Time
Time is the indefinite continued progress of existence and events that occur in irreversible succession through the past, in the present, the future. Time is a component quantity of various measurements used to sequence events, to compare the duration of events or the intervals between them, to quantify rates of change of quantities in material reality or in the conscious experience. Time is referred to as a fourth dimension, along with three spatial dimensions. Time has long been an important subject of study in religion and science, but defining it in a manner applicable to all fields without circularity has eluded scholars. Diverse fields such as business, sports, the sciences, the performing arts all incorporate some notion of time into their respective measuring systems. Time in physics is unambiguously operationally defined as "what a clock reads". See Units of Time. Time is one of the seven fundamental physical quantities in both the International System of Units and International System of Quantities.
Time is used to define other quantities – such as velocity – so defining time in terms of such quantities would result in circularity of definition. An operational definition of time, wherein one says that observing a certain number of repetitions of one or another standard cyclical event constitutes one standard unit such as the second, is useful in the conduct of both advanced experiments and everyday affairs of life; the operational definition leaves aside the question whether there is something called time, apart from the counting activity just mentioned, that flows and that can be measured. Investigations of a single continuum called spacetime bring questions about space into questions about time, questions that have their roots in the works of early students of natural philosophy. Temporal measurement has occupied scientists and technologists, was a prime motivation in navigation and astronomy. Periodic events and periodic motion have long served as standards for units of time. Examples include the apparent motion of the sun across the sky, the phases of the moon, the swing of a pendulum, the beat of a heart.
The international unit of time, the second, is defined by measuring the electronic transition frequency of caesium atoms. Time is of significant social importance, having economic value as well as personal value, due to an awareness of the limited time in each day and in human life spans. Speaking, methods of temporal measurement, or chronometry, take two distinct forms: the calendar, a mathematical tool for organising intervals of time, the clock, a physical mechanism that counts the passage of time. In day-to-day life, the clock is consulted for periods less than a day whereas the calendar is consulted for periods longer than a day. Personal electronic devices display both calendars and clocks simultaneously; the number that marks the occurrence of a specified event as to hour or date is obtained by counting from a fiducial epoch – a central reference point. Artifacts from the Paleolithic suggest that the moon was used to reckon time as early as 6,000 years ago. Lunar calendars were among the first to appear, with years of either 13 lunar months.
Without intercalation to add days or months to some years, seasons drift in a calendar based on twelve lunar months. Lunisolar calendars have a thirteenth month added to some years to make up for the difference between a full year and a year of just twelve lunar months; the numbers twelve and thirteen came to feature prominently in many cultures, at least due to this relationship of months to years. Other early forms of calendars originated in Mesoamerica in ancient Mayan civilization; these calendars were religiously and astronomically based, with 18 months in a year and 20 days in a month, plus five epagomenal days at the end of the year. The reforms of Julius Caesar in 45 BC put the Roman world on a solar calendar; this Julian calendar was faulty in that its intercalation still allowed the astronomical solstices and equinoxes to advance against it by about 11 minutes per year. Pope Gregory XIII introduced a correction in 1582. During the French Revolution, a new clock and calendar were invented in attempt to de-Christianize time and create a more rational system in order to replace the Gregorian calendar.
The French Republican Calendar's days consisted of ten hours of a hundred minutes of a hundred seconds, which marked a deviation from the 12-based duodecimal system used in many other devices by many cultures. The system was abolished in 1806. A large variety of devices have been invented to measure time; the study of these devices is called horology. An Egyptian device that dates to c. 1500 BC, similar in shape to a bent T-square, measured the passage of time from the shadow cast by its crossbar on a nonlinear rule. The T was oriented eastward in the mornings. At noon, the device was turned around so. A sundial uses a gnomon to cast a shadow on a set of markings calibrated to the hour; the position of the shadow marks the hour in local time. The idea to separate the day into smaller parts is credited to Egyptians because of their sundials, which operated on a duodecimal system; the importance of the number 12 is due to the number of lunar cycles in a year and the number of stars used to count the passage of night.
