Newtonian physics give reasonably accurate results as long as you think of the space craft as the frame of reference. The problem arises if you think with respect to the starting point or the destination or a star about 13.7 billion light years away which is receding at 0.999c To reach 0.9991 c (with respect to that star) takes extreme amounts of fuel even if you are at zero speed with respect to Earth's poles.
The other problem is, if you need one ton of fuel to increase your speed by one kilometer per second, you need 100,000 tons of fuel to reach 1/3 the speed of light. Since you have to carry the fuel from the beginning, the fuel better be antimatter, or the one ton was not realistic. Neil