other uses|Trajectory (disambiguation)}}
| | Making the computer run fast is pretty simple. Most computers run slow because they are jammed up with junk files, that Windows has to search from each time it wants to find something. Imagine needing to discover a book in a library, however, all library books are inside a big big pile. That's what it's like for the computer to obtain something, whenever a system is full of junk files.<br><br>Document files enable the user to input information, images, tables plus different elements to enhance the presentation. The just problem with this format compared to additional file types such as .pdf for illustration is its ability to be easily editable. This means which anyone viewing the file will change it by accident. Additionally, this file format is opened by additional programs but it does not guarantee that what you see in the Microsoft Word application will nevertheless be the same when we view it utilizing another program. However, it is nonetheless preferred by many computer consumers for its ease of utilize plus qualities.<br><br>It doesn't matter whether you are not quite obvious regarding what rundll32.exe is. However remember that it plays an important role in keeping the stability of our computers and the integrity of the system. When some software or hardware could not respond usually to your system operation, comes the rundll32 exe error, which can be caused by corrupted files or missing data in registry. Usually, error message will shows up at booting or the beginning of running a program.<br><br>Your computer was quite rapidly whenever you first purchased it. Because your registry was really clean and free of errors. After time, the computer begins to run slow and freezes up now plus then. Because there are mistakes accumulating inside it and certain information is rewritten or even completely deleted by a wrong uninstall of programs, wrong operations, malware or other aspects. That is the reason why the computer performance decreases slowly and become very unstable.<br><br>There are a lot of [http://bestregistrycleanerfix.com/registry-mechanic registry mechanic] s. Which one is the greatest is not convenient to be determined. But when we wish to stand out 1 amidst the multitude we could consider certain products. These are features, scanning speed time, total mistakes detected, total errors repaired, tech help, Boot time performance and price. According to these products Top Registry Cleaner for 2010 is RegCure.<br><br>Windows relies heavily on this database, storing everything from a latest emails to a Internet favorites in there. Because it's thus important, a computer is continually adding plus updating the files inside it. This is ok, however it may make the computer run slow, when a computer accidentally breaks its important registry files. This is a truly well-known issue, plus really makes your computer run slower each day. What arises is that because the computer is frequently using 100's of registry files at when, it occasionally gets confused plus create a few of them unreadable. This then makes the computer run slow, because Windows takes longer to read the files it needs.<br><br>The 'registry' is just the central database which stores all a settings plus choices. It's a certainly significant piece of the XP system, meaning that Windows is regularly adding and updating the files inside it. The issues happen whenever Windows actually corrupts & loses several of these files. This makes your computer run slow, because it tries difficult to locate them again.<br><br>A registry cleaner is a system which cleans the registry. The Windows registry always gets flooded with junk data, info which has not been removed from uninstalled programs, erroneous file association and alternative computer-misplaced entries. These clean little system software tools are very normal today and you are able to find many good ones found on the Internet. The good ones provide we option to maintain, clean, update, backup, and scan the System Registry. When it finds supposedly unwelcome ingredients inside it, the registry cleaner lists them plus recommends the consumer to delete or repair these orphaned entries plus corrupt keys. |
| {{Redirect|Flight path|airline trajectories|Airway (aviation)|other uses|flightpath (disambiguation)}}
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| {{Unreferenced|date=September 2009}}
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| A '''trajectory''' is the path that a moving object follows through space as a function of time. The object might be a [[projectile]] or a [[satellite]], for example. <!-- Strictly speaking '''trajectory''' refers only to that portion of the path during which the body undergoes a transient movement between practically stationary or repetitive motions or until the body eventually stops moving. who says this? cite a source please! --> 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.
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| In [[control theory]] a '''trajectory''' is a time-ordered set of [[state (controls)|state]]s of a [[dynamical system]] (see e.g. [[Poincaré map]]). In [[discrete mathematics]], a '''trajectory''' is a sequence
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| <math>(f^k(x))_{k \in \mathbb{N}}</math> of values calculated by the iterated application of a mapping | |
| <math>f</math> to an element <math>x</math> of its source.
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| [[Image:RiflemansRule.svg|thumb|350px|Illustration showing the trajectory of a bullet fired at an uphill target.]]
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| == Physics of trajectories ==
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| {{confusing|date=November 2011}}
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| 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 gravitational [[Force field (physics)|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 (physics)|drag]] and [[aerodynamics]]). This is the focus of the discipline of [[ballistics]].
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| 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 [[force]]s, such as the [[solar wind]] and [[radiation pressure]], which modify the orbit, and cause the comet to eject material into space.
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| 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.