The most precise timekeeping device of the ancient
Lagrangian mechanics
Lagrangian mechanics is a reformulation of classical mechanics, introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in 1788. In Lagrangian mechanics, the trajectory of a system of particles is derived by solving the Lagrange equations in one of two forms, either the Lagrange equations of the first kind, which treat constraints explicitly as extra equations using Lagrange multipliers. In each case, a mathematical function called the Lagrangian is a function of the generalized coordinates, their time derivatives, time, contains the information about the dynamics of the system. No new physics are introduced in applying Lagrangian mechanics compared to Newtonian mechanics, it is, more mathematically sophisticated and systematic. Newton's laws can include non-conservative forces like friction. Lagrangian mechanics is ideal for systems with conservative forces and for bypassing constraint forces in any coordinate system. Dissipative and driven forces can be accounted for by splitting the external forces into a sum of potential and non-potential forces, leading to a set of modified Euler–Lagrange equations.
Generalized coordinates can be chosen by convenience, to exploit symmetries in the system or the geometry of the constraints, which may simplify solving for the motion of the system. Lagrangian mechanics reveals conserved quantities and their symmetries in a direct way, as a special case of Noether's theorem. Lagrangian mechanics is important not just for its broad applications, but for its role in advancing deep understanding of physics. Although Lagrange only sought to describe classical mechanics in his treatise Mécanique analytique, William Rowan Hamilton developed Hamilton's principle that can be used to derive the Lagrange equation and was recognized to be applicable to much of fundamental theoretical physics as well quantum mechanics and the theory of relativity, it can be applied to other systems by analogy, for instance to coupled electric circuits with inductances and capacitances. Lagrangian mechanics is used to solve mechanical problems in physics and when Newton's formulation of classical mechanics is not convenient.
Lagrangian mechanics applies to the dynamics of particles, while fields are described using a Lagrangian density. Lagrange's equations are used in optimization problems of dynamic systems. In mechanics, Lagrange's equations of the second kind are used much more than those of the first kind. Suppose we have a bead sliding around on a wire, or a swinging simple pendulum, etc. If one tracks each of the massive objects as a particle, calculation of the motion of the particle using Newtonian mechanics would require solving for the time-varying constraint force required to keep the particle in the constrained motion. For the same problem using Lagrangian mechanics, one looks at the path the particle can take and chooses a convenient set of independent generalized coordinates that characterize the possible motion of the particle; this choice eliminates the need for the constraint force to enter into the resultant system of equations. There are fewer equations since one is not directly calculating the influence of the constraint on the particle at a given moment.
For a wide variety of physical systems, if the size and shape of a massive object are negligible, it is a useful simplification to treat it as a point particle. For a system of N point particles with masses m1, m2... mN, each particle has a position vector, denoted r1, r2... rN. Cartesian coordinates are sufficient, so r1 =, r2 = and so on. In three dimensional space, each position vector requires three coordinates to uniquely define the location of a point, so there are 3N coordinates to uniquely define the configuration of the system; these are all specific points in space to locate the particles. The velocity of each particle is how fast the particle moves along its path of motion, is the time derivative of its position, thusIn Newtonian mechanics, the equations of motion are given by Newton's laws; the second law "net force equals mass times acceleration",applies to each particle. For an N particle system in 3 dimensions, there are 3N second order ordinary differential equations in the positions of the particles to solve for.
Instead of forces, Lagrangian mechanics uses the energies in the system. The central quantity of Lagrangian mechanics is the Lagrangian, a function which summarizes the dynamics of the entire system. Overall, the Lagrangian has no single expression for all physical systems. Any function which generates the correct equations of motion, in agreement with physical laws, can be taken as a Lagrangian, it is possible to construct general expressions for large classes of applications. The non-relativistic Lagrangian for a system of particles can be defined by L = T − V where T = 1 2 ∑ k = 1 N m k v k 2 is the total kinetic energy of the system, equalling the sum Σ of the kinetic energies of the particles, V is the potential energy of the system. Kinetic energy is the energy of the system's motion, vk2 = vk
Theory of relativity
The theory of relativity encompasses two interrelated theories by Albert Einstein: special relativity and general relativity. Special relativity applies to elementary particles and their interactions, describing all their physical phenomena except gravity. General relativity explains the law of its relation to other forces of nature, it applies to the astrophysical realm, including astronomy. The theory transformed theoretical physics and astronomy during the 20th century, superseding a 200-year-old theory of mechanics created by Isaac Newton, it introduced concepts including spacetime as a unified entity of space and time, relativity of simultaneity and gravitational time dilation, length contraction. In the field of physics, relativity improved the science of elementary particles and their fundamental interactions, along with ushering in the nuclear age. With relativity and astrophysics predicted extraordinary astronomical phenomena such as neutron stars, black holes, gravitational waves. Albert Einstein published the theory of special relativity in 1905, building on many theoretical results and empirical findings obtained by Albert A. Michelson, Hendrik Lorentz, Henri Poincaré and others.
