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125-422: Kinematics is used in astrophysics to describe the motion of celestial bodies and collections of such bodies. In mechanical engineering , robotics , and biomechanics , kinematics is used to describe the motion of systems composed of joined parts (multi-link systems) such as an engine , a robotic arm or the human skeleton . Geometric transformations, also called rigid transformations , are used to describe

250-416: A B x , a B y , a B z ) {\displaystyle \mathbf {a} _{B}=\left(a_{B_{x}},a_{B_{y}},a_{B_{z}}\right)} then the acceleration of point C relative to point B is the difference between their components: a C / B = a C − a B = (

375-460: A C x − a B x , a C y − a B y , a C z − a B z ) {\displaystyle \mathbf {a} _{C/B}=\mathbf {a} _{C}-\mathbf {a} _{B}=\left(a_{C_{x}}-a_{B_{x}},a_{C_{y}}-a_{B_{y}},a_{C_{z}}-a_{B_{z}}\right)} Alternatively, this same result could be obtained by computing

500-443: A | = | v ˙ | = d v d t . {\displaystyle |\mathbf {a} |=|{\dot {\mathbf {v} }}|={\frac {{\text{d}}v}{{\text{d}}t}}.} A relative position vector is a vector that defines the position of one point relative to another. It is the difference in position of the two points. The position of one point A relative to another point B

625-541: A τ ) d τ = r 0 + v 0 t + 1 2 a t 2 . {\displaystyle \mathbf {r} (t)=\mathbf {r} _{0}+\int _{0}^{t}\mathbf {v} (\tau )\,{\text{d}}\tau =\mathbf {r} _{0}+\int _{0}^{t}\left(\mathbf {v} _{0}+\mathbf {a} \tau \right){\text{d}}\tau =\mathbf {r} _{0}+\mathbf {v} _{0}t+{\tfrac {1}{2}}\mathbf {a} t^{2}.} Additional relations between displacement, velocity, acceleration, and time can be derived. Since

750-436: A r = − v θ , a θ = a , {\displaystyle a_{r}=-v\theta ,\quad a_{\theta }=a,} are called, respectively, the radial and tangential components of acceleration. Astrophysics Astrophysics is a science that employs the methods and principles of physics and chemistry in the study of astronomical objects and phenomena. As one of

875-439: A x x ^ + a y y ^ + a z z ^ . {\displaystyle \mathbf {a} =\lim _{(\Delta t)^{2}\to 0}{\frac {\Delta \mathbf {r} }{(\Delta t)^{2}}}={\frac {{\text{d}}^{2}\mathbf {r} }{{\text{d}}t^{2}}}=a_{x}{\hat {\mathbf {x} }}+a_{y}{\hat {\mathbf {y} }}+a_{z}{\hat {\mathbf {z} }}.} Thus, acceleration

1000-616: A y y ^ + a z z ^ . {\displaystyle \mathbf {a} =\lim _{\Delta t\to 0}{\frac {\Delta \mathbf {v} }{\Delta t}}={\frac {{\text{d}}\mathbf {v} }{{\text{d}}t}}=a_{x}{\hat {\mathbf {x} }}+a_{y}{\hat {\mathbf {y} }}+a_{z}{\hat {\mathbf {z} }}.} Alternatively, a = lim ( Δ t ) 2 → 0 Δ r ( Δ t ) 2 = d 2 r d t 2 =

1125-638: A ¯ x x ^ + a ¯ y y ^ + a ¯ z z ^ {\displaystyle \mathbf {\bar {a}} ={\frac {\Delta \mathbf {\bar {v}} }{\Delta t}}={\frac {\Delta {\bar {v}}_{x}}{\Delta t}}{\hat {\mathbf {x} }}+{\frac {\Delta {\bar {v}}_{y}}{\Delta t}}{\hat {\mathbf {y} }}+{\frac {\Delta {\bar {v}}_{z}}{\Delta t}}{\hat {\mathbf {z} }}={\bar {a}}_{x}{\hat {\mathbf {x} }}+{\bar {a}}_{y}{\hat {\mathbf {y} }}+{\bar {a}}_{z}{\hat {\mathbf {z} }}\,} where Δ v

1250-564: A + v ω ) θ ^ + a z z ^ . {\displaystyle \mathbf {a} _{P}={\frac {\text{d}}{{\text{d}}t}}\left(v{\hat {\mathbf {r} }}+v{\hat {\mathbf {\theta } }}+v_{z}{\hat {\mathbf {z} }}\right)=(a-v\theta ){\hat {\mathbf {r} }}+(a+v\omega ){\hat {\mathbf {\theta } }}+a_{z}{\hat {\mathbf {z} }}.} The term − v θ r ^ {\displaystyle -v\theta {\hat {\mathbf {r} }}} acts toward

1375-486: A t t = 1 2 a t 2 = a t 2 2 {\textstyle A={\frac {1}{2}}BH={\frac {1}{2}}att={\frac {1}{2}}at^{2}={\frac {at^{2}}{2}}} . Adding v 0 t {\displaystyle v_{0}t} and a t 2 2 {\textstyle {\frac {at^{2}}{2}}} results in the equation Δ r {\displaystyle \Delta r} results in

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1500-464: A coordinate frame called the Cartesian frame . Similarly, the position of any point in three-dimensional space can be specified by three Cartesian coordinates , which are the signed distances from the point to three mutually perpendicular planes. More generally, n Cartesian coordinates specify the point in an n -dimensional Euclidean space for any dimension n . These coordinates are

1625-497: A plane is a coordinate system that specifies each point uniquely by a pair of real numbers called coordinates , which are the signed distances to the point from two fixed perpendicular oriented lines , called coordinate lines , coordinate axes or just axes (plural of axis ) of the system. The point where the axes meet is called the origin and has (0, 0) as coordinates. The axes directions represent an orthogonal basis . The combination of origin and basis forms

1750-521: A , b ) to the Cartesian coordinates of every point in the set. That is, if the original coordinates of a point are ( x , y ) , after the translation they will be ( x ′ , y ′ ) = ( x + a , y + b ) . {\displaystyle (x',y')=(x+a,y+b).} To rotate a figure counterclockwise around the origin by some angle θ {\displaystyle \theta }

1875-628: A convenient form. Recall that the trajectory of a particle P is defined by its coordinate vector r measured in a fixed reference frame F . As the particle moves, its coordinate vector r ( t ) traces its trajectory, which is a curve in space, given by: r ( t ) = x ( t ) x ^ + y ( t ) y ^ + z ( t ) z ^ , {\displaystyle \mathbf {r} (t)=x(t){\hat {\mathbf {x} }}+y(t){\hat {\mathbf {y} }}+z(t){\hat {\mathbf {z} }},} where x̂ , ŷ , and ẑ are

