Equilibrium: Inertia & Applied Force

Question

Thank you for taking the time to read this. I am lay and will understand if the question is closed.

My understanding is that in a vacuum, a constant force of $n$ applied to an object of mass $m$, will cause the object to accelerate. However, at some point (relativistic) mass increases and I assume this affects inertia and therefore reduces acceleration.

My question is: Given a certain set of values for $m$ and $n$, would inertia and the applied force ever be in equilibrium, resulting in zero acceleration (constant velocity) or does the object continue to accelerate albeit at an ever decreasing rate?

Answer 1

$\Sigma F = ma$. Therefore, if a net force $n$ is applied to an object, it will accelerate. You are correct that the acceleration will decrease, since $m$ increases, but it will never quite reach zero.

Answer 2

Assuming the force is in the direction of the velocity, the relativistic form of $F=ma$ is $$F=\gamma^3 m_o a.$$

The term relativistic mass has become outdated recently, but it refers to the quantity $M=\gamma m_o$, where $m_o$ is the rest mass.

While not necessary, we can keep this product explicit in the following rearrangement: $$a=\frac{F}{\gamma^2 (\gamma m_o)}$$

The denominator goes to infinity as speed $v$ approaches $c$. (This is because $\gamma=1/\sqrt{1-v^2/c^2}$.) So it appears that the acceleration decreases as speed increases, and in the limit of $v\rightarrow c$, $a\rightarrow 0$. But since this limit can never actually be reached, I would say that acceleration is always non-zero for any physical object.

Other assumptions used: There is only one force acting on the object, denoted $F$.