Max Planck, Hermann Minkowski and others did subsequent work. Einstein developed general relativity between 1907 and 1915, with contributions by many others after 1915; the final form of general relativity was published in 1916. The term "theory of relativity" was based on the expression "relative theory" used in 1906 by Planck, who emphasized how the theory uses the principle of relativity. In the discussion section of the same paper, Alfred Bucherer used for the first time the expression "theory of relativity". By the 1920s, the physics community accepted special relativity, it became a significant and necessary tool for theorists and experimentalists in the new fields of atomic physics, nuclear physics, quantum mechanics. By comparison, general relativity did not appear to be as useful, beyond making minor corrections to predictions of Newtonian gravitation theory, it seemed to offer little potential for experimental test, as most of its assertions were on an astronomical scale. Its mathematics seemed difficult and understandable only by a small number of people.
Around 1960, general relativity became central to astronomy. New mathematical techniques to apply to general relativity streamlined calculations and made its concepts more visualized; as astronomical phenomena were discovered, such as quasars, the 3-kelvin microwave background radiation and the first black hole candidates, the theory explained their attributes, measurement of them further confirmed the theory. Special relativity is a theory of the structure of spacetime, it was introduced in Einstein's 1905 paper "On the Electrodynamics of Moving Bodies". Special relativity is based on two postulates which are contradictory in classical mechanics: The laws of physics are the same for all observers in uniform motion relative to one another; the speed of light in a vacuum is the same for all observers, regardless of their relative motion or of the motion of the light source. The resultant theory copes with experiment better than classical mechanics. For instance, postulate 2 explains the results of the Michelson–Morley experiment.
Moreover, the theory has many counterintuitive consequences. Some of these are: Relativity of simultaneity: Two events, simultaneous for one observer, may not be simultaneous for another observer if the observers are in relative motion. Time dilation: Moving clocks are measured to tick more than an observer's "stationary" clock. Length contraction: Objects are measured to be shortened in the direction that they are moving with respect to the observer. Maximum speed is finite: No physical object, message or field line can travel faster than the speed of light in a vacuum; the effect of Gravity can only travel through space at the speed of light, not faster or instantaneously. Mass -- energy equivalence: E = mc2, energy and mass are transmutable. Relativistic mass, idea used by some researchers; the defining feature of special relativity is the replacement of the Galilean transformations of classical mechanics by the Lorentz transformations.. General relativity is a theory of gravitation developed by Einstein in the years 1907–1915.
The development of general relativity began with the equivalence principle, under which the states of accelerated motion and being at rest in a gravitational field are physically identical. The upshot of this is that free fall is inertial motion: an object in free fall is falling because, how objects move when there is no force being exerted on them, instead of this being due to the force of gravity as is the case in classical mechanics; this is incompatible with classical mechanics and special relativity because in those theories inertially moving objects cannot accelerate with respect to each other, but objects in free fall do so. To resolve this difficulty Einstein first proposed. In 1915, he devised the Einstein field equations which relate the curvature of spacetime with the mass and any momentum within it; some of the consequences of general relativity are: Gravitational time dilation: Clocks run slower in deeper gravitational wells. Precession: Orbits precess in a way unexpected in Newton's theory of gravity
Routhian mechanics
In analytical mechanics, a branch of theoretical physics, Routhian mechanics is a hybrid formulation of Lagrangian mechanics and Hamiltonian mechanics developed by Edward John Routh. Correspondingly, the Routhian is the function which replaces both the Lagrangian and Hamiltonian functions; the Routhian, like the Hamiltonian, can be obtained from a Legendre transform of the Lagrangian, has a similar mathematical form to the Hamiltonian, but is not the same. The difference between the Lagrangian and Routhian functions are their variables. For a given set of generalized coordinates representing the degrees of freedom in the system, the Lagrangian is a function of the coordinates and velocities, while the Hamiltonian is a function of the coordinates and momenta; the Routhian differs from these functions in that some coordinates are chosen to have corresponding generalized velocities, the rest to have corresponding generalized momenta. This choice is arbitrary, can be done to simplify the problem.
