# Where does Energy get lost in the Rocket Ship propulsion vs "direct acceleration" scenario? [closed]

The scenario is as follows. In a hypothetical future drive I spend energy E to accelerate a mass of propellant m to velocity v (e.g., plasma state propellant in the very powerful circular accelerator), so $$v = \sqrt {2E/m}$$.

Then I use my accelerated propellant as exhaust for my jet engine, using all mass in time t, getting thrust:

$$T = vm/t$$

Then my spaceship of mass M will be accelerated to speed (assuming m<<M):

$$V_1 = Tt/M = vm/M = \sqrt {2E/m}*m/M = \sqrt {2Em/M^2}$$

Now, if I spend the same energy E directly to accelerate my ship (by some unknown mechanism but transforming all energy E into kinetic energy of the ship), I get: $$E = MV^2 / 2$$ and $$V_2 = \sqrt {2E/M}$$

So, $$V_2/V_1 = \sqrt{M/m}$$ and my ship's velocity in the 2nd case is much bigger than in the first, while I seem to have spent the same amount of Energy in both excercises.

I am sure I am missing something very obvious but for the life of me can't figure it out, my physics became a bit rusty I guess :( Any pointers appreciated - why this difference? where does the energy "disappear" in the 1st case?

• What's the energy in the mass of the "exhaust"? en.wikipedia.org/wiki/Tsiolkovsky_rocket_equation Oct 3 at 11:21
• This sounds like an illustration of the Oberth effect. Oct 3 at 11:26
• BTW, your second method is unphysical because it doesn't conserve momentum. Oct 3 at 12:33
• @PM2Ring how about I spend energy into curving the geometry of space-time and in effect the ship is accelerated by this artificial gravity field? Or even more realistically, the whole impulse-nuclear-explosion design, where we detonate small nuclear devices and the resulting energy is caught by a "sail" of sorts as here wikiwand.com/en/Nuclear_pulse_propulsion
– J Ho
Oct 3 at 13:18
• @JHo Spacetime curvature is fully determined by the stress-energy-momentum tensor. There are various ways to modify that tensor, but you always have to conserve momentum & energy locally (global energy conservation isn't well-defined in GR). You can shuffle energy around, but you can't make it magically appear or disappear. Rest mass is a very concentrated form of energy, and if you convert mass to a less concentrated form of energy you'll reduce the curvature. Oct 3 at 16:54