2000-449: A diagram ( 3D projection or 2D perspective drawing ) shows the x - and y -axis horizontally and vertically, respectively, then the z -axis should be shown pointing "out of the page" towards the viewer or camera. In such a 2D diagram of a 3D coordinate system, the z -axis would appear as a line or ray pointing down and to the left or down and to the right, depending on the presumed viewer or camera perspective . In any diagram or display,

2125-407: A division of space into eight regions or octants , according to the signs of the coordinates of the points. The convention used for naming a specific octant is to list its signs; for example, (+ + +) or (− + −) . The generalization of the quadrant and octant to an arbitrary number of dimensions is the orthant , and a similar naming system applies. The Euclidean distance between two points of

2250-625: A group of ten associate editors from Europe and the United States, established The Astrophysical Journal: An International Review of Spectroscopy and Astronomical Physics . It was intended that the journal would fill the gap between journals in astronomy and physics, providing a venue for publication of articles on astronomical applications of the spectroscope; on laboratory research closely allied to astronomical physics, including wavelength determinations of metallic and gaseous spectra and experiments on radiation and absorption; on theories of

2375-508: A guide to understanding of other stars. The topic of how stars change, or stellar evolution, is often modeled by placing the varieties of star types in their respective positions on the Hertzsprung–Russell diagram , which can be viewed as representing the state of a stellar object, from birth to destruction. Theoretical astrophysicists use a wide variety of tools which include analytical models (for example, polytropes to approximate

2500-536: A mechanism for a desired range of motion. In addition, kinematics applies algebraic geometry to the study of the mechanical advantage of a mechanical system or mechanism. The term kinematic is the English version of A.M. Ampère 's cinématique , which he constructed from the Greek κίνημα kinema ("movement, motion"), itself derived from κινεῖν kinein ("to move"). Kinematic and cinématique are related to

2625-463: A millisecond timescale ( millisecond pulsars ) or combine years of data ( pulsar deceleration studies). The information obtained from these different timescales is very different. The study of the Sun has a special place in observational astrophysics. Due to the tremendous distance of all other stars, the Sun can be observed in a kind of detail unparalleled by any other star. Understanding the Sun serves as

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2750-561: A model or help in choosing between several alternate or conflicting models. Theorists also try to generate or modify models to take into account new data. In the case of an inconsistency, the general tendency is to try to make minimal modifications to the model to fit the data. In some cases, a large amount of inconsistent data over time may lead to total abandonment of a model. Topics studied by theoretical astrophysicists include stellar dynamics and evolution; galaxy formation and evolution; magnetohydrodynamics; large-scale structure of matter in

2875-433: A number line. For any point P , a line is drawn through P perpendicular to each axis, and the position where it meets the axis is interpreted as a number. The two numbers, in that chosen order, are the Cartesian coordinates of P . The reverse construction allows one to determine the point P given its coordinates. The first and second coordinates are called the abscissa and the ordinate of P , respectively; and

3000-954: A particle trajectory on a circular cylinder occurs when there is no movement along the z axis: r ( t ) = r r ^ + z z ^ , {\displaystyle \mathbf {r} (t)=r{\hat {\mathbf {r} }}+z{\hat {\mathbf {z} }},} where r and z 0 are constants. In this case, the velocity v P is given by: v P = d d t ( r r ^ + z z ^ ) = r ω θ ^ = v θ ^ , {\displaystyle \mathbf {v} _{P}={\frac {\text{d}}{{\text{d}}t}}\left(r{\hat {\mathbf {r} }}+z{\hat {\mathbf {z} }}\right)=r\omega {\hat {\mathbf {\theta } }}=v{\hat {\mathbf {\theta } }},} where ω {\displaystyle \omega }

3125-632: A particle's velocity is the time rate of change of its position. Furthermore, this velocity is tangent to the particle's trajectory at every position along its path. In a non-rotating frame of reference, the derivatives of the coordinate directions are not considered as their directions and magnitudes are constants. The speed of an object is the magnitude of its velocity. It is a scalar quantity: v = | v | = d s d t , {\displaystyle v=|\mathbf {v} |={\frac {{\text{d}}s}{{\text{d}}t}},} where s {\displaystyle s}

3250-1238: A point P can be taken as the distance from P to the plane defined by the other two axes, with the sign determined by the orientation of the corresponding axis. Each pair of axes defines a coordinate plane . These planes divide space into eight octants . The octants are: ( + x , + y , + z ) ( − x , + y , + z ) ( + x , − y , + z ) ( + x , + y , − z ) ( + x , − y , − z ) ( − x , + y , − z ) ( − x , − y , + z ) ( − x , − y , − z ) {\displaystyle {\begin{aligned}(+x,+y,+z)&&(-x,+y,+z)&&(+x,-y,+z)&&(+x,+y,-z)\\(+x,-y,-z)&&(-x,+y,-z)&&(-x,-y,+z)&&(-x,-y,-z)\end{aligned}}} The coordinates are usually written as three numbers (or algebraic formulas) surrounded by parentheses and separated by commas, as in (3, −2.5, 1) or ( t , u + v , π /2) . Thus,

3375-456: A point are ( x , y ) , then its distances from the X -axis and from the Y -axis are | y | and | x |, respectively; where | · | denotes the absolute value of a number. A Cartesian coordinate system for a three-dimensional space consists of an ordered triplet of lines (the axes ) that go through a common point (the origin ), and are pair-wise perpendicular; an orientation for each axis; and

3500-401: A point are usually written in parentheses and separated by commas, as in (10, 5) or (3, 5, 7) . The origin is often labelled with the capital letter O . In analytic geometry, unknown or generic coordinates are often denoted by the letters ( x , y ) in the plane, and ( x , y , z ) in three-dimensional space. This custom comes from a convention of algebra, which uses letters near the end of

3625-819: A reference frame. The position vector of a particle is a vector drawn from the origin of the reference frame to the particle. It expresses both the distance of the point from the origin and its direction from the origin. In three dimensions, the position vector r {\displaystyle {\bf {r}}} can be expressed as r = ( x , y , z ) = x x ^ + y y ^ + z z ^ , {\displaystyle \mathbf {r} =(x,y,z)=x{\hat {\mathbf {x} }}+y{\hat {\mathbf {y} }}+z{\hat {\mathbf {z} }},} where x {\displaystyle x} , y {\displaystyle y} , and z {\displaystyle z} are