It has the consequence that the Routhian equations are the Hamiltonian equations for some coordinates and corresponding momenta, the Lagrangian equations for the rest of the coordinates and their velocities. In each case the Lagrangian and Hamiltonian functions are replaced by a single function, the Routhian; the full set thus has the advantages of both sets of equations, with the convenience of splitting one set of coordinates to the Hamilton equations, the rest to the Lagrangian equations. The Routhian approach may offer no advantage, but one notable case where this is useful is when a system has cyclic coordinates, by definition those coordinates which do not appear in the original Lagrangian; the Lagrangian equations are powerful results, used in theory and practice, since the equations of motion in the coordinates are easy to set up. However, if cyclic coordinates occur there will still be equations to solve for all the coordinates, including the cyclic coordinates despite their absence in the Lagrangian.
The Hamiltonian equations are useful theoretical results, but less useful in practice because coordinates and momenta are related together in the solutions - after solving the equations the coordinates and momenta must be eliminated from each other. The Hamiltonian equations are suited to cyclic coordinates because the equations in the cyclic coordinates trivially vanish, leaving only the equations in the non cyclic coordinates; the Routhian approach has the best of both approaches, because cyclic coordinates can be split off to the Hamiltonian equations and eliminated, leaving behind the non cyclic coordinates to be solved from the Lagrangian equations. Overall fewer equations need to be solved compared to the Lagrangian approach; as with the rest of analytical mechanics, Routhian mechanics is equivalent to Newtonian mechanics, all other formulations of classical mechanics, introduces no new physics. It offers an alternative way to solve mechanical problems. In the case of Lagrangian mechanics, the generalized coordinates q1, q2... and the corresponding velocities dq1/dt, dq2/dt... and time t, enter the Lagrangian, L, q ˙ i = d q i d t, where the overdots denote time derivatives.
In Hamiltonian mechanics, the generalized coordinates q1, q2... and the corresponding generalized momenta p1, p2... and time, enter the Hamiltonian, H = ∑ i q ˙ i p i − L, p i = ∂ L ∂ q ˙ i, where the second equation is the definition of the generalized momentum pi corresponding to the coordinate qi. The velocities dqi/dt are expressed as functions of their corresponding momenta by inverting their defining relation. In this context, pi is said to be the momentum "canonically conjugate" to qi; the Routhian is intermediate between L and H. The choice of
Impulse (physics)
In classical mechanics, impulse is the integral of a force, F, over the time interval, t, for which it acts. Since force is a vector quantity, impulse is a vector in the same direction. Impulse applied to an object produces an equivalent vector change in its linear momentum in the same direction; the SI unit of impulse is the newton second, the dimensionally equivalent unit of momentum is the kilogram meter per second. The corresponding English engineering units are the slug-foot per second. A resultant force causes acceleration and a change in the velocity of the body for as long as it acts. A resultant force applied over a longer time therefore produces a bigger change in linear momentum than the same force applied briefly: the change in momentum is equal to the product of the average force and duration. Conversely, a small force applied for a long time produces the same change in momentum—the same impulse—as a larger force applied briefly. J = F average The impulse is the integral of the resultant force with respect to time: J = ∫ F d t Impulse J produced from time t1 to t2 is defined to be J = ∫ t 1 t 2 F d t where F is the resultant force applied from t1 to t2.
From Newton's second law, force is related to momentum p by F = d p d t Therefore, J = ∫ t 1 t 2 d p d t d t = ∫ p 1 p 2 d p = p 2 − p 1 = Δ p where Δp is the change in linear momentum from time t1 to t2. This is called the impulse-momentum theorem; as a result, an impulse may be regarded as the change in momentum of an object to which a resultant force is applied. The impulse may be expressed in a simpler form when the mass is constant: J = ∫ t 1 t 2 F d t = Δ p = m v 2 − m v 1 where F is the resultant force applied, t1 and t2 are times when the impulse begins and ends m is the mass of the object, v2 is the final velocity of the object at the end of the time interval, v1 is the initial velocity of the object when the time interval begins. Impulse has the same dimensions as momentum. In the International System of Units, these are kg⋅m/s = N⋅s. In English engineering units, they are slug⋅ft/s = lbf⋅s; the term "impulse" is used to refer to a fast-acting force or impact. This type of impulse is idealized so that the change in momentum produced by the force happens with no change in time.
