



Accordingly with Newton's second law of motion, if a net
external force F acts over an object of mass m, this object
accelerates with the acceleration a obtained from the formula
where the
arrows over the quantities means that the force and acceleration are in the same
direction. Understanding that the directions are the same, the acceleration can
be obtained from
.
Notice that in this classical expression when the force is constant, the acceleration is also constant.
For an object accelerating with constant acceleration, the
velocity at any instant of time can be calculated from
which in terms of the force applied to the object becomes
.
Therefore, if the object accelerates for a sufficient long time, the velocity of
the object can increase above the speed of light. However, if at the same time
that the object increases its velocity, it gets harder to increase the object velocity; then,
even when a force is still acting on the the object, the object can not exceed the speed of light. This result is achieved
by redefining the mass (notice that the force depends on an external agent and
it should not be redefined). The
redefinition of the mass when moving at a velocity v with respect to an
observer is:
The value of m_{0} is the measurement of the mass of
the object with respect to an observer at rest with respect to the mass
(measured in a traditional scale). The mass m represents the value
measured for the mass of the object by an observer for whom the mass is moving
at the velocity v.


Most of the relativistic quantities are related to their
classical counter parts through the relativistic factor g,
As it can be observed on the graph below, this factor
is relevant for speeds greater than about 75% of the speed of light.
The
graph above shows the behavior of the relativistic multiplication factor
gamma as a function of the square of the ratio between the speed of the
object and the speed of light. 

Since the factor depends of the square of the
velocity, the factor gamma is independent of the direction of the
velocity. Therefore, an extreme value is obtained when the velocity is
minimal in magnitude, velocity zero. When the velocity increases to
values approaching the speed of light, this ratio becomes closer and
closer to one which makes the g factor
greater and greater reaching infinity, ∞. For
velocities greater than the speed of light, the ratio between the
velocities will be greater than 1. The radical to be calculated will be
the radical of a negative number resulting in a complex quantity result.
As a consequence of these mathematical results, the speed of any mobile
can not be greater or equal to the speed of light. 
The graph above represents a comparison between the
exact factor g and approximation of the
factor for low velocities. Notice that for velocities up to about 25% of
the speed of light, v = 7.53 ×
10^{7} m/s, the two expressions result in about the
same value for the factor
As a way of comparison, the speed of sound is only 3 ×
10^{2}
m/s.
The previous approximation shows the first relativistic correction to
classical values for the different physical quantities.



Derived from electromagnetism, an electromagnetic wave
(light) carrying an amount of energy E has a linear momentum of
.

The diagram on the left represents two instance for a
system made out of a spaceship and a pulse of light. The top portion of
the diagram shows the instant when the pulse of light is emitted at the
back of the spaceship. The coordinate system selected moves with the
center of mass of the system. After the pulse of light reaches the front
of the spaceship the following relations are valid:
Distance travel by the pulse
; and
Distance traveled by the spaceship center of mass
. Since
there is not external force acting on the system, the center of mass of
the system must be unchanged. In this discussion, the pulse of light is
assigned a contribution to the change in the center of mass of the
system in the amount of
. This
amount much the change in the position of the center of mass of the
system coming from the change in the center of mass of the spaceship
given by
. For
the position of the system center of mass to be unchanged due to this
emission, the two changes must be equal to each other,
.
Thus,
.
On the other side, the linear momentum of the system
must be conserved in any inertial frame of reference, including the
frame of reference attached to the system center of mass. In this case,
the linear momentum of the spaceship is just
, to
the left of the diagram while the linear momentum of the pulse of light
is
,
to the right of the diagram. Equating these two momentums, the
following relation is established
with
or


The previous discussion is not a derivation of the formula
that relates the energy with the mass of the object and the square of the speed
of light.
