Velocity-addition formula (original) (raw)
معادلة جمع السرعات في الفيزياء النسبية، هي معادلة ثلاثية الأبعاد تربط سرعات الجسم في إطر مرجعية مختلفة. التطبيقات الأساسبة لمعادلات تركيب السرعات تتضمن انزياح دوبلر، رادار دوبلر، زيوغ الضوء وتباطؤ الضوء في مياه متحركة في تجربة فيزو عام 1851. يدل الرمز u على سرعة الجسم في إطار لورنزي S، وv سرعته في إطار ثانٍ S' و u' السرعة المتحولة للجسم في الإطار الثاني.
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| dbo:abstract | معادلة جمع السرعات في الفيزياء النسبية، هي معادلة ثلاثية الأبعاد تربط سرعات الجسم في إطر مرجعية مختلفة. التطبيقات الأساسبة لمعادلات تركيب السرعات تتضمن انزياح دوبلر، رادار دوبلر، زيوغ الضوء وتباطؤ الضوء في مياه متحركة في تجربة فيزو عام 1851. يدل الرمز u على سرعة الجسم في إطار لورنزي S، وv سرعته في إطار ثانٍ S' و u' السرعة المتحولة للجسم في الإطار الثاني. (ar) Skládáním rychlostí se ve fyzice zpravidla označuje důsledek speciální teorie relativity, přesněji Lorentzovy transformace. Pohybují-li se dva objekty vůči vztažné soustavě S rovnoběžnými rychlostmi , , pak ve vztažné soustavě S' spojené s prvním z nich se bude druhý pohybovat rychlostí Když jsou obě rychlosti malé ve srovnání s rychlostí světla ve vakuu , je jmenovatel zlomku téměř roven jedné, takže rychlosti lze skládat prostým odčítáním (resp. sčítáním, když se tělesa pohybují opačnými směry). Při malých rychlostech tedy dobře funguje klasická fyzika, při velkých rychlostech se začnou projevovat relativistické efekty. Výsledek skládání rychlostí menších než bude podle relativistického vztahu také vždy menší než . Rychlost světla ve vakuu představuje horní mez rychlosti, jakou se mohou tělesa pohybovat. Pro obecné směry rychlostí platí kde (což je Lorentzův faktor). Zajímavé je, že existuje fyzikální veličina podobná rychlosti, která také popisuje míru pohybu, ale není shora omezená a umožňuje skládání obyčejným sčítáním. Nazývá se rapidita. (cs) Das Relativistische Additionstheorem für Geschwindigkeiten besagt, wie die Geschwindigkeit eines Objekts in einem bestimmten Bezugssystem zu bestimmen ist, wenn sich das Objekt mit einer Geschwindigkeit gegenüber einem zweiten Bezugssystem bewegt, das sich selbst gegenüber dem ersten mit einer Geschwindigkeit bewegt. Das Theorem kann aus der Lorentztransformation für gegeneinander bewegte Inertialsysteme hergeleitet werden. In der klassischen Mechanik werden Geschwindigkeiten vektoriell addiert und haben daher keine obere Schranke. Da aber nach der speziellen Relativitätstheorie die Geschwindigkeit eines Objekts die Lichtgeschwindigkeit nicht überschreiten kann, können die klassischen Gleichungen nur eine Näherung sein. Unterschiede machen sich bemerkbar, wenn eine oder beide der zu addierenden Geschwindigkeiten nicht mehr vernachlässigbar klein gegenüber der Lichtgeschwindigkeit sind. Das Relativistische Additionstheorem für Geschwindigkeiten ist durch Messungen bestätigt worden. (de) In fisica, la composizione delle velocità è un insieme di equazioni che descrivono il legame tra le velocità di un oggetto in due sistemi di riferimento diversi, l'uno in moto rettilineo uniforme rispetto all'altro. Nella teoria della relatività ristretta esse tengono conto, in particolare, dell'insuperabilità della velocità della luce e della sua costanza indipendentemente dal sistema di riferimento inerziale scelto. (it) In relativistic physics, a velocity-addition formula is a three-dimensional equation that relates the velocities of objects in different reference frames. Such formulas apply to successive Lorentz transformations, so they also relate different frames. Accompanying velocity addition is a kinematic effect known as Thomas precession, whereby successive non-collinear Lorentz boosts become equivalent to the composition of a rotation of the coordinate system and a boost. Standard applications of velocity-addition formulas include the Doppler shift, Doppler navigation, the aberration of light, and the dragging of light in moving water observed in the 1851 Fizeau experiment. The notation employs u as velocity of a body within a Lorentz frame S, and v as velocity of a second frame S′, as measured in S, and u′ as the transformed velocity of the body within the second frame. (en) 速度の加法則（そくどのかほうそく、英：velocity-addition formula）は、人 A が 人 B からみて速度 v で走っており、人 B が人 C からみて速度 w で走っている際、人 A が人 C からみてどのような速度をもって走っているようにみえるかを与える公式である。 