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| Consider a particle of [[mass]] <math>m</math>, moving in a [[potential field]] <math>V</math>. Physically speaking, mass represents [[inertia]], and the field <math>V</math> represents external forces, of a particular kind known as "conservative". That is, given <math>V</math> 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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| The motion of the particle is described by the second-order [[differential equation]]
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| :<math> m \frac{\mathrm{d}^2 \vec{x}(t)}{\mathrm{d}t^2} = -\nabla V(\vec{x}(t)) </math> with <math>\vec{x} = (x, y, z)</math>
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| On the right-hand side, the force is given in terms of <math>\nabla V</math>, the [[gradient]] 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.
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| ==Examples==
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| === Uniform gravity, no drag or wind===
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| [[File:Inclinedthrow.gif|thumb|400px|Trajectories of a mass thrown at an angle of 70°:<br>
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| {{color box|black}} without [[Drag (physics)|drag]]<br>
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| {{color box|blue}} with [[Stokes'_law|Stokes drag]]<br>
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| {{color box|green}} with [[Newtonian_fluid|Newton drag]]]]
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| 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]]. 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 [[Europe]]. Nevertheless, by anticipating the existence of the [[vacuum]], later to be demonstrated on Earth by his collaborator [[Evangelista Torricelli]]{{Citation needed|date=March 2009}}, Galileo was able to initiate the future science of [[mechanics]].{{Citation needed|date=March 2009}} And in a near vacuum, as it turns out for instance on the [[Moon]], his simplified parabolic trajectory proves essentially correct.
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| In the analysis that follows we derive the equation of motion of a projectile as measured from an inertial frame, at rest with respect to the ground, to which frame is associated a right-hand co-ordinate system - the origin of which coincides with the point of launch of the projectile. The x-axis is parallel to the ground and the y axis perpendicular to it ( parallel to the gravitational field lines ). Let <math>g</math> be the [[standard gravity|acceleration of gravity]]. Relative to the flat terrain, let the initial horizontal speed be <math>v_h = v \cos(\theta)</math> and the initial vertical speed be <math>v_v = v \sin(\theta)</math>. It will also be shown that, the [[range of a projectile|range]] is <math>2v_h v_v/g</math>, and the maximum altitude is <math>v_v^2/2g</math>; The maximum range, for a given initial speed <math>v</math>, is obtained when <math>v_h=v_v</math>, i.e. the initial angle is 45 degrees. This range is <math>v^2/g</math>, and the maximum altitude at the maximum range is a quarter of that.
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| ====Derivation of the equation of motion====
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| 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 [[principle of equivalence]]) would be <math>y = x \tan(\theta)</math>. The co-ordinates of this free-fall frame, with respect to our inertial frame would be <math>y = - gt^2/2</math>. That is, <math>y = - g(x/v_h)^2/2</math>.
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| Now translating back to the inertial frame the co-ordinates of the projectile becomes <math>y = x \tan(\theta)- g(x/v_h)^2/2</math> That is:
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| <math>y=-{g\sec^2\theta\over 2v_0^2}x^2+x\tan\theta</math>,
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| (where ''v''<sub>0</sub> is the initial velocity, <math>\theta</math> is the angle of elevation, and ''g'' is the acceleration due to gravity).
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| ====Range and height====
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| [[Image:Ideal_projectile_motion_for_different_angles.svg|thumb|350px|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<sup>2</sup>. Points are at 0.05 s intervals and length of their tails is linearly proportional to their speed. ''t'' = time from launch, ''T'' = time of flight, ''R'' = range and ''H'' = highest point of trajectory (indicated with arrows).]]
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| The '''range''', ''R'', is the greatest distance the object travels along the [[x-axis]] in the I sector. The '''initial velocity''', ''v<sub>i</sub>'', is the speed at which said object is launched from the point of origin. The '''initial angle''', ''θ<sub>i</sub>'', is the angle at which said object is released. The ''g'' is the respective gravitational pull on the object within a null-medium.
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| :<math>R={v_i^2\sin2\theta_i\over g}</math>
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| The '''height''', ''h'', is the greatest parabolic height said object reaches within its trajectory
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| :<math>h={v_i^2\sin^2\theta_i\over 2g}</math>
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| ====Angle of elevation====
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| In terms of angle of elevation <math>\theta</math> and initial speed <math>v</math>:
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| :<math>v_h=v \cos \theta,\quad v_v=v \sin \theta \;</math>
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| giving the range as
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| :<math>R= 2 v^2 \cos(\theta) \sin(\theta) / g = v^2 \sin(2\theta) / g\,.</math>
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| This equation can be rearranged to find the angle for a required range
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| :<math> { \theta } = \frac 1 2 \sin^{-1} \left( { {g R} \over { v^2 } } \right) </math> (Equation II: angle of projectile launch)
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| Note that the [[sine]] function is such that there are two solutions for <math>\theta</math> for a given range <math>d_h</math>. The angle <math>\theta</math> giving the maximum range can be found by considering the derivative or <math>R</math> with respect to <math>\theta</math> and setting it to zero.