This sort of change is a step change, is not physically possible. However, this is a useful model for computing the effects of ideal collisions. Additionally, in rocketry, the term "total impulse" is used and is considered synonymous with the term "impulse"; the application of Newton's second law for variable mass allows impulse and momentum to be used as analysis tools for jet- or rocket-propelled vehicles. In the case of rockets, the impulse imparted can be normalized by unit of propellant expended, to create a performance parameter, specific impulse; this fact can be used to derive the Tsiolkovsky rocket equation, which relates the vehicle's propulsive change in velocity to the engine's specific impulse and the vehicle's propellant-mass ratio. Wave–particle duality defines the impulse of a wave collision; the preservation of momentum in the collision is called phase matching. Applications include: Compton effect Nonlinear optics Acousto-optic modulator Electron phonon scattering Serway, Raymond A..
Physics for Scientists and Engineers. Brooks/Cole. ISBN 0-534-40842-7. Tipler, Paul. Physics for Scientists and Engineers: Mechanics and Waves, Thermodynamics. W. H. Freeman. ISBN 0-7167-0809-4. Dynamics
Quantum mechanics
Quantum mechanics, including quantum field theory, is a fundamental theory in physics which describes nature at the smallest scales of energy levels of atoms and subatomic particles. Classical physics, the physics existing before quantum mechanics, describes nature at ordinary scale. Most theories in classical physics can be derived from quantum mechanics as an approximation valid at large scale. Quantum mechanics differs from classical physics in that energy, angular momentum and other quantities of a bound system are restricted to discrete values. Quantum mechanics arose from theories to explain observations which could not be reconciled with classical physics, such as Max Planck's solution in 1900 to the black-body radiation problem, from the correspondence between energy and frequency in Albert Einstein's 1905 paper which explained the photoelectric effect. Early quantum theory was profoundly re-conceived in the mid-1920s by Erwin Schrödinger, Werner Heisenberg, Max Born and others; the modern theory is formulated in various specially developed mathematical formalisms.
In one of them, a mathematical function, the wave function, provides information about the probability amplitude of position and other physical properties of a particle. Important applications of quantum theory include quantum chemistry, quantum optics, quantum computing, superconducting magnets, light-emitting diodes, the laser, the transistor and semiconductors such as the microprocessor and research imaging such as magnetic resonance imaging and electron microscopy. Explanations for many biological and physical phenomena are rooted in the nature of the chemical bond, most notably the macro-molecule DNA. Scientific inquiry into the wave nature of light began in the 17th and 18th centuries, when scientists such as Robert Hooke, Christiaan Huygens and Leonhard Euler proposed a wave theory of light based on experimental observations. In 1803, Thomas Young, an English polymath, performed the famous double-slit experiment that he described in a paper titled On the nature of light and colours.
This experiment played a major role in the general acceptance of the wave theory of light. In 1838, Michael Faraday discovered cathode rays; these studies were followed by the 1859 statement of the black-body radiation problem by Gustav Kirchhoff, the 1877 suggestion by Ludwig Boltzmann that the energy states of a physical system can be discrete, the 1900 quantum hypothesis of Max Planck. Planck's hypothesis that energy is radiated and absorbed in discrete "quanta" matched the observed patterns of black-body radiation. In 1896, Wilhelm Wien empirically determined a distribution law of black-body radiation, known as Wien's law in his honor. Ludwig Boltzmann independently arrived at this result by considerations of Maxwell's equations. However, it underestimated the radiance at low frequencies. Planck corrected this model using Boltzmann's statistical interpretation of thermodynamics and proposed what is now called Planck's law, which led to the development of quantum mechanics. Following Max Planck's solution in 1900 to the black-body radiation problem, Albert Einstein offered a quantum-based theory to explain the photoelectric effect.
Around 1900–1910, the atomic theory and the corpuscular theory of light first came to be accepted as scientific fact. Among the first to study quantum phenomena in nature were Arthur Compton, C. V. Raman, Pieter Zeeman, each of whom has a quantum effect named after him. Robert Andrews Millikan studied the photoelectric effect experimentally, Albert Einstein developed a theory for it. At the same time, Ernest Rutherford experimentally discovered the nuclear model of the atom, for which Niels Bohr developed his theory of the atomic structure, confirmed by the experiments of Henry Moseley. In 1913, Peter Debye extended Niels Bohr's theory of atomic structure, introducing elliptical orbits, a concept introduced by Arnold Sommerfeld; this phase is known as old quantum theory. According to Planck, each energy element is proportional to its frequency: E = h ν, where h is Planck's constant. Planck cautiously insisted that this was an aspect of the processes of absorption and emission of radiation and had nothing to do with the physical reality of the radiation itself.
