Suppose I am travelling 0.7c with respect to the Earth. My ships engines and fuel are also travelling at this speed. Shouldn't it be easy to accelerate to 0.8c without too much effort? How is this more energetically expensive than accelerating from rest to 0.1c? From the ships and the engines point of view, the ship is at rest. I'm confused about the difficulties of getting closer to light speed. With respect to whom?
Speeds are indeed relative, so you must compute the change in speed from the frame of the spaceship. In this case you wish to accelerate from 0.7c to 0.8c, both measured with respect to the Earth. By the relativistic velocity addition law, the velocity change is not 0.1c in the frame of the spaceship, but is 0.23c. In the spaceship frame, the energy required is given by the corresponding boost factor, which in this case is 1.028. The additional energy needed is thus 0.028mc2. The change in boost factor as measured from the Earth frame is Gamma(.8c) = 1.67 - Gamma(.7c)=1.40; the difference is 0.27, so the additional energy required will be 0.27mc2, as measured in Earth's frame. Why the difference? The straightforward answer is that a measurement of energy is relative. However, in either frame the amount of energy required to accelerate by the same factor will increase as the ship's speed increases, because of the dependence of the boost factor upon the square of the speed. Accelerating from 0.8c to 0.9c in the Earth's frame corresponds to an increase of 0.35c in the spaceship's frame. In the spaceship frame, the energy required for the increase is 0.06mc2, while in the Earth frame it is 0.68mc2; in both cases substantially more energy is required to go from 0.8c to 0.9c than to go from 0.7c to 0.8c.