Momentum
is a vector quantity. It has magnitude and direction. In ordinary
circumstances, the momentum (p) of an object is defined as the product of
the mass (m) of the object with the (vector) velocity (v) of the object.
The momentum of the object has the same direction as the velocity of the
object.
Momentum is a property of an object which remains well-defined even when the object moves past us at speeds near the speed of light. No object can move past us faster than light; objects have a "speed limit" described by Einstein's theory of relativity. Although there is an upper limit of the speed of an object, there is no upper limit on the momentum of an object.
Both assertions (the speed limit, and the lack of a momentum limit) are verified by laboratory observation of fast particles. We must conclude from these observations that the relation between momentum and velocity is not as simple as it is in ordinary circumstances.
The complete relation is shown in the figure. For the relativistic momentum equation the mass of the object, m, must be measured by an observer who is at rest with the object. For this equation, we have chosen a coordinate system with the x axis parallel to the velocity, so that the velocity vector may be represented by an algebraic symbol, v. In the equation, the letter c represents the speed of light in a vacuum.
Note that in the limit of low velocity, the relativistic momentum has the same form as the ordinary momentum.