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| :<math>{\mathrm{d}R\over \mathrm{d}\theta}={2v^2\over g} \cos(2\theta)=0</math>
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| which has a nontrivial solution at <math>2\theta=\pi/2=90^\circ</math>, or <math>\theta=45^\circ</math>.
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| The maximum range is then <math>R_{max} = v^2/g\,</math>. At this angle <math>sin(\pi/2)=1</math>, so the maximum height obtained is <math>{v^2 \over 4g}</math>.
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| To find the angle giving the maximum height for a given speed calculate the derivative of the maximum height <math>H=v^2 sin^2(\theta) /(2g)</math> with respect to <math>\theta</math>, that is
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| <math>{\mathrm{d}H\over \mathrm{d}\theta}=v^2 2\cos(\theta)\sin(\theta) /(2g)</math>
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| which is zero when <math>\theta=\pi/2=90^\circ</math>. So the maximum height <math>H_{max}={v^2\over 2g}</math> is obtained when the projectile is fired straight up.
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| ===Uphill/downhill in uniform gravity in a vacuum===
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| {{Disputed-section|date=March 2008}}
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| Given a hill angle <math>\alpha</math> and launch angle <math>\theta</math> as before, it can be shown that the range along the hill <math>R_s</math> forms a ratio with the original range <math>R</math> along the imaginary horizontal, such that:
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| :<math>\frac{R_s} {R}=(1-\cot \theta \tan \alpha)\sec \alpha </math> (Equation 11)
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| In this equation, downhill occurs when <math>\alpha</math> is between 0 and -90 degrees. For this range of <math>\alpha</math> we know: <math>\tan(-\alpha)=-\tan \alpha</math> and <math>\sec ( - \alpha ) = \sec \alpha</math>. Thus for this range of <math>\alpha</math>,
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| <math>R_s/R=(1+\tan \theta \tan \alpha)\sec \alpha </math>. Thus <math>R_s/R</math> 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.
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| 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 <math>R_s/R=1</math> (i.e. the slant range is equal to the level terrain range) and solving for the "critical angle" <math>\theta_{cr}</math>:
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| :<math>1=(1-\tan \theta \tan \alpha)\sec \alpha \quad \; </math>
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| :<math>\theta_{cr}=\arctan((1-\csc \alpha)\cot \alpha) \quad \; </math>
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| Equation 11 may also be used to develop the "[[rifleman's rule]]" for small values of <math>\alpha</math> and <math>\theta</math> (i.e. close to horizontal firing, which is the case for many firearm situations). For small values, both <math>\tan \alpha</math> and <math>\tan \theta</math> 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:
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| :<math>\frac{R_s} {R}=(1-0)\sec \alpha </math>
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| And solving for level terrain range, <math>R</math>
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| :<math>R=R_s \cos \alpha \ </math> "Rifleman's rule"
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| 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]
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| ====Derivation based on equations of a parabola====
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| 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 <math>m</math> in standard linear form at coordinates <math>(x,y)</math>: | |
| :<math>y=mx+b \;</math> (Equation 12) where in this case, <math>y=d_v</math>, <math>x=d_h</math> and <math>b=0</math>
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| Substituting the value of <math>d_v=m d_h</math> into Equation 10:
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| :<math>m x=-\frac{g}{2v^2{\cos}^2 \theta}x^2 + \frac{\sin \theta}{\cos \theta} x</math>
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| :<math>x=\frac{2v^2\cos^2\theta}{g}\left(\frac{\sin \theta}{\cos \theta}-m\right)</math> (Solving above x)
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| This value of x may be substituted back into the linear equation 12 to get the corresponding y coordinate at the intercept:
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| :<math>y=mx=m \frac{2v^2\cos^2\theta}{g} \left(\frac{\sin \theta}{\cos \theta}-m\right)</math>
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| Now the slant range <math>R_s</math> is the distance of the intercept from the origin, which is just the [[hypotenuse]] of x and y:
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| :<math>R_s=\sqrt{x^2+y^2}=\sqrt{\left(\frac{2v^2\cos^2\theta}{g}\left(\frac{\sin \theta}{\cos \theta}-m\right)\right)^2+\left(m \frac{2v^2\cos^2\theta}{g} \left(\frac{\sin \theta}{\cos \theta}-m\right)\right)^2}</math>
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| ::<math>=\frac{2v^2\cos^2\theta}{g} \sqrt{\left(\frac{\sin \theta}{\cos \theta}-m\right)^2+m^2 \left(\frac{\sin \theta}{\cos \theta}-m\right)^2}</math>
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| ::<math>=\frac{2v^2\cos^2\theta}{g} \left(\frac{\sin \theta}{\cos \theta}-m\right) \sqrt{1+m^2}</math>
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| Now <math>\alpha</math> is defined as the angle of the hill, so by definition of [[tangent (trigonometric function)|tangent]], <math>m=\tan \alpha</math>. This can be substituted into the equation for <math>R_s</math>:
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| :<math>R_s=\frac{2v^2\cos^2\theta}{g} \left(\frac{\sin \theta}{\cos \theta}-\tan \alpha\right) \sqrt{1+\tan^2 \alpha}</math>
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| Now this can be refactored and the [[trigonometric identity]] for <math>\sec \alpha = \sqrt {1 + \tan^2 \alpha}</math> may be used:
