In mathematical physics, the equation of motion is an equation that describes the behavior of the physical system in terms of its movement as a function of time. More specifically, the motion equation describes the behavior of a physical system as a set of mathematical functions in terms of dynamic variables: usually spatial coordinates and time are used, but others are also possible, such as the momentum and time components. The most common choices are common coordinates that can be variables that match the characteristics of the physical system. Functions are defined in Euclidean space in classical mechanics, but are replaced by arc space in relativity. If the dynamics of a system are known, the equations are solutions to differential equations that describe the motion dynamics.
There are two main descriptions of motion: dynamics and kinematics. Dynamics are general, because the momentum, force and energy of the particles are taken into account. In this case, sometimes this term refers to a differential equation that satisfies the system (eg, Newton's second law or Euler-Lagrange equation), and sometimes for solutions to the equation.
However, kinematics is simpler because it involves only the variable that comes from the position of the object, and time. In a constant acceleration state, this simpler equation of motion is usually referred to as the SUVAT equation, which arises from the definition of kinematic quantities: displacement (span> s ), initial velocity ( < i> u ), end speed ( v ), acceleration ( a ), and time ( t ).
Equations of motion can therefore be grouped under the main grouping of this motion. In all cases, the main types of movement are translation, rotation, oscillation, or any combination of these.
A differential motion equation, usually identified as some physical law and applying the physical quantity definition, is used to set the equations for the problem. Solving differential equations will lead to a common solution with arbitrary constants, arbitrariness associated with the solution's family. Certain solutions can be obtained by setting an initial value, which improves the value of the constant.
To express this formally, in general the motion equation M is the function of the r position of the object, its speed (derivative first r , v = d r / dt ), and the acceleration (second instance r , a = d 2 r / dt 2 ), and the time t . Euclidean vectors in 3D are denoted in bold. This is equivalent to saying the motion equation in r is a regular two-order differential equation (ODE) in r ,
di mana t adalah waktu, dan setiap overdot menandakan satu kali turunan. Kondisi awal diberikan oleh nilai konstan pada t = 0 ,
The solution of r ( t ) on the motion equation, with the default value specified, describes the system for all time t after t = 0 . Other dynamic variables like the p momentum of the object, or the number coming from r and p like angular momentum, can be used instead of r as the solvable quantity of some equations of motion, although the position of the object at t is by far the most searched quantity.
Sometimes, the equation will be linear and more likely to be solved appropriately. In general, the equations will be non-linear, and can not be solved appropriately so various approaches should be used. Solutions for nonlinear equations can show chaotic behavior depending on how sensitive the system is to the initial state.
Video Equations of motion
Histori
Historically, equations of motion first appeared in classical mechanics to describe the movement of large objects, an important application is the celestial mechanics to predict planetary movements as if they orbited like clockwork (this is how Neptune predicted before its invention), and also investigated stability of the solar system.
It is important to observe that a large body of work involving the kinematics, dynamics and mathematical models of the universe developed in infant steps - falter, wake and self-correct - over three millennia and includes contributions from both known names and others that have since faded from historical history.
In ancient times, apart from the success of priests, astrologers and astronomers in predicting solar and lunar eclipses, the solstices and turning points of the Sun and the Moon, there is nothing but a collection of algorithms to help them. Although great strides were made in the development of geometries made by Ancient Greeks and surveys in Rome, we had to wait another thousand years before the first equation of the movement arrived.
European exposure to works collected by Muslims from Greek, Indian and Islamic scholars, such as Euclid's Elements , Archimedes' works, and Al-Khw's treatise? Rizm? Started in Spain, and scholars from all over Europe went to Spain, reading, copying, and translating learning into Latin. European exposure to Arabic numerals and their ease of calculations led the first scholars to study them and then traders and refresh the spread of knowledge across Europe.
In the thirteenth century, universities in Oxford and Paris had emerged, and scholars were now studying mathematics and philosophy with lesser worries about worldly tasks - they were not so obvious as they were in modern times. Of these, compendiums and editors, such as Johannes Campanus, of Euclid and Aristotle, deal with scholars with the notion of infinity and elemental ratio theory as a means of expressing the relationship between the various numbers involved with the moving body. These studies lead to a body of new knowledge now known as physics.
From these institutes, Merton College protects a group of scholars who devote themselves to the natural sciences, especially physics, astronomy and mathematics, similar to intellectuals at the University of Paris. Thomas Bradwardine, one of the scholars, expanded the number of Aristoteles such as distance and speed, and gave them intensity and expansion. Bradwardine suggests an exponential law involving strength, resistance, distance, speed and time. Nicholas Oresme further extends Bradwardine's argument. The Merton School proves that the quantity of gestures that undergo uniform acceleration movements equals the quantity of uniform movement at the speed achieved in the middle of the path through accelerated motion.
For writers on kinematics before Galileo, since small time intervals can not be measured, the affinity between time and motion is unclear. They use time as a function of distance, and free fall, greater speed as a result of greater improvement. Only Domingo de Soto, a Spanish theologian, in his commentary on Aristotle Physics published in 1545, after defining a "uniformly difform" movement (which is a uniform acceleration movement) - word velocity not used - as proportional with time, it is properly stated that this type of movement can be identified with a fallen body and projectile, without which it proves this proposition or suggests a formula that links time, velocity and distance. De Soto's comment is really surprising about the definition of acceleration (acceleration is the rate of change of motion (speed) in time) and the observation that during the acceleration movement the climbing acceleration will become negative.
Such discourse spread throughout Europe and certainly influenced Galileo and others, and helped in laying the foundations of kinematics. Galileo summed up the equation s = 1 / 2 gt 2 in his work geometrically, using the Merton rule, now known as the special case of one kinematic equation. He can not use familiar mathematical reasoning. The relationship between speed, distance, time and acceleration was not known at the time.
