So it all goes back to the original propeller/wheel vehicle that violates electro's misunderstood laws of physics.
I see. Thanks for letting me know!
This reminds me of the simple thought experiment on conservation of momentum and kinetic energy in elastic collisions.
Let's say you have a spaceship of mass M traveling at velocity V. There is a projectile of mass m and velocity v on the same trajectory (v > V, both in the same exact direction). They impact, but elastically, so that neither deforms, they just bounce without any losses. What are the resulting velocities V' and v'?
Conservation of momentum says that MV + mv = MV' + mv'. In a perfectly elastic collision, kinetic energy is also conserved, MV^2/2 + mv^2/2 = MV'^2/2 + mv'^2/2. Solving the system of two equations for V' and v' yields two answers: one is V'=V, v'=v, i.e. no change. The other is V' = (2mv+MV-mV)/(m+M), v' = (2MV+mv-Mv)/(m+M). (Feel free to check, e.g.
here.)
What happens when the projectile goes twice as fast as the ship, and weighs twice as much as the ship, i.e. m=2M, v=2V?
You work out the math, and out comes the unintuitive but physically correct and easily verifiable (using e.g. an air track) V'=7/3V≃2.333V, v'=4/3V≃1.333V.
In other words, the projectile loses one third of its velocity, and the ship gains four thirds; and the ship will end up traveling faster than the incoming projectile originally was.
If the velocities are a significant fraction of light speed, then one needs to switch to
generalized momentum, (Newtonian momentum multiplied by the Lorentz factor γ, p = γmv) and
relativistic kinetic energy (E=(1-γ)mc²) that are conserved, but at v<<c, the two yield the same answer to within rounding error.
This also answers the question,
"Can you accelerate a spaceship to a velocity higher than at which you can lob boulders at it?", with
"Yes. Just use boulders with more mass than the ship has."This is also the reason why one wants solar sails to be reflective, and not absorb the photons. If the solar sail absorbs the photon, the craft gains the momentum of the photon. However, if the solar sail reflects the photon, the craft gains up to two times the momentum of the original photon, depending on the angle of reflection, with maximum achieved when the photon is reflected back the way it came from.