In physics the notion of mass arises in three different contexts: 1. In the relation between force and acceleration, 2. In the relation between momentum and velocity, 3. In the relation between kinetic energy and velocity squared.

In Newtonian mechanics these relations are embodied in the following formulas F is force, a is acceleration, p is momentum, v is velocity, and K is kinetic energy:

• In the relation between force and acceleration

  m = F/a

• In the relation between momentum and velocity,

  m = p/v

• In the relation between kinetic energy and velocity squared

  m = 2K/v²

  .

Because the same value of m satisfies all three relationships it is taken to be an intrinsic property.

There is also the force between two bodies of mass m₁ and m₂ being

F = Gm₁m₂/s²

where s is separation distance and G is a constant.

Intrinsic Properties and
Contextual Characteristics

Under Special Relativity a situation involves an observer at rest and a vehicle traveling at a velocity v with respect to the observer. In the vehicle there are; three objects a stick of length l₀, a clock that runs at a rate of t₀ and a mass of m₀. These values were established at a time when the vehicle was motionless with respect to the observer. When the vehicle is moving at a velocity of v with respect to the observer the characteristics of the three objects appear to be

l₀(1−β²)^½
t₀/(1−β²)^½
m₀/(1−β²)^½

where β is velocity with respect to the speed of light v/c.

These quantities are not intrinsic properties of the object; they are characteristics of the situation in which they are observed.

Relativistic Mechanics

In Relativistic Mechanics it is said that the mass of a moving body is 

m = m₀/(1−β²)^½

as though this an intrinsic property of the object. This is only a contextual characteristic with respect an observer.

It is then said that that the relativistic momentum is

p* = mv = m₀v/(1−β²)^½

But Lagranian analysis says the momentum with respect to a location variable x is

p = (∂K/∂v)

where v=(dx/dt).

The relativistic kinetic energy is given by

K= mc² − m₀c²
or, equivalently
K = m₀c²[1/(1−β²)^½ −1]

Evaluation of (∂K/∂v) gives

p = mv/(1−β²) = m₀v/(1−β²)^3/2

The expression m₀/(1−β²)^3/2 has a historical precedence. In cerca 1881 J.J. Thomson noticed a resistance of charged bodies to acceleration. This resistance to acceleration he found was as though its mass was

m₀v/(1−β²)^3/2

This quantity came to be called longitudial mass m_L. Thus relativistic momentum p can be computed as

p = m_Lv

However just because m_L is used for mass in the momentum equation does not mean m_L should be used for mass in the kinetic energy equation or other quantities. Mass and other variables under relativity are not intrinsic quantities. They depend upon the context of the situation.

Thus for computing relativistic kinetic energy mass is equal to

m₀/(1−β²)^½

But for computing momentum mass is

m₀/(1−β²)^3/2

This maintains the formula of momentum being equal to mass times velocity. One can equally maintain the mass concept and vary the formula; i.e.,

momentum = mv/(1−β²)