3750-472: A single unit of length for all three axes. As in the two-dimensional case, each axis becomes a number line. For any point P of space, one considers a plane through P perpendicular to each coordinate axis, and interprets the point where that plane cuts the axis as a number. The Cartesian coordinates of P are those three numbers, in the chosen order. The reverse construction determines the point P given its three coordinates. Alternatively, each coordinate of

3875-542: A substantial amount of work in the realms of theoretical and observational physics. Some areas of study for astrophysicists include their attempts to determine the properties of dark matter , dark energy , black holes , and other celestial bodies ; and the origin and ultimate fate of the universe . Topics also studied by theoretical astrophysicists include Solar System formation and evolution ; stellar dynamics and evolution ; galaxy formation and evolution ; magnetohydrodynamics ; large-scale structure of matter in

Kinematics - Misplaced Pages Continue

4000-551: A time interval is defined as the ratio. a ¯ = Δ v ¯ Δ t = Δ v ¯ x Δ t x ^ + Δ v ¯ y Δ t y ^ + Δ v ¯ z Δ t z ^ =

4125-635: Is mainly concerned with finding out the measurable implications of physical models . It is the practice of observing celestial objects by using telescopes and other astronomical apparatus. Most astrophysical observations are made using the electromagnetic spectrum . Other than electromagnetic radiation, few things may be observed from the Earth that originate from great distances. A few gravitational wave observatories have been constructed, but gravitational waves are extremely difficult to detect. Neutrino observatories have also been built, primarily to study

4250-436: Is measured along the z -axis, then the coordinate vector to the top of the tower is r = (0 m, −50 m, 50 m). In the most general case, a three-dimensional coordinate system is used to define the position of a particle. However, if the particle is constrained to move within a plane, a two-dimensional coordinate system is sufficient. All observations in physics are incomplete without being described with respect to

4375-514: Is obtained by projecting the point onto one axis along a direction that is parallel to the other axis (or, in general, to the hyperplane defined by all the other axes). In such an oblique coordinate system the computations of distances and angles must be modified from that in standard Cartesian systems, and many standard formulas (such as the Pythagorean formula for the distance) do not hold (see affine plane ). The Cartesian coordinates of

4500-602: Is simply the difference between their positions which is the difference between the components of their position vectors. If point A has position components r A = ( x A , y A , z A ) {\displaystyle \mathbf {r} _{A}=\left(x_{A},y_{A},z_{A}\right)} and point B has position components r B = ( x B , y B , z B ) {\displaystyle \mathbf {r} _{B}=\left(x_{B},y_{B},z_{B}\right)} then

4625-882: Is simply the difference between their velocities v A / B = v A − v B {\displaystyle \mathbf {v} _{A/B}=\mathbf {v} _{A}-\mathbf {v} _{B}} which is the difference between the components of their velocities. If point A has velocity components v A = ( v A x , v A y , v A z ) {\displaystyle \mathbf {v} _{A}=\left(v_{A_{x}},v_{A_{y}},v_{A_{z}}\right)} and point B has velocity components v B = ( v B x , v B y , v B z ) {\displaystyle \mathbf {v} _{B}=\left(v_{B_{x}},v_{B_{y}},v_{B_{z}}\right)} then

4750-593: Is the angular velocity of the unit vector θ around the z axis of the cylinder. The acceleration a P of the particle P is now given by: a P = d ( v θ ^ ) d t = a θ ^ − v θ r ^ . {\displaystyle \mathbf {a} _{P}={\frac {{\text{d}}(v{\hat {\mathbf {\theta } }})}{{\text{d}}t}}=a{\hat {\mathbf {\theta } }}-v\theta {\hat {\mathbf {r} }}.} The components

4875-879: Is the Cartesian version of Pythagoras's theorem . In three-dimensional space, the distance between points ( x 1 , y 1 , z 1 ) {\displaystyle (x_{1},y_{1},z_{1})} and ( x 2 , y 2 , z 2 ) {\displaystyle (x_{2},y_{2},z_{2})} is d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 + ( z 2 − z 1 ) 2 , {\displaystyle d={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}},} which can be obtained by two consecutive applications of Pythagoras' theorem. The Euclidean transformations or Euclidean motions are

5000-417: Is the arc-length measured along the trajectory of the particle. This arc-length must always increase as the particle moves. Hence, d s / d t {\displaystyle {\text{d}}s/{\text{d}}t} is non-negative, which implies that speed is also non-negative. The velocity vector can change in magnitude and in direction or both at once. Hence, the acceleration accounts for both

5125-410: Is the area under a velocity–time graph. We can take Δ r {\displaystyle \Delta r} by adding the top area and the bottom area. The bottom area is a rectangle, and the area of a rectangle is the A ⋅ B {\displaystyle A\cdot B} where A {\displaystyle A} is the width and B {\displaystyle B}

Kinematics - Misplaced Pages Continue

5250-441: Is the average velocity and Δ t is the time interval. The acceleration of the particle is the limit of the average acceleration as the time interval approaches zero, which is the time derivative, a = lim Δ t → 0 Δ v Δ t = d v d t = a x x ^ +

5375-948: Is the displacement vector during the time interval Δ t {\displaystyle \Delta t} . In the limit that the time interval Δ t {\displaystyle \Delta t} approaches zero, the average velocity approaches the instantaneous velocity, defined as the time derivative of the position vector, v = lim Δ t → 0 Δ r Δ t = d r d t = v x x ^ + v y y ^ + v z z ^ . {\displaystyle \mathbf {v} =\lim _{\Delta t\to 0}{\frac {\Delta \mathbf {r} }{\Delta t}}={\frac {{\text{d}}\mathbf {r} }{{\text{d}}t}}=v_{x}{\hat {\mathbf {x} }}+v_{y}{\hat {\mathbf {y} }}+v_{z}{\hat {\mathbf {z} }}.} Thus,

5500-406: Is the first derivative of the velocity vector and the second derivative of the position vector of that particle. In a non-rotating frame of reference, the derivatives of the coordinate directions are not considered as their directions and magnitudes are constants. The magnitude of the acceleration of an object is the magnitude | a | of its acceleration vector. It is a scalar quantity: |

5625-423: Is the height. In this case A = t {\displaystyle A=t} and B = v 0 {\displaystyle B=v_{0}} (the A {\displaystyle A} here is different from the acceleration a {\displaystyle a} ). This means that the bottom area is t v 0 {\displaystyle tv_{0}} . Now let's find

5750-454: Is the set of all real numbers. In the same way, the points in any Euclidean space of dimension n be identified with the tuples (lists) of n real numbers; that is, with the Cartesian product R n {\displaystyle \mathbb {R} ^{n}} . The concept of Cartesian coordinates generalizes to allow axes that are not perpendicular to each other, and/or different units along each axis. In that case, each coordinate

