"the traveling object. It equals the object's momentum p divided by its rest mass ""m"", and is made up of the space-like components of the object's four-vector velocity. William Shurcliff's monograph mentioned its early use in the Sears and Brehme text. Fraundorf has explored its pedagogical value while Ungar, Baylis and Hestenes have examined its relevance from group theory and geometric algebra perspectives. Proper velocity is sometimes referred to as celerity. Unlike the more familiar coordinate velocity v, proper velocity is synchrony-free (does not require synchronized clocks) and is useful for describing both super-relativistic and sub-relativistic motion. Like coordinate velocity"
"as inertial mass, and that the acceleration of gravity is a result of a ""valley"" or slope in the space-time continuum that masses ""fell down"". Dennis Sciama later showed that the reaction force produced by the combined gravity of all matter in the universe upon an accelerating object is mathematically equal to the object's inertia, but this would only be a workable physical explanation if, by some mechanism, the gravitational effects operated instantaneously. At any non-zero speed, relativistic mass always exceeds gravitational mass. If the mass is made to travel close to the speed of light, its ""inertial mass"" (relativistic)"
"acceleration is equal to the sum of the forces on the mass. Newton's law of gravity says that the gravitational force felt on mass by a single mass is given by where is the gravitational constant and is the magnitude of the distance between and (metric induced by the norm). Summing over all masses yields the -body equations of motion: where is the ""self-potential"" energy Defining the momentum to be , Hamilton's equations of motion for the -body problem become where the Hamiltonian function is and is the kinetic energy Hamilton's equations show that the -body problem is a system"
"be moving away from the center of mass at the same speed. Adding the speed of the center of mass to both, we find that the body that was moving is now stopped and the other is moving away at speed . The bodies have exchanged their velocities. Regardless of the velocities of the bodies, a switch to the center of mass frame leads us to the same conclusion. Therefore, the final velocities are given by In general, when the initial velocities are known, the final velocities are given by If one body has much greater mass than the other,"
"from Tsiolkovsky's rocket equation: where This equation can be rewritten in the following equivalent form: The fraction on the left-hand side of this equation is the rocket's mass ratio by definition. This equation indicates that a Δv of formula_4 times the exhaust velocity requires a mass ratio of formula_5. For instance, for a vehicle to achieve a formula_6 of 2.5 times its exhaust velocity would require a mass ratio of formula_7 (approximately 12.2). One could say that a ""velocity ratio"" of formula_4 requires a mass ratio of formula_5. Sutton defines the mass ratio inversely as: In this case, the values"
"equation that describes how the vector r = x − x between the masses changes with time. The solutions of these independent one-body problems can be combined to obtain the solutions for the trajectories x(""t"") and x(""t""). Let formula_3 be the position of the center of mass (barycenter) of the system. Addition of the force equations (1) and (2) yields where we have used Newton's third law F = −F and where The resulting equation: shows that the velocity V = ""d""R/""dt"" of the center of mass is constant, from which follows that the total momentum ""m"" v + ""m"""
"mass ""m"" varies with time by either ejecting or accreting mass is obtained by applying the second law to the entire, constant-mass system consisting of the body and its ejected/accreted mass; the result is where u is the velocity of the escaping or incoming mass relative to the body. From this equation one can derive the equation of motion for a varying mass system, for example, the Tsiolkovsky rocket equation. Under some conventions, the quantity u d""m""/d""t"" on the left-hand side, which represents the advection of momentum, is defined as a force (the force exerted on the body by the"
"to see that the inertia of the central particle is reduced by the total energy emitted. This allowed Einstein to conclude that the inertia of a heavy particle is increased or diminished according to the energy it absorbs or emits. After Einstein first made his proposal, it became clear that the word mass can have two different meanings. Some denote the ""relativistic mass"" with an explicit index: This mass is the ratio of momentum to velocity, and it is also the relativistic energy divided by (it is not Lorentz-invariant, in contrast to formula_7). The equation holds for moving objects. When"