In fact, he considered his quantum hypothesis a mathematical trick to get the right answer rather than a sizable discovery. However, in 1905 Albert Einstein interpreted Planck's quantum hypothesis realistically and used it to explain the photoelectric effect, in which shining light on certain materials can eject electrons from the material, he won the 1921 Nobel Prize in Physics for this work. Einstein further developed this idea to show that an electromagnetic wave such as light could be described as a particle, with a discrete quantum of energy, dependent on its frequency; the foundations of quantum mechanics were established during the first half of the 20th century by Max Planck, Niels Bohr, Werner Heisenberg, Louis de Broglie, Arthur Compton, Albert Einstein, Erwin Schrödinger, Max Born, John von Neumann, Paul Dirac, Enrico Fermi, Wolfgang Pauli, Max von Laue, Freeman Dyson, David Hilbert, Wi
Momentum
In Newtonian mechanics, linear momentum, translational momentum, or momentum is the product of the mass and velocity of an object. It is a vector quantity, possessing a direction in three-dimensional space. If m is an object's mass and v is the velocity the momentum is p = m v, In SI units, it is measured in kilogram meters per second. Newton's second law of motion states that a body's rate of change in momentum is equal to the net force acting on it. Momentum depends on the frame of reference, but in any inertial frame it is a conserved quantity, meaning that if a closed system is not affected by external forces, its total linear momentum does not change. Momentum is conserved in special relativity and, in a modified form, in electrodynamics, quantum mechanics, quantum field theory, general relativity, it is an expression of one of the fundamental symmetries of time: translational symmetry. Advanced formulations of classical mechanics and Hamiltonian mechanics, allow one to choose coordinate systems that incorporate symmetries and constraints.
In these systems the conserved quantity is generalized momentum, in general this is different from the kinetic momentum defined above. The concept of generalized momentum is carried over into quantum mechanics, where it becomes an operator on a wave function; the momentum and position operators are related by the Heisenberg uncertainty principle. In continuous systems such as electromagnetic fields and deformable bodies, a momentum density can be defined, a continuum version of the conservation of momentum leads to equations such as the Navier–Stokes equations for fluids or the Cauchy momentum equation for deformable solids or fluids. Momentum is a vector quantity: it has both magnitude and direction. Since momentum has a direction, it can be used to predict the resulting direction and speed of motion of objects after they collide. Below, the basic properties of momentum are described in one dimension; the vector equations are identical to the scalar equations. The momentum of a particle is conventionally represented by the letter p.
It is the product of two quantities, the particle's mass and its velocity: p = m v. The unit of momentum is the product of the units of velocity. In SI units, if the mass is in kilograms and the velocity is in meters per second the momentum is in kilogram meters per second. In cgs units, if the mass is in grams and the velocity in centimeters per second the momentum is in gram centimeters per second. Being a vector, momentum has direction. For example, a 1 kg model airplane, traveling due north at 1 m/s in straight and level flight, has a momentum of 1 kg⋅m/s due north measured with reference to the ground; the momentum of a system of particles is the vector sum of their momenta. If two particles have respective masses m1 and m2, velocities v1 and v2, the total momentum is p = p 1 + p 2 = m 1 v 1 + m 2 v 2; the momenta of more than two particles can be added more with the following: p = ∑ i m i v i. A system of particles has a center of mass, a point determined by the weighted sum of their positions: r cm = m 1 r 1 + m 2 r 2 + ⋯ m 1 + m 2 + ⋯ = ∑ i m i r i ∑ i m i.
If all the particles are moving, the center of mass will be moving as well. If the center of mass is moving at velocity vcm, the momentum is: p = m v cm; this is known as Euler's first law. If the net force applied to a particle is a constant F, is applied for a time interval Δt, the momentum of the particle changes by an amount Δ p = F Δ t. In differential form, this is Newton's second law. If the net force experienced by a particle changes as a function of time, F, the change in momentum between times t1 and t2 is Δ p = J = ∫ t 1