5875-501: Is usually named after the coordinate which is measured along it; so one says the x-axis , the y-axis , the t-axis , etc. Another common convention for coordinate naming is to use subscripts, as ( x 1 , x 2 , ..., x n ) for the n coordinates in an n -dimensional space, especially when n is greater than 3 or unspecified. Some authors prefer the numbering ( x 0 , x 1 , ..., x n −1 ). These notations are especially advantageous in computer programming : by storing

6000-546: The Cartesian coordinates and x ^ {\displaystyle {\hat {\mathbf {x} }}} , y ^ {\displaystyle {\hat {\mathbf {y} }}} and z ^ {\displaystyle {\hat {\mathbf {z} }}} are the unit vectors along the x {\displaystyle x} , y {\displaystyle y} , and z {\displaystyle z} coordinate axes, respectively. The magnitude of

6125-516: The Lambda-CDM model , are the Big Bang , cosmic inflation , dark matter, dark energy and fundamental theories of physics. The roots of astrophysics can be found in the seventeenth century emergence of a unified physics, in which the same laws applied to the celestial and terrestrial realms. There were scientists who were qualified in both physics and astronomy who laid the firm foundation for

6250-619: The cosmic microwave background . Emissions from these objects are examined across all parts of the electromagnetic spectrum , and the properties examined include luminosity , density , temperature , and chemical composition. Because astrophysics is a very broad subject, astrophysicists apply concepts and methods from many disciplines of physics, including classical mechanics , electromagnetism , statistical mechanics , thermodynamics , quantum mechanics , relativity , nuclear and particle physics , and atomic and molecular physics . In practice, modern astronomical research often involves

6375-473: The dot product , which is appropriate as the products are scalars rather than vectors. 2 ( r − r 0 ) ⋅ a = | v | 2 − | v 0 | 2 . {\displaystyle 2\left(\mathbf {r} -\mathbf {r} _{0}\right)\cdot \mathbf {a} =|\mathbf {v} |^{2}-|\mathbf {v} _{0}|^{2}.} The dot product can be replaced by

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6500-842: The unit vectors along the x , y and z axes of the reference frame F , respectively. Consider a particle P that moves only on the surface of a circular cylinder r ( t ) = constant, it is possible to align the z axis of the fixed frame F with the axis of the cylinder. Then, the angle θ around this axis in the x – y plane can be used to define the trajectory as, r ( t ) = r cos ⁡ ( θ ( t ) ) x ^ + r sin ⁡ ( θ ( t ) ) y ^ + z ( t ) z ^ , {\displaystyle \mathbf {r} (t)=r\cos(\theta (t)){\hat {\mathbf {x} }}+r\sin(\theta (t)){\hat {\mathbf {y} }}+z(t){\hat {\mathbf {z} }},} where

6625-405: The xy -plane, yz -plane, and xz -plane. In mathematics, physics, and engineering contexts, the first two axes are often defined or depicted as horizontal, with the third axis pointing up. In that case the third coordinate may be called height or altitude . The orientation is usually chosen so that the 90-degree angle from the first axis to the second axis looks counter-clockwise when seen from

6750-416: The z -coordinate is sometimes called the applicate . The words abscissa , ordinate and applicate are sometimes used to refer to coordinate axes rather than the coordinate values. The axes of a two-dimensional Cartesian system divide the plane into four infinite regions, called quadrants , each bounded by two half-axes. These are often numbered from 1st to 4th and denoted by Roman numerals : I (where

6875-457: The ( bijective ) mappings of points of the Euclidean plane to themselves which preserve distances between points. There are four types of these mappings (also called isometries): translations , rotations , reflections and glide reflections . Translating a set of points of the plane, preserving the distances and directions between them, is equivalent to adding a fixed pair of numbers (

7000-408: The 17th century, natural philosophers such as Galileo , Descartes , and Newton began to maintain that the celestial and terrestrial regions were made of similar kinds of material and were subject to the same natural laws . Their challenge was that the tools had not yet been invented with which to prove these assertions. For much of the nineteenth century, astronomical research was focused on

7125-406: The Cartesian system, commonly learn the order to read the values before cementing the x -, y -, and z -axis concepts, by starting with 2D mnemonics (for example, 'Walk along the hall then up the stairs' akin to straight across the x -axis then up vertically along the y -axis). Computer graphics and image processing , however, often use a coordinate system with the y -axis oriented downwards on

7250-517: The French word cinéma, but neither are directly derived from it. However, they do share a root word in common, as cinéma came from the shortened form of cinématographe, "motion picture projector and camera", once again from the Greek word for movement and from the Greek γρᾰ́φω grapho ("to write"). Particle kinematics is the study of the trajectory of particles. The position of a particle is defined as

7375-477: The Sun, Moon, planets, comets, meteors, and nebulae; and on instrumentation for telescopes and laboratories. Around 1920, following the discovery of the Hertzsprung–Russell diagram still used as the basis for classifying stars and their evolution, Arthur Eddington anticipated the discovery and mechanism of nuclear fusion processes in stars , in his paper The Internal Constitution of the Stars . At that time,

7500-430: The Sun. Cosmic rays consisting of very high-energy particles can be observed hitting the Earth's atmosphere. Observations can also vary in their time scale. Most optical observations take minutes to hours, so phenomena that change faster than this cannot readily be observed. However, historical data on some objects is available, spanning centuries or millennia . On the other hand, radio observations may look at events on

7625-424: The acceleration a P , which is the time derivative of the velocity v P , is given by: a P = d d t ( v r ^ + v θ ^ + v z z ^ ) = ( a − v θ ) r ^ + (

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7750-729: The acceleration is constant, a = Δ v Δ t = v − v 0 t {\displaystyle \mathbf {a} ={\frac {\Delta \mathbf {v} }{\Delta t}}={\frac {\mathbf {v} -\mathbf {v} _{0}}{t}}} can be substituted into the above equation to give: r ( t ) = r 0 + ( v + v 0 2 ) t . {\displaystyle \mathbf {r} (t)=\mathbf {r} _{0}+\left({\frac {\mathbf {v} +\mathbf {v} _{0}}{2}}\right)t.} A relationship between velocity, position and acceleration without explicit time dependence can be had by solving

7875-406: The alphabet for unknown values (such as the coordinates of points in many geometric problems), and letters near the beginning for given quantities. These conventional names are often used in other domains, such as physics and engineering, although other letters may be used. For example, in a graph showing how a pressure varies with time , the graph coordinates may be denoted p and t . Each axis