特殊相対性理論での速度の合成は、古典力学的な場合の常識的な速度の合成からずれる。以下、簡単のため、A, B, C は共通の一直線上を動いているものとする。 (ja) Relativistische snelheid is een begrip dat samenhangt met de speciale relativiteitstheorie en waarmee wordt aangegeven hoe de snelheid van een beweging wordt waargenomen door waarnemers in verschillende inertiaalstelsels. Dit wordt berekend met behulp van de lorentztransformatie. Het begrip relativistische snelheid is nodig om tegenstrijdigheden tussen elektromagnetisme en klassieke mechanica uit de wereld te helpen. (nl) Під час розгляду складного руху (коли точка або тіло рухається в одній системі відліку, а ця система відліку в свою чергу рухається відносно іншої системи) виникає питання про зв'язок швидкостей у двох системах відліку. (uk) 物理學中，速度加成式是將不同參考系下個別描述同一移動物體速度的關聯方程式。 (zh) При рассмотрении сложного движения (когда точка или тело движется в одной системе отсчёта, а эта система отсчёта в свою очередь движется относительно другой системы) возникает вопрос о связи скоростей в двух системах отсчёта. (ru) |
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| dbp:equation | and in the forwards direction (en) |
| dbp:indent | : (en) |
| dbp:proof | Reverse formula found by using standard procedure of swapping for and for . (en) Since a relativistic transformation rotates space and time into each other much as geometric rotations in the plane rotate the - and -axes, it is convenient to use the same units for space and time, otherwise a unit conversion factor appears throughout relativistic formulae, being the speed of light. In a system where lengths and times are measured in the same units, the speed of light is dimensionless and equal to . A velocity is then expressed as fraction of the speed of light. To find the relativistic transformation law, it is useful to introduce the four-velocities , which is the motion of the ship away from the shore, as measured from the shore, and which is the motion of the fly away from the ship, as measured from the ship. The four-velocity is defined to be a four-vector with relativistic length equal to , future-directed and tangent to the world line of the object in spacetime. Here, corresponds to the time component and to the component of the ship's velocity as seen from the shore. It is convenient to take the -axis to be the direction of motion of the ship away from the shore, and the -axis so that the plane is the plane spanned by the motion of the ship and the fly. This results in several components of the velocities being zero: The ordinary velocity is the ratio of the rate at which the space coordinates are increasing to the rate at which the time coordinate is increasing: Since the relativistic length of is , so The Lorentz transformation matrix that converts velocities measured in the ship frame to the shore frame is the inverse of the transformation described on the Lorentz transformation page, so the minus signs that appear there must be inverted here: This matrix rotates the pure time-axis vector to , and all its columns are relativistically orthogonal to one another, so it defines a Lorentz transformation. If a fly is moving with four-velocity in the ship frame, and it is boosted by multiplying by the matrix above, the new four-velocity in the shore frame is , Dividing by the time component and substituting for the components of the four-vectors and in terms of the components of the three-vectors and gives the relativistic composition law as The form of the relativistic composition law can be understood as an effect of the failure of simultaneity at a distance. For the parallel component, the time dilation decreases the speed, the length contraction increases it, and the two effects cancel out. The failure of simultaneity means that the fly is changing slices of simultaneity as the projection of onto . Since this effect is entirely due to the time slicing, the same factor multiplies the perpendicular component, but for the perpendicular component there is no length contraction, so the time dilation multiplies by a factor of . (en) |
