The science
The suicide burn
A booster falling at speed v must shed it before the ground. With net upward acceleration a - g at full thrust, the stopping distance is h = v^2 / (2 (a - g)) where a = T/m. Lighting the engine as late as possible saves propellant (fuel-optimal soft landing is maximum thrust last: Meditch, 1964) but leaves no margin: a half-second error is about 100 m of altitude.
Why it is not a simple formula
Mass falls as propellant burns, so a grows (Tsiolkovsky: dv = Isp g0 ln(m0/m1)). Engine thrust rises as air thins (T = T_sl + (p_0 - p) A_e, modelled as an Isp rise from sea level to vacuum at constant mass flow). Drag helps early in the burn. The engine takes about 0.7 s to spool up (thrust ramp), which costs about v/(2 rate) of altitude. The lab's "corrected formula" adds these and agrees with the simulation to about 1%; the textbook formula over-estimates by roughly 20%.
Throttle floor and hover-slam
A nearly empty booster weighs little, so even one engine at its minimum throttle has thrust-to-weight above 1 (here about 1.4 at 40%). It cannot hover; it can only slow down and, if it stops too early, climb. The guidance therefore tracks a braking curve v_ref(h) = sqrt(v_td^2 + 2 a_c h) with constant planned deceleration a_c = u T/m - g (u = planned throttle) and corrects with T/m = g + a_c + k (v - v_ref). On the phase plane: right of the red full-throttle curve you cannot stop; left of the amber floor curve the engine over-brakes and you hover-slam. Three engines have three times the floor thrust, so the usable window almost vanishes.
Air and wind
International Standard Atmosphere (1976) density and pressure. Drag 0.5 rho v^2 Cd A (Cd 1.05, A 10.75 m^2 in the engines-first attitude; terminal speed sqrt(2 m g / (rho Cd A)) is checked in the tests). A body at angle of attack feels a normal force that pushes against tilt (CN_alpha = 2/rad plus a cross-flow term) acting ahead of the centre of mass, which makes the descent weather-vane stable. Wind rises with height as a power law and is gusty.
Steering
Attitude uses a PD controller torque = I (Kp e - Kd w) shared out between grid fins (authority proportional to dynamic pressure, fading as the air thins), engine gimbal (needs thrust, limited to 5 degrees) and cold-gas thrusters. Sideways guidance is a damped position/velocity law that commands tilt; this is a teaching stand-in for the convex-optimisation guidance (G-FOLD, Acikmese and Ploen 2007; Acikmese et al. 2013; Blackmore 2016) used on real vehicles, which plans the whole burn rather than reacting.
Legs versus catch
Legs tolerate about 2 m/s down, 2.5 m/s sideways, 6 degrees tilt within 12 m of the deck centre. A tower catch must hold the grid-fin pins inside a box a few metres wide: this lab uses 3 m, under 1 m/s, under 2 degrees. The first booster caught by tower arms was reported in October 2024.
What is simplified
- Generic vehicle, round numbers; not any real booster's data.
- Constant gravity, no Earth rotation, no deck motion (a real barge heaves and pitches).
- No roll, no structural flexing, no engine start failures, no sensor noise.
- Landing burn only, starting at 3.5 km and 250 m/s; entry and boostback are not simulated.
Experiments
Ignition optimum: a full-throttle burn is bisected for the altitude that reaches the ground at zero speed and compared with both formulas; an independent explicit-Euler 1-D model (not sharing code) gives the same altitude to 0.02%. Monte-Carlo: random ignition timing, start position, thrust (1.5%), air density (4%) and gusts.