8000-739: The average acceleration for time and substituting and simplifying t = v − v 0 a {\displaystyle t={\frac {\mathbf {v} -\mathbf {v} _{0}}{\mathbf {a} }}} ( r − r 0 ) ⋅ a = ( v − v 0 ) ⋅ v + v 0 2   , {\displaystyle \left(\mathbf {r} -\mathbf {r} _{0}\right)\cdot \mathbf {a} =\left(\mathbf {v} -\mathbf {v} _{0}\right)\cdot {\frac {\mathbf {v} +\mathbf {v} _{0}}{2}}\ ,} where ⋅ {\displaystyle \cdot } denotes

8125-494: The behaviors of a star) and computational numerical simulations . Each has some advantages. Analytical models of a process are generally better for giving insight into the heart of what is going on. Numerical models can reveal the existence of phenomena and effects that would otherwise not be seen. Theorists in astrophysics endeavor to create theoretical models and figure out the observational consequences of those models. This helps allow observers to look for data that can refute

8250-635: The case of acceleration always in the direction of the motion and the direction of motion should be in positive or negative, the angle between the vectors ( α ) is 0, so cos ⁡ 0 = 1 {\displaystyle \cos 0=1} , and | v | 2 = | v 0 | 2 + 2 | a | | r − r 0 | . {\displaystyle |\mathbf {v} |^{2}=|\mathbf {v} _{0}|^{2}+2\left|\mathbf {a} \right|\left|\mathbf {r} -\mathbf {r} _{0}\right|.} This can be simplified using

8375-460: The center of curvature of the path at that point on the path, is commonly called the centripetal acceleration . The term v ω θ ^ {\displaystyle v\omega {\hat {\mathbf {\theta } }}} is called the Coriolis acceleration . If the trajectory of the particle is constrained to lie on a cylinder, then the radius r is constant and

8500-412: The components of their accelerations. If point C has acceleration components a C = ( a C x , a C y , a C z ) {\displaystyle \mathbf {a} _{C}=\left(a_{C_{x}},a_{C_{y}},a_{C_{z}}\right)} and point B has acceleration components a B = (

8625-407: The computer display. This convention developed in the 1960s (or earlier) from the way that images were originally stored in display buffers . For three-dimensional systems, a convention is to portray the xy -plane horizontally, with the z -axis added to represent height (positive up). Furthermore, there is a convention to orient the x -axis toward the viewer, biased either to the right or left. If

8750-2770: The constant distance from the center is denoted as r , and θ ( t ) is a function of time. The cylindrical coordinates for r ( t ) can be simplified by introducing the radial and tangential unit vectors, r ^ = cos ⁡ ( θ ( t ) ) x ^ + sin ⁡ ( θ ( t ) ) y ^ , θ ^ = − sin ⁡ ( θ ( t ) ) x ^ + cos ⁡ ( θ ( t ) ) y ^ . {\displaystyle {\hat {\mathbf {r} }}=\cos(\theta (t)){\hat {\mathbf {x} }}+\sin(\theta (t)){\hat {\mathbf {y} }},\quad {\hat {\mathbf {\theta } }}=-\sin(\theta (t)){\hat {\mathbf {x} }}+\cos(\theta (t)){\hat {\mathbf {y} }}.} and their time derivatives from elementary calculus: d r ^ d t = ω θ ^ . {\displaystyle {\frac {{\text{d}}{\hat {\mathbf {r} }}}{{\text{d}}t}}=\omega {\hat {\mathbf {\theta } }}.} d 2 r ^ d t 2 = d ( ω θ ^ ) d t = α θ ^ − ω r ^ . {\displaystyle {\frac {{\text{d}}^{2}{\hat {\mathbf {r} }}}{{\text{d}}t^{2}}}={\frac {{\text{d}}(\omega {\hat {\mathbf {\theta } }})}{{\text{d}}t}}=\alpha {\hat {\mathbf {\theta } }}-\omega {\hat {\mathbf {r} }}.} d θ ^ d t = − θ r ^ . {\displaystyle {\frac {{\text{d}}{\hat {\mathbf {\theta } }}}{{\text{d}}t}}=-\theta {\hat {\mathbf {r} }}.} d 2 θ ^ d t 2 = d ( − θ r ^ ) d t = − α r ^ − ω 2 θ ^ . {\displaystyle {\frac {{\text{d}}^{2}{\hat {\mathbf {\theta } }}}{{\text{d}}t^{2}}}={\frac {{\text{d}}(-\theta {\hat {\mathbf {r} }})}{{\text{d}}t}}=-\alpha {\hat {\mathbf {r} }}-\omega ^{2}{\hat {\mathbf {\theta } }}.} Using this notation, r ( t ) takes

8875-589: The constant tangential acceleration is applied along that path, so v 2 = v 0 2 + 2 a Δ r . {\displaystyle v^{2}=v_{0}^{2}+2a\Delta r.} This reduces the parametric equations of motion of the particle to a Cartesian relationship of speed versus position. This relation is useful when time is unknown. We also know that Δ r = ∫ v d t {\textstyle \Delta r=\int v\,{\text{d}}t} or Δ r {\displaystyle \Delta r}

9000-432: The coordinate vector from the origin of a coordinate frame to the particle. For example, consider a tower 50 m south from your home, where the coordinate frame is centered at your home, such that east is in the direction of the x -axis and north is in the direction of the y -axis, then the coordinate vector to the base of the tower is r = (0 m, −50 m, 0 m). If the tower is 50 m high, and this height

9125-413: The coordinates both have positive signs), II (where the abscissa is negative − and the ordinate is positive +), III (where both the abscissa and the ordinate are −), and IV (abscissa +, ordinate −). When the axes are drawn according to the mathematical custom, the numbering goes counter-clockwise starting from the upper right ("north-east") quadrant. Similarly, a three-dimensional Cartesian system defines

9250-455: The coordinates of a point as an array , instead of a record , the subscript can serve to index the coordinates. In mathematical illustrations of two-dimensional Cartesian systems, the first coordinate (traditionally called the abscissa ) is measured along a horizontal axis, oriented from left to right. The second coordinate (the ordinate ) is then measured along a vertical axis, usually oriented from bottom to top. Young children learning

9375-441: The coordinates of points of the shape. For example, a circle of radius 2, centered at the origin of the plane, may be described as the set of all points whose coordinates x and y satisfy the equation x + y = 4 ; the area , the perimeter and the tangent line at any point can be computed from this equation by using integrals and derivatives , in a way that can be applied to any curve. Cartesian coordinates are

