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| [[File:ParabolicWaterTrajectory.jpg|thumb|250px|Parabolic water trajectory]]
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| [[File:Ferde hajitas1.svg|thumb|250px|Initial velocity of parabolic throwing]]
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| [[File:Ferde hajitas2.svg|thumb|250px|Components of initial velocity of parabolic throwing]]
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| '''Projectile motion''' is a form of [[motion (physics)|motion]] in which an object or particle (called a [[projectile]]) is thrown obliquely near the earth's surface, and ''it moves along a curved path under the action of [[gravity]] only.''The path followed by a projectile motion is called its [[trajectory]]. Projectile motion only occurs when there is one force applied at the beginning of the trajectory, after which there is no force in operation apart from gravity.
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| == The initial velocity ==
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| If the projectile is launched with an initial velocity {{math|'''''v'''''<sub>0</sub>}}, then it can be written as
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| :<math> \mathbf{v}_0 = v_{0x}\mathbf{i} + v_{0y}\mathbf{j}</math>.
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| The components {{math|''v''<sub>0''x''</sub>}} and {{math|''v''<sub>0''y''</sub>}} can be found if the angle, {{math|''θ''}} is known:
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| :<math> v_{0x} = v_0\cos\theta</math>,
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| :<math> v_{0y} = v_0\sin\theta</math>.
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| If the projectile's range, launch angle, and drop height are known, launch velocity can be found using Newton's formula
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| :<math> V_0 = \sqrt{{R^2 g} \over {R \sin 2\theta + 2h \cos^2\theta}} </math>.
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| The launch angle is usually expressed by the symbol theta, but often the symbol alpha is used.
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| == Kinematic quantities of projectile motion .==
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| In projectile motion, the horizontal motion and the vertical motion are independent of each other; that is, neither motion affects the other.
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| === Acceleration ===
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| Since there is no acceleration in the horizontal direction, the velocity in the horizontal direction is constant, being equal to {{math|'''''v'''''<sub>0</sub> cos ''θ''}}. The vertical motion of the projectile is the motion of a particle during its free fall. Here the [[acceleration]] is constant, being equal to {{math|''g''}}.<ref>The {{math|''g''}} is the [[Standard gravity|acceleration due to gravity]]. ({{math|9.81 ''m''/''s''<sup>2</sup>}} near the surface of the Earth).</ref> The components of the acceleration are:
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| :<math> a_x = 0 </math>, | |
| :<math> a_y = -g </math>.
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| === Velocity ===
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| The horizontal component of the [[velocity]] of the object remains unchanged throughout the motion. The vertical component of the velocity increases linearly, because the acceleration due to gravity is constant. The accelerations in the x and y directions can be integrated to solve for the components of velocity at any time ''t'', as follows:
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| :<math> v_x=v_0 \cos(\theta) </math>,
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| :<math> v_y=v_0 \sin(\theta) - gt </math>.
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| The magnitude of the velocity (under the Pythagorean theorem):
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| :<math> v=\sqrt{v_x^2 + v_y^2 \ } </math>.
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| === Displacement ===
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| [[File:Ferde hajitas3.svg|thumb|250px|Displacement and coordinates of parabolic throwing]]
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| At any time ''t'', the projectile's horizontal and vertical [[displacement (vector)|displacement]]:
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| :<math> x = v_0 t \cos(\theta) </math>,
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| :<math> y = v_0 t \sin(\theta) - \frac{1}{2}gt^2 </math>.
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| The magnitude of the displacement:
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| :<math> \Delta r=\sqrt{x^2 + y^2 \ } </math>.
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| ==Parabolic trajectory==
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| {{Main|Trajectory of a projectile}}
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| Consider the equations,
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| :<math> x = v_0 t \cos(\theta) </math>,
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| :<math> y = v_0 t \sin(\theta) - \frac{1}{2}gt^2 </math>.
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| If ''t'' is eliminated between these two equations the following equation is obtained:
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| :<math>y=\tan(\theta) \cdot x-\frac{g}{2v^2_{0}\cos^2 \theta} \cdot x^2</math>,
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| This equation is the equation of a parabola.
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| Since ''g'', θ, and ''v''<sub>0</sub> are constants, the above equation is of the form
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| :<math>y=ax+bx^2</math>,
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| in which ''a'' and ''b'' are constants. This is the equation of a parabola, so the path is parabolic. The axis of the parabola is vertical.
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| == The maximum height of projectile ==
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| [[File:Ferde hajitas4.svg|thumb|250px|Maximum height of projectile]]
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| The highest height which the object will reach is known as the peak of the object's motion.
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| The increase of the height will last, until <math>v_y=0</math>, that is,
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| :<math> 0=v_0 \sin(\theta) - gt_h </math>.
