Rocket Mass Ratio from Delta-v

This is the rocket equation asked from the direction a designer actually works in. The mission states the delta-v; the engine states the exhaust velocity; the question is what fraction of the vehicle has to be propellant. Invert the logarithm and $MR = e^{\Delta v / v_e}$, and the exponential is where the bad news lives.

Work an example. A kerolox engine at $v_e = 3000$ m/s asked for 9000 m/s of delta-v needs $MR = e^{3} = 20.1$: the vehicle must be 95% propellant, leaving 5% for tanks, engine, plumbing, avionics, structure AND payload combined. Nobody has ever flown that in a single stage. Ask the same engine for 12,000 m/s and the ratio becomes $e^{4} = 54.6$, which is 98.2% propellant — a number that is not merely difficult but physically absurd, because the tanks alone weigh more than that.

Notice which lever moves the answer. Improving the engine to $v_e = 4400$ m/s, the hydrolox figure, drops the 9000 m/s requirement from a ratio of 20.1 to $e^{2.045} = 7.7$ — from 95% propellant to 87%, which is a hard vehicle rather than an impossible one. Structural cleverness cannot do that, because it fights inside the exponent while a better engine changes the exponent itself. This asymmetry is why a few seconds of specific impulse are worth arguing about for years.

Two cautions when reading the answer. The mass ratio is $m_0/m_f$, not the propellant fraction — the propellant fraction is $1 - 1/MR$, and confusing the two turns a ratio of 10 into a claimed 10% propellant when it is really 90%. And the delta-v you feed in must already include the gravity and drag losses of the ascent, or the ratio comes back optimistic by the 1.5 to 2 km/s those losses cost.