There is a spaceship, whose mass is $100 \, \mathrm{kg}$. The thrust of its rocket is $300 \, \mathrm{N}$.
How is it possible to calculate the maximal speed that the spaceship can reach, and the time it takes to approach it? We can assume that the mass of the spaceship remains the same, and the rocket lasts forever.
I know objects cannot accelerate forever, but I have no idea how to find the limits.
If the rocket maintains the same rest mass and also the same thrust in the center of mass system, its terminal velocity can be as close to the speed of light as you wish. It will just take forever.
It is interesting to ask when a given velocity is reached. Say the thrust remains the same all the time in a stationary frame. The mass of the rocket will grow like $m_0 \gamma$ where $\gamma = 1/\sqrt{1 - v^2/c^2}$ is the Lorentz factor. Then you can write $\dot p = F$ where $p$ is the momentum and $F$ is the thrust. Also $p = m v = \gamma(v) m_0 v$. This leads to the equation $$F = \dot p = m_0 \frac{\mathrm d}{\mathrm d t} \gamma(v(t)) \, v(t) \,.$$ Inserting the values, one has the differential equation $$\frac{\mathrm d}{\mathrm d t} \gamma(v(t)) \, v(t) = \frac{100 \, \mathrm N}{100 \, \mathrm{kg}} := a_0 $$ which one has to solve for $v(t)$ with $v(0) = 0$. As the other side does not depend on time explicitly, one can just integrate both sides from $t' = 0$ to $t' = t$. Mathematica gave the this solution: $$ v(t) = \frac{a_0 c t}{\sqrt{a_0^2 + c^2 t^2}} \,. $$
This is the code that I have used:
gamma[v_] = 1/Sqrt[1 - v^2/c^2]
DSolve[{D[gamma[v[t]] v[t], t] == a0, v[0] == 0}, v, t]
This is the corresponding plot:
You can see that the acceleration is constant (linear part) at first. Then your rocket drive has diminishing returns and it will asymptotically approach the speed of light.
It depends on what other forces act on the spaceship.
If the spaceship starts from rest on the surface of the earth, pointing up, its maximum speed will be 0, no matter how long it thrusts. That's because the force of gravity that pulls the spaceship down to earth is 980 Newtons (read the definition of N in wikipedia), which is greater than the thruster.
If the thrust of the rocket is the strongest force acting on the spaceship, then the spaceship will accelerate and approach arbitrarily close to the speed of light but never reach it.
