WDS: THIS WHOLE "BUSSARD RAMJET" IDEA IS JUST IRREDEEMABLY STUPID. Schroeppel: Your argument assumes that we stop the hydrogen, react it, and use the reaction energy to eject helium. However, if you can capture the energy of the impinging hydrogen, and apply it to the departing helium, you have a net gain in the ship's momentum.
--Suppose the hydrogen is stopped and its energy is not lost, instead only a fraction F of its energy is lost. We then fuse the H to get He, which we eject out the back using both the fusion energy and the non-lost (1-F) fraction. Ship velocity = Vship. Outgoing velocity (if F=1) is Vfusion. Outgoing velocity (general F with 0<F<=1) is Vout. Assume relativistic effects small enough so we can use Newton laws. Vout^2 = Vfusion^2 + (1-F)*Vship^2 . The ship will keep accelerating while Vout>Vsip, but will stop accelerating when Vship=Vout. Solving, we find this happens when Vship[max possible] = Vfusion / squareroot(F). My old claim was implicitly assuming F=1. However, F=0 is not possible if perfect efficiency not possible. With any fixed F with 0<F<=1 there is an uppermost speed. I think 90% efficiency would be rather optimistic (the best heat engines humanity has only about 40% efficient, steam turbines, right?)... and would give ship at most sqrt(10)*Vfusion. Well actually, I think even 1% efficiency would be amazing and have no idea how to reach it. But anyhow: there is an upper limit. Meanwhile rockets, unlike jets, have no upper limit (if make the fuel tank big enough).