| dbp:title | Trigonometry (en) Decomposition into parallel and perpendicular components in terms of (en) Detailed proof (en) Details for u (en) Details in derivation (en) The algebra (en) Transformation of velocity (en) A proof using -vectors and Lorentz transformation matrices (en) |
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| dct:subject | dbc:Equations dbc:Velocity dbc:Special_relativity dbc:Kinematics |
| gold:hypernym | dbr:Equation |
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| rdfs:comment | معادلة جمع السرعات في الفيزياء النسبية، هي معادلة ثلاثية الأبعاد تربط سرعات الجسم في إطر مرجعية مختلفة. التطبيقات الأساسبة لمعادلات تركيب السرعات تتضمن انزياح دوبلر، رادار دوبلر، زيوغ الضوء وتباطؤ الضوء في مياه متحركة في تجربة فيزو عام 1851. يدل الرمز u على سرعة الجسم في إطار لورنزي S، وv سرعته في إطار ثانٍ S' و u' السرعة المتحولة للجسم في الإطار الثاني. (ar) In fisica, la composizione delle velocità è un insieme di equazioni che descrivono il legame tra le velocità di un oggetto in due sistemi di riferimento diversi, l'uno in moto rettilineo uniforme rispetto all'altro. Nella teoria della relatività ristretta esse tengono conto, in particolare, dell'insuperabilità della velocità della luce e della sua costanza indipendentemente dal sistema di riferimento inerziale scelto. (it) 速度の加法則（そくどのかほうそく、英：velocity-addition formula）は、人 A が 人 B からみて速度 v で走っており、人 B が人 C からみて速度 w で走っている際、人 A が人 C からみてどのような速度をもって走っているようにみえるかを与える公式である。 特殊相対性理論での速度の合成は、古典力学的な場合の常識的な速度の合成からずれる。以下、簡単のため、A, B, C は共通の一直線上を動いているものとする。 (ja) Relativistische snelheid is een begrip dat samenhangt met de speciale relativiteitstheorie en waarmee wordt aangegeven hoe de snelheid van een beweging wordt waargenomen door waarnemers in verschillende inertiaalstelsels. Dit wordt berekend met behulp van de lorentztransformatie. Het begrip relativistische snelheid is nodig om tegenstrijdigheden tussen elektromagnetisme en klassieke mechanica uit de wereld te helpen. (nl) Під час розгляду складного руху (коли точка або тіло рухається в одній системі відліку, а ця система відліку в свою чергу рухається відносно іншої системи) виникає питання про зв'язок швидкостей у двох системах відліку. (uk) 物理學中，速度加成式是將不同參考系下個別描述同一移動物體速度的關聯方程式。 (zh) При рассмотрении сложного движения (когда точка или тело движется в одной системе отсчёта, а эта система отсчёта в свою очередь движется относительно другой системы) возникает вопрос о связи скоростей в двух системах отсчёта. (ru) Skládáním rychlostí se ve fyzice zpravidla označuje důsledek speciální teorie relativity, přesněji Lorentzovy transformace. Pohybují-li se dva objekty vůči vztažné soustavě S rovnoběžnými rychlostmi , , pak ve vztažné soustavě S' spojené s prvním z nich se bude druhý pohybovat rychlostí Pro obecné směry rychlostí platí kde (což je Lorentzův faktor). Zajímavé je, že existuje fyzikální veličina podobná rychlosti, která také popisuje míru pohybu, ale není shora omezená a umožňuje skládání obyčejným sčítáním. Nazývá se rapidita. (cs) Das Relativistische Additionstheorem für Geschwindigkeiten besagt, wie die Geschwindigkeit eines Objekts in einem bestimmten Bezugssystem zu bestimmen ist, wenn sich das Objekt mit einer Geschwindigkeit gegenüber einem zweiten Bezugssystem bewegt, das sich selbst gegenüber dem ersten mit einer Geschwindigkeit bewegt. Das Theorem kann aus der Lorentztransformation für gegeneinander bewegte Inertialsysteme hergeleitet werden. Das Relativistische Additionstheorem für Geschwindigkeiten ist durch Messungen bestätigt worden. (de) In relativistic physics, a velocity-addition formula is a three-dimensional equation that relates the velocities of objects in different reference frames. Such formulas apply to successive Lorentz transformations, so they also relate different frames. Accompanying velocity addition is a kinematic effect known as Thomas precession, whereby successive non-collinear Lorentz boosts become equivalent to the composition of a rotation of the coordinate system and a boost. (en) |
| rdfs:label | معادلة جمع السرعات (ar) Skládání rychlostí (cs) Relativistisches Additionstheorem für Geschwindigkeiten (de) Composizione delle velocità (it) 速度の加法則 (ja) Snelheidstransformatie (nl) Сложение скоростей (ru) Velocity-addition formula (en) Додавання швидкостей (uk) 速度加成式 (zh) |
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