Galileo is the first to show that the projectile path is a parabola. Galileo has an understanding of centrifugal force and provides the exact momentum definition. This momentum emphasis as fundamental quantity in dynamics is very important. He measured momentum with speed and weight products; The mass is a concept developed by Huygens and Newton. In a simple pendulum swing, Galileo says in Discourse that "any momentum obtained in a derivative along the arc equals that causes the same body motion to rise through the same arc." His analysis of the projectiles showed that Galileo had understood the first law and the second law of motion. He does not generalize and make it applicable to bodies that are not subject to earth's gravity. The move was Newton's contribution.
The term "inertia" was used by Kepler who applied it to the body at rest. (The first law of motion is now often called the law of inertia.)
Galileo did not fully understand the third law of motion, the law of equations of action and reaction, although he corrected some of Aristotle's mistakes. With Stevin and others, Galileo also wrote about statics. He formulates the principle of the parallelogram of forces, but he does not fully recognize the scope.
Galileo was also attracted by the laws of the pendulum, his first observation being as a young man. In 1583, when he prayed at the cathedral in Pisa, his attention was held back by the movement of a large lamp that flashed and swung left, referring to his own pulse to save time. For him the period appeared the same, even after the movement was greatly reduced, discovering the pendulum isocronism.
Careful experiments were carried out by him later, and described in discourses, revealing varying oscillation periods with long square roots but not dependent on pendulum mass.
So we arrive at Renà © Descartes, Isaac Newton, Gottfried Leibniz, et al. and the evolving forms of equations of motion that are beginning to be recognized as modern.
Then the equations of motion also appear in electrodynamics, when describing the motion of charged particles in electric and magnetic fields, the Lorentz force is a general equation that functions as the definition of what is meant by electric fields and magnetic fields. With the advent of special relativity and general relativity, theoretical modifications to spacetime mean the equations of classical motion are also modified to account for the limited speed of light, and the curvature of spacetime. In all these cases the differential equation is in terms of a function that describes the particle trajectory in terms of space and time coordinates, since it is influenced by force or energy transformation.
However, the equations of quantum mechanics can also be regarded as "equations of motion", since they are the differential equations of wave functions, which illustrate how quantum states behave analogously using the coordinates of space and time of particles. There is an analog equation of motion in other physics, for the collection of physical phenomena that can be considered waves, fluids, or fields.
Maps Equations of motion
The kinematic equation for one particle
Quantity kinematic
Dari posisi seketika r = r ( t ) , arti seketika pada nilai waktu instan t , kecepatan sesaat v = v ( t ) dan percepatan a = a ( t ) memiliki definisi umum, koordinat-independen;
Note that the velocity always points toward the movement, in other words for the curved path it is a tangent vector. Loosely, first-order derivatives are tangent to the curve. Still for the curved path, the acceleration is directed to the center of the curvature of the road. Again, loosely, second-order derivatives are related to curvature.
Analog rotasi adalah "vektor sudut" (sudut partikel berputar sekitar beberapa sumbu) ? = ? ( t ) , kecepatan sudut ? = ? ( t ) , dan percepatan sudut ? = ? ( t ) :
where n? is the unit vector in the direction of the rotation axis, and ? is the angle of the object turns about its axis.
Hubungan berikut berlaku untuk partikel seperti titik, yang mengorbit tentang beberapa sumbu dengan kecepatan sudut ? :
where r is the position vector of the particle (radial of the rotation axis) and v the tangential velocity of the particle. For the continuous rigid body of the continuum, these relations apply to every point in a rigid body.
Uniform acceleration
The motion differential equation for a constant or uniform particle acceleration in a straight line is simple: the acceleration is constant, so the second derivative of the object position is constant. The results of this case are summarized below.
Constant translational acceleration in a straight line
Where:
- r 0 is the starting position of the particle
- r is the end position of the particle
- v 0 is the initial velocity of the particle
- v is the final velocity of the particle
- a is particle acceleration
- t is the time interval
Here a is constant acceleration , or in the case of objects moving under the influence of gravity, the standard gravity g is used. Note that each equation contains four of the five variables, so in this situation it is enough to know three of the five variables to calculate the two remaining.
where u has replaced v 0 , s replace r , and s 0 = 0 . They are often referred to as the SUVAT equation, where "SUVAT" is an acronym of the variable: s = displacement ( s 0 = initial displacement), u = initial velocity, v = final speed, a = acceleration, t = time.
Constant linear acceleration in all directions
Basic examples and often in kinematics involve projectiles, such as balls being thrown up into the air. With the initial velocity u , one can calculate how high the ball will move before it starts to fall. Acceleration is the local gravity acceleration g . At this point we must remember that while this number seems to be scalar, the direction of displacement, velocity and acceleration are important. They can actually be considered as a directional vector. Selecting s to measure from ground, acceleration a must be actually -g , because the force of gravity acts down and therefore also acceleration on the ball because of it.
Pada titik tertinggi, bola akan beristirahat: oleh karena itu v = 0 . Menggunakan persamaan [4] di set di atas, kami memiliki:
Mengganti dan membatalkan tanda minus memberi:
Percepatan melingkar konstan
where ? is a constant angular acceleration, ? is the angular velocity, ? 0 is the initial angular velocity, ? is the rotating angle (angle of movement), ? 0 is the starting angle, and t is the time it takes to rotate from the initial state to the last state.
General planar motion
This is a kinematic equation for particles that cross the path in the field, described by the position r = r ( t ) . They are only derivative of position vectors in plane polar coordinates using defin
Source of the article : Wikipedia
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