9500-574: The cosine of the angle α between the vectors (see Geometric interpretation of the dot product for more details) and the vectors by their magnitudes, in which case: 2 | r − r 0 | | a | cos ⁡ α = | v | 2 − | v 0 | 2 . {\displaystyle 2\left|\mathbf {r} -\mathbf {r} _{0}\right|\left|\mathbf {a} \right|\cos \alpha =|\mathbf {v} |^{2}-|\mathbf {v} _{0}|^{2}.} In

9625-510: The current science of astrophysics. In modern times, students continue to be drawn to astrophysics due to its popularization by the Royal Astronomical Society and notable educators such as prominent professors Lawrence Krauss , Subrahmanyan Chandrasekhar , Stephen Hawking , Hubert Reeves , Carl Sagan and Patrick Moore . The efforts of the early, late, and present scientists continue to attract young people to study

9750-518: The discovery. The French cleric Nicole Oresme used constructions similar to Cartesian coordinates well before the time of Descartes and Fermat. Both Descartes and Fermat used a single axis in their treatments and have a variable length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes' La Géométrie was translated into Latin in 1649 by Frans van Schooten and his students. These commentators introduced several concepts while trying to clarify

9875-443: The end of the 20th century, studies of astronomical spectra had expanded to cover wavelengths extending from radio waves through optical, x-ray, and gamma wavelengths. In the 21st century, it further expanded to include observations based on gravitational waves . Observational astronomy is a division of the astronomical science that is concerned with recording and interpreting data, in contrast with theoretical astrophysics , which

10000-448: The equation Δ r = v 0 t + a t 2 2 {\textstyle \Delta r=v_{0}t+{\frac {at^{2}}{2}}} . This equation is applicable when the final velocity v is unknown. It is often convenient to formulate the trajectory of a particle r ( t ) = ( x ( t ), y ( t ), z ( t )) using polar coordinates in the X – Y plane. In this case, its velocity and acceleration take

10125-449: The first axis is usually defined or depicted as horizontal and oriented to the right, and the second axis is vertical and oriented upwards. (However, in some computer graphics contexts, the ordinate axis may be oriented downwards.) The origin is often labeled O , and the two coordinates are often denoted by the letters X and Y , or x and y . The axes may then be referred to as the X -axis and Y -axis. The choices of letters come from

10250-780: The form, r ( t ) = r r ^ + z ( t ) z ^ . {\displaystyle \mathbf {r} (t)=r{\hat {\mathbf {r} }}+z(t){\hat {\mathbf {z} }}.} In general, the trajectory r ( t ) is not constrained to lie on a circular cylinder, so the radius R varies with time and the trajectory of the particle in cylindrical-polar coordinates becomes: r ( t ) = r ( t ) r ^ + z ( t ) z ^ . {\displaystyle \mathbf {r} (t)=r(t){\hat {\mathbf {r} }}+z(t){\hat {\mathbf {z} }}.} Where r , θ , and z might be continuously differentiable functions of time and

10375-487: The foundation of analytic geometry , and provide enlightening geometric interpretations for many other branches of mathematics, such as linear algebra , complex analysis , differential geometry , multivariate calculus , group theory and more. A familiar example is the concept of the graph of a function . Cartesian coordinates are also essential tools for most applied disciplines that deal with geometry, including astronomy , physics , engineering and many more. They are

10500-451: The founders of the discipline, James Keeler , said, astrophysics "seeks to ascertain the nature of the heavenly bodies, rather than their positions or motions in space– what they are, rather than where they are", which is studied in celestial mechanics . Among the subjects studied are the Sun ( solar physics ), other stars , galaxies , extrasolar planets , the interstellar medium and

10625-972: The function notation is dropped for simplicity. The velocity vector v P is the time derivative of the trajectory r ( t ), which yields: v P = d d t ( r r ^ + z z ^ ) = v r ^ + r ω θ ^ + v z z ^ = v ( r ^ + θ ^ ) + v z z ^ . {\displaystyle \mathbf {v} _{P}={\frac {\text{d}}{{\text{d}}t}}\left(r{\hat {\mathbf {r} }}+z{\hat {\mathbf {z} }}\right)=v{\hat {\mathbf {r} }}+r\mathbf {\omega } {\hat {\mathbf {\theta } }}+v_{z}{\hat {\mathbf {z} }}=v({\hat {\mathbf {r} }}+{\hat {\mathbf {\theta } }})+v_{z}{\hat {\mathbf {z} }}.} Similarly,

10750-427: The history and science of astrophysics. The television sitcom show The Big Bang Theory popularized the field of astrophysics with the general public, and featured some well known scientists like Stephen Hawking and Neil deGrasse Tyson . Cartesian coordinates In geometry , a Cartesian coordinate system ( UK : / k ɑːr ˈ t iː zj ə n / , US : / k ɑːr ˈ t iː ʒ ə n / ) in

10875-479: The ideas contained in Descartes's work. The development of the Cartesian coordinate system would play a fundamental role in the development of the calculus by Isaac Newton and Gottfried Wilhelm Leibniz . The two-coordinate description of the plane was later generalized into the concept of vector spaces . Many other coordinate systems have been developed since Descartes, such as the polar coordinates for

11000-401: The length unit, and center at the origin), the unit square (whose diagonal has endpoints at (0, 0) and (1, 1) ), the unit hyperbola , and so on. The two axes divide the plane into four right angles , called quadrants . The quadrants may be named or numbered in various ways, but the quadrant where all coordinates are positive is usually called the first quadrant . If the coordinates of

11125-431: The line and assigning them to two distinct real numbers (most commonly zero and one). Other points can then be uniquely assigned to numbers by linear interpolation . Equivalently, one point can be assigned to a specific real number, for instance an origin point corresponding to zero, and an oriented length along the line can be chosen as a unit, with the orientation indicating the correspondence between directions along

11250-465: The line and positive or negative numbers. Each point corresponds to its signed distance from the origin (a number with an absolute value equal to the distance and a + or − sign chosen based on direction). A geometric transformation of the line can be represented by a function of a real variable , for example translation of the line corresponds to addition, and scaling the line corresponds to multiplication. Any two Cartesian coordinate systems on

11375-414: The line can be related to each-other by a linear function (function of the form x ↦ a x + b {\displaystyle x\mapsto ax+b} ) taking a specific point's coordinate in one system to its coordinate in the other system. Choosing a coordinate system for each of two different lines establishes an affine map from one line to the other taking each point on one line to

