## Does a body with constant acceleration exceed the speed of light?

Consider a body $\mathcal{B}$ that has a constant acceleration. Does body $\mathcal{B}$ exceeds the speed of light? Let the constant acceleration be $a$. Assume that body $\mathcal{B}$ starts at rest, then the travelled distance is given by $\displaystyle s(t) = \frac{1}{2} a t^2$ and $v(t) = a t$. So the critical time is given by $\displaystyle t_c = \frac{c}{a}$. So it looks like as if such a body would exceed the speed of light.

But HOW can body $\mathcal{B}$ have a constant acceleration? The force acting on a body is given by $\displaystyle \vec{F} = \frac{d\vec{p}}{dt}$, and $\displaystyle \vec{p} = \frac{m_0 \vec{v}}{\sqrt{1 - v^2/c^2}}$. If the force is acting in the direction of the velocity, then we have

(1) $\displaystyle F = \frac{m_0 a}{\sqrt{1 - v^2/c^2}^3}$

Meaning that the force becomes infinity at the speed of light and the force becomes complex if the speed of light is exceded. However, an infinite force would require an infinite amount of energy; so the speed limit cannot be exceded. Due to relativity; the acceleration cannot be maintained constant. So we cannot say that a body has a constant acceleration for ever; only during some time interval that requires a finite amount of energy.