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| :<math>R_s=\frac{2v^2\cos\theta\sin\theta}{g}\left(1-\frac{\sin\theta}{\cos\theta}\tan\alpha\right)\sec\alpha</math>
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| Now the flat range <math>R=v^2\sin 2 \theta / g = 2v^2\sin\theta\cos\theta / g</math> by the previously used [[trigonometric identity]] and <math>\cos\theta/\sin\theta=cotan\theta</math> so:
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| :<math>R_s=R(1-\cot\theta\tan\alpha)\sec\alpha \;</math>
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| :<math>\frac{R_s}{R}=(1-\cot\theta\tan\alpha)\sec\alpha</math>
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| ===Orbiting objects===
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| If instead of a uniform downwards gravitational force we consider
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| two bodies orbiting with the mutual gravitation between them, we obtain
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| [[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]].
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| ==Catching balls==
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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 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.
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| '''Proof'''
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| Suppose the ball starts with a vertical component of velocity of <math>v,</math> upward, and a horizontal component of velocity of <math>h</math> toward the player who wants to catch it. Its altitude above the ground is given by:
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| :<math>a=vt-\frac{1}{2}gt^2,</math> where <math>t</math> is the time since the ball was hit, and <math>g</math> is the acceleration due to gravity.
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| The total time for the flight, until the ball is back down to the ground, from which it started, is given by: | |
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| :<math>a=0</math>
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| :<math> \therefore T=\frac{2v}{g}.</math>
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| The horizontal component of the ball's distance from the catcher at time <math>t</math> is:
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| :<math>d=h(T-t) = \frac{2hv}{g}-ht</math>
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| The tangent of the angle of elevation of the ball, as seen by the catcher, is:
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| :<math>\tan(e)=\frac{a}{d}</math>
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| :<math>=\frac{vt-\frac{gt^2}{2}}{\frac{2hv}{g}-ht}</math>
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| :<math>=\frac{2gvt-g^2t^2}{4hv-2ght}</math>
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| :<math>=\frac{gt(2v-gt)}{2h(2v-gt)}</math>
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| The quantity in the brackets will not be zero except when the ball is on the ground, therefore, while the ball is in flight:
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| :<math>\tan(e)=\left(\frac{g}{2h}\right)t</math>
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| The bracket in this last expression is constant for a given flight of the ball. Therefore the tangent of the angle of elevation of the ball, as seen by the player who is properly positioned to catch it, is directly proportional to the time since the ball was hit.
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| ==See also==
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| *[[Aft-crossing trajectory]]
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| *[[Orbit (dynamics)]]
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| *[[Orbit (group theory)]]
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| *[[Planetary orbit]]
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| *[[Porkchop plot]]
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| *[[Range of a projectile]]
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| *[[Rigid body]]
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| *[[Trajectory of a projectile]]
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| ==External links==
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| {{Wikibooks|High school physics|Projectile motion}}
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| * [http://www.physics-lab.net/applets/projectile-motion Projectile Motion Flash Applet]
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| * [http://hyperphysics.phy-astr.gsu.edu/hbase/traj.html Trajectory calculator]
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| * [http://www.phy.hk/wiki/englishhtm/ThrowABall.htm An interactive simulation on projectile motion]
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| * [http://publicliterature.org/tools/projectile_motion/ Projectile Motion Simulator, java applet]
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| * [http://www.thewritingpot.com/projectilelab/ Projectile Lab, JavaScript trajectory simulator]
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| * [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]].
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| * [http://scienceworld.wolfram.com/physics/Trajectory.html Trajectory], ScienceWorld.
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| *[http://www.geogebra.org/en/upload/files/nikenuke/projectile06d.html Java projectile-motion simulation, with first-order air resistance.]
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| *[http://www.geogebra.org/en/upload/files/nikenuke/projTARGET01.html Java projectile-motion simulation; targeting solutions, parabola of safety.]
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| {{Use dmy dates|date=September 2010}}
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| [[Category:Ballistics]]
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| [[Category:Mechanics]]
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