# About accelerating particles [duplicate]

Can a particle moving below the speed of light be accelerated more and more until it is travelling at say c/2? IF so does it behave like electro-magnetic radiation?

No, it cannot. The momentum of the particle must be finite in any reference frame where the momentum (for a massive particle) is given by

$$\vec p = \frac{m\vec v}{\sqrt{1 - \frac{v^2}{c^2}}}$$

Note that the momentum is undefined for $v = c$ or, to put it another way, the momentum goes to infinity as $v \rightarrow c$.

So, just as it would be impossible in Newtonian mechanics to accelerate a particle to 'infinite' momentum, it is impossible in relativistic mechanics.

The difference being that, in Newtonian mechanics, momentum and velocity are proportional whereas in special relativity, they aren't.

• Can a particle traveling less than c be accelerated to arbitrarily near c without reaching c? Apr 18, 2014 at 2:27
• @user128932, a particle at rest in one frame of reference is travelling arbitrarily close to $c$ in another. Motion is relative, not absolute. Apr 18, 2014 at 2:43
• Well, can a particle traveling at about (1/2) c be accelerated arbitrarily close to c all within the same frame of reference? Apr 18, 2014 at 5:21
• @user128932, to accelerate a particle arbitrarily close to c requires arbitrarily large energy. Assuming the final speed is ultra-relativistic, the required energy is approximately $\Delta E \approx mc^2 \frac{\beta}{\sqrt{1 - \beta^2}}$ which is unbounded as $\beta \rightarrow 1$ where $\beta = \frac{v}{c}$. Apr 18, 2014 at 13:38
• Would this mean for all particles that are not leptons (if that's the right term) the speed of light is not the limiting speed of most physical events but that limit is slightly less than c? (as it wounld require arbitrarily large energy to get near c) Apr 19, 2014 at 4:58