"Mass Mass is both a property of a physical body and a measure of its resistance to acceleration (a change in its state of motion) when a net force is applied. The object's mass also determines the strength of its gravitational attraction to other bodies. The basic SI unit of mass is the kilogram (kg). In physics, mass is not the same as weight, even though mass is often determined by measuring the object's weight using a spring scale, rather than balance scale comparing it directly with known masses. An object on the Moon would weigh less than it does"
"This quantity formula_5 is frame invariant and velocity independent. However, some texts group the Lorentz factor with the mass factor to define a new quantity called the ""relativistic mass"", formula_7. This quantity is evidently velocity dependent, and from it arises the notion that ""mass increases with speed"". This construction is optional, however, and adds little insight into the dynamics of Special Relativity. Since the electron mass determines a number of observed effects in atomic physics, there are potentially many ways to determine its mass from an experiment, if the values of other physical constants are already considered known. Historically, the"
"""σ"" = 1, 2, ..., ""k""; ""λ"" = ""k""+2, ""k""+3, ..., ""k""+""n"". Here ""δ"" is the Kronecker delta. These transformations are generalization of the Lorentz boost in a fixed space direction (""x"") in the field of the multidimensional time and multidimensional space. Let us denote formula_11 and formula_12 where ""σ"" = 1, 2, ..., ""k""; ""η"" = ""k""+1, ""k""+2, ..., ""k""+""n"". The velocity-addition formula is then given by where ""σ"" = 1, 2, ..., ""k""; ""λ"" = ""k""+2, ""k""+3, ..., ""k""+""n"". For simplicity, let us consider only one spatial dimension ""x"" and the two time dimensions ""x"" and ""x"". (E."
"Added mass In fluid mechanics, added mass or virtual mass is the inertia added to a system because an accelerating or decelerating body must move (or deflect) some volume of surrounding fluid as it moves through it. Added mass is a common issue because the object and surrounding fluid cannot occupy the same physical space simultaneously. For simplicity this can be modeled as some volume of fluid moving with the object, though in reality ""all"" the fluid will be accelerated, to various degrees. The dimensionless added mass coefficient is the added mass divided by the displaced fluid mass – i.e."
"force, ""m"" is the inertial mass, and ""a"" is the acceleration. By this formula, the greater its mass, the less a body accelerates under a given force. The masses defined by the above formulas are equal because the latter formula is a consequence of the former, if mass does not depend on time and velocity. Thus, ""mass is the quantitative or numerical measure of a body’s inertia, that is of its resistance to being accelerated"". This meaning of a ""body's inertia"" therefore is altered from the popular meaning as ""a tendency to maintain momentum"" to a description of the measure"
"the mass while proportional to the ""square"" of the velocity. The velocity of the bullet is a more important determinant of muzzle energy. For a constant velocity, if the mass is doubled, the energy is doubled; however, for a constant mass, if the velocity is doubled, the muzzle energy increases ""four"" times. In the SI system the above ""E"" will be in unit joule if the mass, ""m"", is in kilogram, and the speed, ""v"", is in metre per second. It must be stressed that muzzle energy is dependent upon the factors previously listed and that even velocity is highly"
"work is gained from a loss of potential energy. The gravitational force between two bodies of mass ""M"" and ""m"" separated by a distance ""r"" is given by where formula_19 is a vector of length 1 pointing from ""M"" to ""m"" and ""G"" is the gravitational constant. Let the mass ""m"" move at the velocity v then the work of gravity on this mass as it moves from position r(t) to r(t) is given by Notice that the position and velocity of the mass ""m"" are given by where e and e are the radial and tangential unit vectors directed"
"Another three base units (for temperature, amount of substance and luminous intensity) were added later. The early metric systems defined a unit of weight as a base unit, while the SI defines an analogous unit of mass. In everyday use, these are mostly interchangeable, but in scientific contexts the difference matters. Mass, strictly the inertial mass, represents a quantity of matter. It relates the acceleration of a body to the applied force via Newton's law, : force equals mass times acceleration. A force of 1 N (newton) applied to a mass of 1 kg will accelerate it at 1 m/s."
