Ultimate Speed of A Material Particle – Denying the Concept of Infinite mass – Photons and More

In classical mechanics, there being no upper limit to velocity it is possible that as a particle is given more and more acceleration, its speed may go on increasing progressively and may well become greater than $c$, –in fact, it may have any velocity whatever.

This is firmly denied by the theory of relativity. It may legitimately be asked, therefore, as to what will happen if the particle is continually accelerated. Certainly, its velocity v goes on increasing and hence also its mass in accordance with the mass-velocity relation $m= \frac {m_0} { \sqrt {1- \frac {v^2} {c^2} } } = \gamma m_0$ . But as $v$ approaches $c$, $\frac {v^2} {c^2} \longrightarrow 1$ and therefore $\sqrt {1- \frac {v^2} {c^2} } \longrightarrow 0$ and hence the mass of the particle $m \longrightarrow \infty$, as shown graphically, from which it is clear that for velocities right up to 50% of $c$, the increase in mass from the value of the rest mass or inertial mass $m_0$ is quite inappreciable.
Beyond this, however, the mass increases relatively more rapidly and tends to become infinite as $v \longrightarrow c$.

INFINITE MASS ?

The idea of infinite mass, however, makes on sense for a number of reasons. Thus, for example

• It would require an infinite force to accelerate the particle to the speed of light c.
• Due to Lorentz – Fitzgerald Contraction, its length in its direction of motion must be zero.
• Its volume, too must, therefore, be zero.
• It must exert an infinite gravitational pull on all other bodies in the universe.
etc. etc, etc, etc.

So that, discarding the notion of an infinite mass, we interpret the above relationship to mean that no material body can equal or exceed the speed of light in free space, viz. $3\times 10^8 m/s$.

Photon?

Now, it may perhaps be cited in apparent refutation of this statement, that a photon, in free space, travels with the free – space speed c of light. So, of course, it does, but then a photon is not a material particle since it possesses no rest mass.

Other Velocities?

Also, there are, what are called phase velocities, which may even exceed the free space speed c of light, but these are, again, no material particles but only abstract mathematical functions.
Finally, some of the atomic particles may travel in media like air, glass or water with velocities higher than that of light in those media but by no means higher than its speed in free space.

Thus , we come to the conclusion that the ultimate speed for any material particle is the speed of light in free-space, which it can never obtain or exceed.