Length contraction | MY DIGITAL NOTEBOOK

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. Continue reading →