"of the fluid (continuous phase), formula_5 is the volume of the particle, and D/D""t"" denotes the material derivative. The origin of the notion ""virtual mass"" becomes evident when we take a look at the momentum equation for the particle. where formula_7 is the sum of all other force terms on the particle, such as gravity, pressure gradient, drag, lift, Basset force, etc. Moving the derivative of the particle velocity from the right hand side of the equation to the left we get so the particle is accelerated as if it had an added mass of half the fluid it displaces,"
"as ""v""), and Δ""v"" is the maximum change of speed of the vehicle (when no external forces are acting). Variable-mass system In mechanics, a variable-mass system is a collection of matter whose mass varies with time. It can be confusing to try to apply Newton's second law of motion directly to such a system. Instead, the time dependence of the mass ""m"" can be calculated by rearranging Newton's second law and adding a term to account for the momentum carried by mass entering or leaving the system. The general equation of variable-mass motion is written as where F is the"
"(at the maxima of displacement), the magnitude of the force on the mass, which is the vectorial sum of the spring force and the kinetic frictional force, changes by twice the magnitude of the frictional force, since the spring force is continuous and the frictional force ""reverses"" its direction when the velocity does. Therefore the acceleration jumps by this amount divided by the mass. That is, the mass experiences a discontinuous acceleration and the jerk contains a Dirac delta, each time the mass passes through the (decreasing) maximal displacements, until it comes to a halt, because the static friction force"
"get: As can be expected, the solution is invariant under adding a constant to all velocities, which is like using a frame of reference with constant translational velocity. The velocity of the center of mass does not change by the collision: The center of mass at time formula_13 before the collision and at time formula_14 after the collision is given by two equations: Hence, the velocities of the center of mass before and after the collision are: The numerator of formula_19 is the total momentum before the collision, and numerator of formula_20 is the total momentum after the collision. Since"
"a physical object maintains the same density and is scaled up, its volume and mass are increased by the cube of the multiplier while its surface area increases only by the square of said multiplier. This would mean that when the larger version of the object is accelerated at the same rate as the original, more pressure would be exerted on the surface of the larger object. Consider a simple example of a body of mass, M, having an acceleration, a, and surface area, A, of the surface upon which the accelerating force is acting. The force due to acceleration,"
"mass () times the speed of light () squared, or . The mass-energy relation can be used to predict how much energy will be released or consumed by nuclear reactions; one simply measures the mass of all constituents and the mass of all the products and multiplies the difference between the two by . The result shows how much energy will be released or consumed, usually in the form of light or heat. When applied to certain nuclear reactions, the equation shows that an extraordinarily large amount of energy will be released, millions of times as much as in the"
"mass. This was clear at the beginning of the 20th century, before the advent of the theory of relativity. Mass, ""m"", denoted something like an amount of substance or quantity of matter. At the same time, mass was the quantitative measure of inertia of a body. The mass of a body determines the momentum, formula_1, of the body at given velocity, formula_2; it is a proportionality factor in the formula: The factor ""m"" is referred to as inertial mass. But mass, as related to the ""inertia"" of a body, can also be defined by the formula: where ""F"" is the"
"When the mass density difference is caused by a temperature difference, Ra is, by definition, the ratio of the timescale for thermal transport due to thermal diffusion, to the timescale for thermal transport due to fluid falling at speed formula_1under gravity formula_2 This means it is a type of Péclet number. For a volume of fluid a size formula_3across (in all three dimensions), with a mass density difference formula_4, then the force of gravity is of order formula_5, for formula_6 the acceleration due to gravity. From the Stokes equation, when the volume of fluid is falling at speed formula_1the viscous"
"body by mass and volume (mass divided by volume gives the mass density of the body). In a gravitational (or other body force) field, and in a vacuum, the body experiences a transfer of momentum giving its weight. If the vacuum is substituted by a medium (water, air), then the body loses the weight of the displaced medium's volume (otherwise that volume of medium replaced into the remaining medium would not remain force-free). Reconsider the body and the body's volume made up by medium: Subtracting the respective vacuum-weights (body minus medium) gives the resultant, effective, net-weight of the body in"
