The problem:
(A) ==> <== (B)
3/4 c 3/4 c
What is the relative speed of A and B?
In classic physics, when you throw a rock at 5 mph out of a car moving at 55 mph, the rock is moving at 60 mph to a stationary observer, while it moves at 5 mph to the observer in the car. By this transformation (known as the Galilean transformation for velocities), it would appear that (A) and (B) are moving at 1.5 c with respect to each other - which is greater than the speed of light.
The problem here is that the Galilean transformation is no longer valid and instead the Lorentz transformation is necessary. For one dimension with respect to A:
v' = VA - VB
--------
1 - VAVB
----
C2
v' is the speed which A perceives B is moving at.
V
x is the
velocity from the external frame of reference
c is the speed of light.
thus:
v' = .75c - -.75c
------------
1 - -.56c2
------
c2
which is: 0.96c.
0.96c is the velocity that A will perceive B is moving at - not 1.5c.
At low speeds, this reduces to the Galilean transformation that we
are quite familiar with and is in line with common sense.
As for time dilation (which really isn't part of "Why matter cannot
reach the speed of light", however, it has been brought up),
this has been experimentally demonstrated: