jbriggs444 said:
No. This is not correct.
The context here is the question of how a rocket can succeed in adding kinetic energy to the payload at a higher rate when going from 100 mph to 200 mph than it can when going from 0 mph to 100 mph.
The fuel consumption is identical for both situations.
... to do the calculations. We have a rocket of mass ##M+m##, which expels a small mass ##m## at speed ##v## relative to the rocket. By conservation of momentum, the rocket will accelerate by approximately ##\frac m M v##. And the total gain in KE is $$\Delta KE = \frac 1 2 mv^2(1 + \frac m M) \approx \frac 1 2 mv^2$$
If the rocket expels a second mass ##m## with the same relative speed, then we have the same increase in speed and the same gain in KE. In this case, we can see this by changing to the new rest frame of the rocket and observing that we have an identical scenario to the first expulsion. To be precise the acceleration will be slightly greater, as the mass of the rocket has reduced to ##M##. As the rocket reduces in mass, these continual mass expulsions will become ever more efficient.
The difference from the previous analysis where the Earth was involved is that we do not have the external large mass to take into account when we change frames.
We can check by doing the analysis in a frame of reference where the rocket has any speed ##u##.
Initial momentum ##p_0 = (M+m) u## and initial KE ##E_0 = \frac 1 2 (M+m) u^2##.
After the expulsion, we have momentum ##p = M(u + \Delta u) - m(v-u)##, hence by conservation of momentum we have (as before) $$\Delta u = \frac m M v$$ And the change in KE is:
$$\Delta E = \frac 1 2 M (u + \Delta u)^2 + \frac 1 2 m(v-u)^2 - \frac 1 2 (M+m) u^2$$
Which after some simplification gives, as before:
$$\Delta E = \frac 1 2 mv^2(1 + \frac m M)$$
Again we see that changing the reference frame does not change the increase in KE.