ABSTRACT
Thus ( ) ( )0m w m vγ= and this gives the functional form of the dependence of inertial mass on speed. It holds for any value of and the limit 0v ( ) ( ) 000 0gives0 mmvmv =→→ which can be taken as the Newtonian mass or rest mass of the particle. Hence the inertial mass of the particle depends on its speed as
( ) ( ) 0 2 2
0 1 /
mm v m v c
γ= = −
(5.1)
and the momentum of a particle moving with arbitrary velocity vG is
vm cv
vmp G GG =
− =
/1 (5.2)
We can now find an energy-velocity relation, as a consequence of Newton's law of motion (2.9), which transforms according to
vdpvpdrd dt pdrdFdE =⋅=⋅=⋅= GGG GGG (5.3)
where the parallelism of pG and vG , see Figure 5.2, allows us to write
vdpvpdvpd =α=⋅ cosGGG In other words, a change in energy requires a change in the magnitude of the momentum. A change in just the direction of pG , as given by a transverse force, leaves E constant.
