Let's start with kinetic energy (from los Wikipedias)
The kinetic energy of an object is the energy which it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes. The same amount of work is done by the body in decelerating from its current speed to a state of rest. In classical mechanics, the kinetic energy of a non-rotating object of mass m traveling at a speed $v$ is $\frac{1}{2}mv^2$.
Let's say you & your bike have mass of 100kgs, then your kinetic energy at 10m/s would be
$$ E_a = 1/2 \times 100 \times 10^2 = 5000J = 5kJ$$
If you apply another 5kJ of energy, you don't get to 20m/s though, you only get to:
$$ E_b = 10000J =1/2 \times 100 \times V_b^2$$
$$\implies V_b = \sqrt{10000 / (1/2 \times 100)}
= √200
= 14.14m/s$$
Let's say you and a buddy are both coasting along at 10m/s though, from their perspective you've just burned 5kJ but only accelerated 4.1m/s, even though you seemed stationary.
Imagine you and your mate are in space drifting along together, at an unknown speed. Your mate fires his burners and accelerates away from you. There's a big screen on his ship showing how many joules of energy he just burned, and you can measure his resulting relative velocity just fine.
The question is, Will 5kJ of energy always produce 10m/s of relative velocity,,assuming 100kg spaceships?
If 5kJ always produces 10m/s, Why does the second 5kJ only produce 4.1m/s? What is going on here?
Best Answer
Without reading the rest of your question, I must first reply that one has nothing to do with the other.
Kinetic energy is frame dependent, just as velocity is.
Momentum is proportional to velocity and is frame dependent too, just as velocity is.
Now, looking at the body of your question:
Unknown speed relative to what? Unknown speed relative to Earth? Unknown speed relative to the solar system? Unknown speed relative to the CMB?
Relative to what? Relative to the initial inertial frame of reference before the acceleration? Or relative to some frame of reference in some arbitrary relative motion?
(The point of all these questions is to prompt you to think more clearly about your question in the hope that you'll come to the answer yourself...)