11500-502: The most common coordinate system used in computer graphics , computer-aided geometric design and other geometry-related data processing . The adjective Cartesian refers to the French mathematician and philosopher René Descartes , who published this idea in 1637 while he was resident in the Netherlands . It was independently discovered by Pierre de Fermat , who also worked in three dimensions, although Fermat did not publish

11625-433: The movement of components in a mechanical system , simplifying the derivation of the equations of motion. They are also central to dynamic analysis . Kinematic analysis is the process of measuring the kinematic quantities used to describe motion. In engineering, for instance, kinematic analysis may be used to find the range of movement for a given mechanism and, working in reverse, using kinematic synthesis to design

11750-437: The notation for the magnitudes of the vectors | a | = a , | v | = v , | r − r 0 | = Δ r {\displaystyle |\mathbf {a} |=a,|\mathbf {v} |=v,|\mathbf {r} -\mathbf {r} _{0}|=\Delta r} where Δ r {\displaystyle \Delta r} can be any curvaceous path taken as

11875-412: The orientation of the three axes, as a whole, is arbitrary. However, the orientation of the axes relative to each other should always comply with the right-hand rule , unless specifically stated otherwise. All laws of physics and math assume this right-handedness , which ensures consistency. For 3D diagrams, the names "abscissa" and "ordinate" are rarely used for x and y , respectively. When they are,

12000-429: The origin has coordinates (0, 0, 0) , and the unit points on the three axes are (1, 0, 0) , (0, 1, 0) , and (0, 0, 1) . Standard names for the coordinates in the three axes are abscissa , ordinate and applicate . The coordinates are often denoted by the letters x , y , and z . The axes may then be referred to as the x -axis, y -axis, and z -axis, respectively. Then the coordinate planes can be referred to as

12125-407: The original convention, which is to use the latter part of the alphabet to indicate unknown values. The first part of the alphabet was used to designate known values. A Euclidean plane with a chosen Cartesian coordinate system is called a Cartesian plane . In a Cartesian plane, one can define canonical representatives of certain geometric figures, such as the unit circle (with radius equal to

12250-410: The particle's position as a function of time. The velocity of a particle is a vector quantity that describes the direction as well as the magnitude of motion of the particle. More mathematically, the rate of change of the position vector of a point with respect to time is the velocity of the point. Consider the ratio formed by dividing the difference of two positions of a particle ( displacement ) by

12375-455: The physicist, Gustav Kirchhoff , and the chemist, Robert Bunsen , had demonstrated that the dark lines in the solar spectrum corresponded to bright lines in the spectra of known gases, specific lines corresponding to unique chemical elements . Kirchhoff deduced that the dark lines in the solar spectrum are caused by absorption by chemical elements in the Solar atmosphere. In this way it

12500-520: The plane with Cartesian coordinates ( x 1 , y 1 ) {\displaystyle (x_{1},y_{1})} and ( x 2 , y 2 ) {\displaystyle (x_{2},y_{2})} is d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 . {\displaystyle d={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}}.} This

12625-446: The plane, and the spherical and cylindrical coordinates for three-dimensional space. An affine line with a chosen Cartesian coordinate system is called a number line . Every point on the line has a real-number coordinate, and every real number represents some point on the line. There are two degrees of freedom in the choice of Cartesian coordinate system for a line, which can be specified by choosing two distinct points along

12750-468: The point (0, 0, 1) ; a convention that is commonly called the right-hand rule . Since Cartesian coordinates are unique and non-ambiguous, the points of a Cartesian plane can be identified with pairs of real numbers ; that is, with the Cartesian product R 2 = R × R {\displaystyle \mathbb {R} ^{2}=\mathbb {R} \times \mathbb {R} } , where R {\displaystyle \mathbb {R} }

12875-423: The point on the other line with the same coordinate. A Cartesian coordinate system in two dimensions (also called a rectangular coordinate system or an orthogonal coordinate system ) is defined by an ordered pair of perpendicular lines (axes), a single unit of length for both axes, and an orientation for each axis. The point where the axes meet is taken as the origin for both, thus turning each axis into

13000-401: The point where the axes meet is called the origin of the coordinate system. The coordinates are usually written as two numbers in parentheses, in that order, separated by a comma, as in (3, −10.5) . Thus the origin has coordinates (0, 0) , and the points on the positive half-axes, one unit away from the origin, have coordinates (1, 0) and (0, 1) . In mathematics, physics, and engineering,

13125-558: The position of point A relative to point B is the difference between their components: r A / B = r A − r B = ( x A − x B , y A − y B , z A − z B ) {\displaystyle \mathbf {r} _{A/B}=\mathbf {r} _{A}-\mathbf {r} _{B}=\left(x_{A}-x_{B},y_{A}-y_{B},z_{A}-z_{B}\right)} The velocity of one point relative to another

13250-439: The position vector | r | {\displaystyle \left|\mathbf {r} \right|} gives the distance between the point r {\displaystyle \mathbf {r} } and the origin. | r | = x 2 + y 2 + z 2 . {\displaystyle |\mathbf {r} |={\sqrt {x^{2}+y^{2}+z^{2}}}.} The direction cosines of

13375-993: The position vector provide a quantitative measure of direction. In general, an object's position vector will depend on the frame of reference; different frames will lead to different values for the position vector. The trajectory of a particle is a vector function of time, r ( t ) {\displaystyle \mathbf {r} (t)} , which defines the curve traced by the moving particle, given by r ( t ) = x ( t ) x ^ + y ( t ) y ^ + z ( t ) z ^ , {\displaystyle \mathbf {r} (t)=x(t){\hat {\mathbf {x} }}+y(t){\hat {\mathbf {y} }}+z(t){\hat {\mathbf {z} }},} where x ( t ) {\displaystyle x(t)} , y ( t ) {\displaystyle y(t)} , and z ( t ) {\displaystyle z(t)} describe each coordinate of

13500-399: The rate of change of the magnitude of the velocity vector and the rate of change of direction of that vector. The same reasoning used with respect to the position of a particle to define velocity, can be applied to the velocity to define acceleration. The acceleration of a particle is the vector defined by the rate of change of the velocity vector. The average acceleration of a particle over

13625-405: The routine work of measuring the positions and computing the motions of astronomical objects. A new astronomy, soon to be called astrophysics, began to emerge when William Hyde Wollaston and Joseph von Fraunhofer independently discovered that, when decomposing the light from the Sun, a multitude of dark lines (regions where there was less or no light) were observed in the spectrum . By 1860

13750-402: The second time derivative of the relative position vector r B/A . Assuming that the initial conditions of the position, r 0 {\displaystyle \mathbf {r} _{0}} , and velocity v 0 {\displaystyle \mathbf {v} _{0}} at time t = 0 {\displaystyle t=0} are known, the first integration yields

