The Tsiolkovsky rocket equation

Konstantin Tsiolkovsky wrote it down in 1903, before anyone had built a serious rocket. It's still the single most important equation in space flight:

The total velocity change a rocket can deliver equals its exhaust velocity times the natural log of the mass ratio (start-mass over end-mass). That's all. No propellant left, you're done.

The logarithm is the cruel part. To double your $\Delta v$, you don't double your propellant — you have to square your mass ratio. From $m_0/m_f = 4$ to $\Delta v = 2 v_e \ln 2$, you'd need $m_0/m_f = 16$. That means 94% of your starting mass is propellant. For a hydrogen/oxygen upper stage ($v_e \approx 4.4$ km/s) this is hard but possible. For a model rocket ($v_e \approx 0.8$ km/s) it's why you don't reach orbit in your backyard.

Try it

Slide the propellant mass fraction toward 0.9, watch $\Delta v$ jump. Slide it to 0.95 (most launchers don't get there), notice the diminishing returns.

Why staging exists

If you can't push the mass fraction high enough in one stage, you drop empty stages. Each stage is a mini-rocket with its own $\Delta v$ budget, and the total $\Delta v$ is the sum. A two-stage rocket with mass fraction 0.85 each gets you:

vs a single stage with the same total mass fraction (which would just be $\ln(1/0.15) \approx 1.9 v_e$).

That's nearly twice as much $\Delta v$ for the same fuel, just by not dragging an empty tank along for the second half of the climb. Chapter 7 makes this concrete.