# Mathematical relationship between mass and velocity

### Relativistic mass

There isn't any relation between time and mass in speical relativity: mass is a Lorentz The expressions for the energy and the momentum, that contain the relative velocity between the two frames, .. time has only a mathematical existence. Your understanding that the orbital velocity decreases as the radius increases is correct. Yet, as the article states, we see that orbiting stars. Mass in special relativity incorporates the general understandings from the laws of motion of The more general invariant mass (calculated with a more complicated mechanics does not bear a precise relationship to the concept in relativity. in circles with a relativistic velocity, the mass of the cyclotron+electron system is.

### Relation between mass and velocity | Physics Forums

The phrase "The rest mass of a photon is zero" sounds nonsensical because the photon can never be at rest but this is just a misfortunate accident of terminology. In modern physics texts the term mass when unqualified means invariant mass and photons are said to be "massless" see Physics FAQ What is the mass of the photon? Teaching experience shows that this avoids most sources of confusion.

Despite the general usage of an invariant mass in the scientific literature, the use of the word mass to mean relativistic mass is still found in many popular science books. For example, Stephen Hawking in "A Brief History of Time" writes "Because of the equivalence of energy and mass, the energy which an object has due to its motion will add to its mass.

The standard convention followed by some physicists seems to be: It is a curious dichotomy of terminology which inevitably leads to confusion. A common example is the mistaken belief that a fast moving particle must form a black hole because of its increase in mass see relativity FAQ article If you go too fast do you become a black hole?

Looking more deeply into what is going on we find that there are two equivalent ways of formulating special relativity. Einstein's original mechanical formalism is described in terms of inertial reference frames, velocities, forces, length contraction and time dilation. Relativistic mass fits naturally into this mechanical framework but it is not essential. The second formulation is the more mathematical one introduced a year later by Minkowski.

It is described in terms of space-time, energy-momentum four vectors, world lines, light cones, proper time and invariant mass. This version is harder to relate to ordinary intuition because force and velocity are less useful in their 4-vector forms.

On the other hand, it is much easier to generalise this formalism to the curved space-time of general relativity where global inertial frames do not usually exist. It may seem that Einstein's original mechanical formalism should be easier to learn because it retains many equations from the familiar Newtonian mechanics.

In Minkowski's geometric formalism simple concepts such as velocity and force are replaced with worldlines and four-vectors. Yet the mechanical formalism often proves harder to swallow and is at the root of many peoples failure to get over the paradoxes which are so often discussed.

Once students have been taught about Minkowski space they invariably see things more clearly.

### relation between mass and velocity - Physics - Science Forums

The paradoxes are revealed for what they are and calculations also become simpler. It is debatable whether or not the relativistic mechanical formalism should be avoided altogether. It can still provide the correspondence between the new physics and the old which is important to grasp at the early stages.

The step from the mechanical formalism to the geometric can then be easier. The alternative modern teaching method is to translate Newtonian mechanics into a geometric formalism using Galilean relativity in 4 dimensional space-time then modify the geometric picture to Minkowski space.

A closed container of gas closed to energy as well has a system "rest mass" in the sense that it can be weighed on a resting scale, even while it contains moving components. This mass is the invariant mass, which is equal to the total relativistic energy of the container including the kinetic energy of the gas only when it is measured in the center of momentum frame.

Just as is the case for single particles, the calculated "rest mass" of such a container of gas does not change when it is in motion, although its "relativistic mass" does change. The container may even be subjected to a force which gives it an overall velocity, or else equivalently it may be viewed from an inertial frame in which it has an overall velocity that is, technically, a frame in which its center of mass has a velocity.

In this case, its total relativistic mass and energy increase. However, in such a situation, although the container's total relativistic energy and total momenta increase, these energy and momentum increases subtract out in the invariant mass definition, so that the moving container's invariant mass will be calculated as the same value as if it were measured at rest, on a scale.

## Relation between mass and velocity

Closed meaning totally isolated systems[ edit ] All conservation laws in special relativity for energy, mass, and momentum require isolated systems, meaning systems that are totally isolated, with no mass-energy allowed in or out, over time.

If a system is isolated, then both total energy and total momentum in the system are conserved over time for any observer in any single inertial frame, though their absolute values will vary, according to different observers in different inertial frames. The invariant mass of the system is also conserved, but does not change with different observers. This is also the familiar situation with single particles: Conservation of invariant mass also requires the system to be enclosed so that no heat and radiation and thus invariant mass can escape.

As in the example above, a physically enclosed or bound system does not need to be completely isolated from external forces for its mass to remain constant, because for bound systems these merely act to change the inertial frame of the system or the observer. Though such actions may change the total energy or momentum of the bound system, these two changes cancel, so that there is no change in the system's invariant mass.

This is just the same result as with single particles: On the other hand, for systems which are unbound, the "closure" of the system may be enforced by an idealized surface, inasmuch as no mass-energy can be allowed into or out of the test-volume over time, if conservation of system invariant mass is to hold during that time.

If a force is allowed to act on do work on only one part of such an unbound system, this is equivalent to allowing energy into or out of the system, and the condition of "closure" to mass-energy total isolation is violated.

A team that has the momentum is on the move and is going to take some effort to stop. A team that has a lot of momentum is really on the move and is going to be hard to stop. Momentum is a physics term; it refers to the quantity of motion that an object has. A sports team that is on the move has the momentum. If an object is in motion on the move then it has momentum.

Momentum can be defined as "mass in motion. The amount of momentum that an object has is dependent upon two variables: Momentum depends upon the variables mass and velocity. In terms of an equation, the momentum of an object is equal to the mass of the object times the velocity of the object. The equation illustrates that momentum is directly proportional to an object's mass and directly proportional to the object's velocity. The units for momentum would be mass units times velocity units.