13875-421: The signed distances from the point to n mutually perpendicular fixed hyperplanes . Cartesian coordinates are named for René Descartes , whose invention of them in the 17th century revolutionized mathematics by allowing the expression of problems of geometry in terms of algebra and calculus . Using the Cartesian coordinate system, geometric shapes (such as curves ) can be described by equations involving

14000-487: The solar spectrum with any known elements. He thus claimed the line represented a new element, which was called helium , after the Greek Helios , the Sun personified. In 1885, Edward C. Pickering undertook an ambitious program of stellar spectral classification at Harvard College Observatory , in which a team of woman computers , notably Williamina Fleming , Antonia Maury , and Annie Jump Cannon , classified

14125-627: The source of stellar energy was a complete mystery; Eddington correctly speculated that the source was fusion of hydrogen into helium, liberating enormous energy according to Einstein's equation E = mc . This was a particularly remarkable development since at that time fusion and thermonuclear energy, and even that stars are largely composed of hydrogen (see metallicity ), had not yet been discovered. In 1925 Cecilia Helena Payne (later Cecilia Payne-Gaposchkin ) wrote an influential doctoral dissertation at Radcliffe College , in which she applied Saha's ionization theory to stellar atmospheres to relate

14250-516: The spectra recorded on photographic plates. By 1890, a catalog of over 10,000 stars had been prepared that grouped them into thirteen spectral types. Following Pickering's vision, by 1924 Cannon expanded the catalog to nine volumes and over a quarter of a million stars, developing the Harvard Classification Scheme which was accepted for worldwide use in 1922. In 1895, George Ellery Hale and James E. Keeler , along with

14375-408: The spectral classes to the temperature of stars. Most significantly, she discovered that hydrogen and helium were the principal components of stars, not the composition of Earth. Despite Eddington's suggestion, discovery was so unexpected that her dissertation readers (including Russell ) convinced her to modify the conclusion before publication. However, later research confirmed her discovery. By

14500-404: The time derivative of the relative position vector r B/A . The acceleration of one point C relative to another point B is simply the difference between their accelerations. a C / B = a C − a B {\displaystyle \mathbf {a} _{C/B}=\mathbf {a} _{C}-\mathbf {a} _{B}} which is the difference between

14625-1152: The time interval. This ratio is called the average velocity over that time interval and is defined as v ¯ = Δ r Δ t = Δ x Δ t x ^ + Δ y Δ t y ^ + Δ z Δ t z ^ = v ¯ x x ^ + v ¯ y y ^ + v ¯ z z ^ {\displaystyle \mathbf {\bar {v}} ={\frac {\Delta \mathbf {r} }{\Delta t}}={\frac {\Delta x}{\Delta t}}{\hat {\mathbf {x} }}+{\frac {\Delta y}{\Delta t}}{\hat {\mathbf {y} }}+{\frac {\Delta z}{\Delta t}}{\hat {\mathbf {z} }}={\bar {v}}_{x}{\hat {\mathbf {x} }}+{\bar {v}}_{y}{\hat {\mathbf {y} }}+{\bar {v}}_{z}{\hat {\mathbf {z} }}\,} where Δ r {\displaystyle \Delta \mathbf {r} }

14750-470: The top area (a triangle). The area of a triangle is 1 2 B H {\textstyle {\frac {1}{2}}BH} where B {\displaystyle B} is the base and H {\displaystyle H} is the height. In this case, B = t {\displaystyle B=t} and H = a t {\displaystyle H=at} or A = 1 2 B H = 1 2

14875-499: The universe; origin of cosmic rays ; general relativity , special relativity , quantum and physical cosmology (the physical study of the largest-scale structures of the universe), including string cosmology and astroparticle physics . Astronomy is an ancient science, long separated from the study of terrestrial physics. In the Aristotelian worldview, bodies in the sky appeared to be unchanging spheres whose only motion

15000-489: The universe; origin of cosmic rays; general relativity and physical cosmology, including string cosmology and astroparticle physics. Relativistic astrophysics serves as a tool to gauge the properties of large-scale structures for which gravitation plays a significant role in physical phenomena investigated and as the basis for black hole ( astro )physics and the study of gravitational waves . Some widely accepted and studied theories and models in astrophysics, now included in

15125-803: The velocity and acceleration vectors simplify. The velocity of v P is the time derivative of the trajectory r ( t ), v P = d d t ( r r ^ + z z ^ ) = r ω θ ^ + v z z ^ = v θ ^ + v z z ^ . {\displaystyle \mathbf {v} _{P}={\frac {\text{d}}{{\text{d}}t}}\left(r{\hat {\mathbf {r} }}+z{\hat {\mathbf {z} }}\right)=r\omega {\hat {\mathbf {\theta } }}+v_{z}{\hat {\mathbf {z} }}=v{\hat {\mathbf {\theta } }}+v_{z}{\hat {\mathbf {z} }}.} A special case of

15250-676: The velocity of point A relative to point B is the difference between their components: v A / B = v A − v B = ( v A x − v B x , v A y − v B y , v A z − v B z ) {\displaystyle \mathbf {v} _{A/B}=\mathbf {v} _{A}-\mathbf {v} _{B}=\left(v_{A_{x}}-v_{B_{x}},v_{A_{y}}-v_{B_{y}},v_{A_{z}}-v_{B_{z}}\right)} Alternatively, this same result could be obtained by computing

15375-682: The velocity of the particle as a function of time. v ( t ) = v 0 + ∫ 0 t a d τ = v 0 + a t . {\displaystyle \mathbf {v} (t)=\mathbf {v} _{0}+\int _{0}^{t}\mathbf {a} \,{\text{d}}\tau =\mathbf {v} _{0}+\mathbf {a} t.} A second integration yields its path (trajectory), r ( t ) = r 0 + ∫ 0 t v ( τ ) d τ = r 0 + ∫ 0 t ( v 0 +

15500-406: Was proved that the chemical elements found in the Sun and stars were also found on Earth. Among those who extended the study of solar and stellar spectra was Norman Lockyer , who in 1868 detected radiant, as well as dark lines in solar spectra. Working with chemist Edward Frankland to investigate the spectra of elements at various temperatures and pressures, he could not associate a yellow line in

15625-448: Was uniform motion in a circle, while the earthly world was the realm which underwent growth and decay and in which natural motion was in a straight line and ended when the moving object reached its goal . Consequently, it was held that the celestial region was made of a fundamentally different kind of matter from that found in the terrestrial sphere; either Fire as maintained by Plato , or Aether as maintained by Aristotle . During

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