"related concepts, provide an often convenient means by which to quantify these properties. In galaxy dynamics, the mass of a galaxy is often inferred by measuring the rotation velocity of its gas and stars, assuming circular Keplerian orbits. Using the virial theorem, the dispersion velocity can be used in a similar way. Taking the kinetic energy (per particle) of the system as , and the potential energy (per particle) as we can write Here formula_50 is the radius at which the velocity dispersion is being measured, and is the mass within that radius. The virial mass and radius are generally"
"momentum is conserved, we have formula_21. With respect to the center of mass both velocities are reversed by the collision: in the case of particles of different mass, a heavy particle moves slowly toward the center of mass, and bounces back with the same low speed, and a light particle moves fast toward the center of mass, and bounces back with the same high speed. From the equations for formula_22 and formula_23 above we see that in the case of a large formula_24, the value of formula_25 is small if the masses are approximately the same: hitting a much lighter"
"describe the same properties of physical bodies at the qualitative and quantitative level respectively, by other words, the mass quantitatively describes the inertia. According to Newton's second law of motion, if a body of fixed mass ""m"" is subjected to a single force ""F"", its acceleration ""a"" is given by ""F""/""m"". A body's mass also determines the degree to which it generates or is affected by a gravitational field. If a first body of mass ""m"" is placed at a distance ""r"" (center of mass to center of mass) from a second body of mass ""m"", each body is subject"
"is defined as the ratio of the momentum of an object to its velocity. Relativistic mass depends on the motion of the object, so that different observers in relative motion see different values for it. If the object is moving slowly, the relativistic mass is nearly equal to the rest mass and both are nearly equal to the usual Newtonian mass. If the object is moving quickly, the relativistic mass is greater than the rest mass by an amount equal to the mass associated with the kinetic energy of the object. As the object approaches the speed of light, the"
"substances mass, conservation of mass is expressed by where formula_7 is the square of the ""speed of sound"". For low Mach numbers and incompressible media formula_8 is assumed to be infinite, which is the reason for the above continuity equation to reduce to The way of obtaining a velocity field satisfying the above, is to compute a pressure which when substituted into the momentum equation leads to the desired correction of a preliminary computed intermediate velocity. Applying the divergence operator to the compressible momentum equation yields formula_11 then provides the governing equation for pressure computation. The idea of pressure-correction also"
"Mass–energy equivalence In physics, mass–energy equivalence states that anything having mass has an equivalent amount of energy and vice versa, with these fundamental quantities directly relating to one another by Albert Einstein's famous formula: formula_1 This formula states that the equivalent energy () can be calculated as the mass () multiplied by the speed of light ( = about m/s) squared. Similarly, anything having energy exhibits a corresponding mass given by its energy divided by the speed of light squared . Because the speed of light is a very large number in everyday units, the formula implies that even an"
"""a"", ""a"") can be calculated from Newton's Law of Gravitation. Each body attracts each other body, the total acceleration being the sum of all these attractions. Next one chooses a small time-step Δ""t"" and applies Newton's Second Law of Motion. The acceleration multiplied with Δ""t"" gives a correction to the velocity. The velocity multiplied with Δ""t"" gives a correction to the position. This procedure is repeated for all other bodies. The result is a new value for position and velocity for all bodies. Then, using these new values one starts over the whole calculation for the next time-step Δ""t"". Repeating"
"(1 m/s) from a state of rest, when it achieves the speed of 5 m/s after 5 seconds and 10 m/s after 10 seconds. The average acceleration can be calculated by dividing the speed (m/s) by the time (s), so the average acceleration in the first example would be calculated formula_1. Newton's Second Law states that force equals mass multiplied by acceleration. The unit of force is the newton (N), and mass has the SI unit kilogram (kg). One newton equals one kilogram metre per second squared. Therefore, the unit metre per second squared is equivalent to newton per kilogram,"