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| Time to reach the maximum height:
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| :<math> t_h = {v_0 \sin(\theta) \over g} </math>.
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| From the vertical displacement of the maximum height of projectile:
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| :<math> h = v_0 t_h \sin(\theta) - \frac{1}{2} gt^2_h </math>
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| :<math> h = {v_0^2 \sin^2(\theta) \over {2g}} </math> .
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| {{clear}}
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| == Relation between horizontal range and maximum height ==
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| The relation between the range ({{math|''R''}}) on the horizontal plane and the maximum height ({{math|''h''}}) reached at {{math|{{sfrac|''t<sub>d</sub>''|2}}}} is:
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| :<math>h={R\tan\theta \over 4}</math>
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| == The maximum distance of projectile ==
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| {{main|Range of a projectile}}
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| [[File:Ferde hajitas5.svg|thumb|250px|The maximum distance of projectile]]It is important to note that the Range and the Maximum height of the Projectile does not depend upon mass of the trajected body. Hence Range and Max. Height are equal for all those bodies which are thrown by same velocity and direction. Air resistance does not affect displacement of projectile.
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| The horizontal range ''d'' of the projectile is the horizontal distance the projectile has travelled when it returns to its initial height (''y'' = 0).
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| :<math> 0 = v_0 t_d \sin(\theta) - \frac{1}{2}gt_d^2 </math>. | |
| Time to reach ground:
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| :<math> t_d = {2v_0 \sin(\theta) \over g} </math>.
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| From the horizontal displacement the maximum distance of projectile:
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| :<math> d = v_0 t_d \cos(\theta) </math>,
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| so<ref>2·sin(α)·cos(α) = sin(2α)</ref>
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| :<math> d = \frac{v_0^2}{g}\sin(2\theta) </math>.
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| Note that ''d'' has its maximum value when
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| :<math>\sin 2\theta=1</math>,
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| which necessarily corresponds to
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| :<math>2\theta=90^\circ</math>,
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| or
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| :<math>\theta=45^\circ</math>.
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| == Application of the work energy theorem ==
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| According to the [[Work (physics)#Work and kinetic energy|work-energy theorem]] the vertical component of velocity:
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| :<math> v_y^2 = (v_0 \sin \theta)^2-2gy </math>.
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| ==Projectile motion in art==
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| * The sixth panel of Hwaseonghaenghaengdo Byeongpun (화성행행도 병풍) describes King Shooting Arrows at Deukjung Pavilion, 1795-02-14. According to palace records, [[Lady Hyegyeong]], the King's mother, was so pleased to be presented with this 8-panels screen of such magnificent scale and stunning precision that she rewarded each of the seven artists who participated in its production. The artists were [[Choe Deuk-hyeon]], [[Kim Deuk-sin]], [[Yi Myeong-gyu]], [[Jang Han-jong]] (1768 - 1815), [[Yun Seok-keun]], [[Heo Sik]] (1762 - ?) and [[Yi In-mun]].
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| {{multiple image | align = center
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| |footer=화성행행도 병풍 <br> Hwaseonghaenghaengdo Byeongpun
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| | footer_align = center
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| | footer_background = #dddddd
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| |width1=200
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| |image1= Blue6-Haenghaeng-deugjungjeongeosa.jpg
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| |caption1 = <div align=center>Arrows <br>득중정어사</div>
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| }}
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| ==References==
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| * Budó Ágoston: ''Kísérleti fizika I.'',Budapest, Tankönyvkiadó, 1986. ISBN 963 17 8772 9 {{hu}}
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| * Ifj. Zátonyi Sándor: ''Fizika 9.'',Budapest, Nemzeti Tankönyvkiadó, 2009. ISBN 978-963-19-6082-2 {{hu}}
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| * Hack Frigyes: ''Négyjegyű függvénytáblázatok, összefüggések és adatok'', Budapest, Nemzeti Tankönyvkiadó, 2004. ISBN 963-19-3506-X {{hu}}
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| == Notes ==
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| {{Reflist}}
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| *Since the value of g is not specific the body with high velocity over g limit cannot be measured using the concept of the projectile motion
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| ==External links==
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| * [http://www.phy.ntnu.edu.tw/ntnujava/htmltag.php?code=users.sgeducation.lookang.Projectile02_pkg.Projectile02Applet.class&name=Projectile02&muid=14019 Open Source Physics computer model]
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| * [http://www.geogebra.org/en/upload/files/nikenuke/projectile06d.html A Java simulation of projectile motion, including first-order air resistance]
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| {{DEFAULTSORT:Projectile Motion}}
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| [[Category:Mechanics]]
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| [[ja:飛翔経路]]
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| [[zh:拋體運動]]
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