Rocket Landing Burn and Tower Catch Simulator

The burn uses NASA Glenn's definitions $v_e=I_{sp}g_0$ and $q=F/v_e$, with selected constant thrust $F=uT_{full}$ and $g_0=9.80665\,\mathrm{m/s^2}$. Mass is $m(t)=m_0-qt$. Positive speed is downward.

During the zero-thrust delay, $v_{ign}=v_0+g_0t_d$ and the fall distance is $v_0t_d+g_0t_d^2/2$. During powered braking,

A bracketed root finds $v(t_b)=v_c$. Integrating this velocity gives the selected-thrust ignition height:

A small-$z$ series prevents cancellation. Command height adds the delay distance. Propellant use is $qt_b$. The initial-mass comparison is $h_{cm}=(v_{ign}^2-v_c^2)/(2a_{start})$.

At contact, thrust stops and mass stays fixed. Work-energy gives mean net stopping acceleration $a_c=v_c^2/(2s)$ and stroke-averaged support $F_{mean}=m_c(g_0+a_c)$. Support work is $\tfrac12m_cv_c^2+m_cg_0s$. The assumed peak support is $m_c(g_0+k_{peak}a_c)$, shared equally between two arms. Felt load is support or thrust divided by $mg_0$; upward net acceleration is smaller by $g_0$.

The catch animation uses constant mean deceleration, $t_c=2s/v_c$; the peak factor estimates a load envelope only. At zero contact speed no stroke is used and support equals weight. The exact distance integral and catch model are derived here from Newton's law and work-energy. Blackmore (2016), pp. 16–18 (attached article pp. 2–4), supplies precision-landing context, not tower parameters or this simplified trajectory algorithm.

• Illustrative booster study: ignition-command mass 230,000 kg; explore 180,000–350,000 kg. Usable propellant 35,000 kg; explore 20,000–60,000 kg. These are analyst assumptions, not SpaceX measurements. Mass includes all propellant; usable propellant excludes protected reserves.
• Combined full-throttle thrust 7,355 kN approximates three engines at SpaceX's published 250 tf per engine (about 2.45 MN each). Explore 6–9 MN. Selecting engines, throttle limits, and operating conditions remains the user's responsibility; the published engine rating does not establish landing thrust.
• Selected throttle 0.65; illustrative sweep 0.4–1.0. Effective specific impulse 330 s; explore 300–350 s. Neither is a measured flight profile. Constant thrust and specific impulse are assumed throughout one final burn.
• Downward command speed 180 m/s; study 100–250 m/s. Delay 0.5 s; study 0–1 s. Delay consumes no propellant and ends with an instantaneous thrust step.
• Contact speed 2 m/s; study 0–5 m/s. Effective vertical stroke 0.5 m; study 0.1–1 m. Peak-to-stroke-mean net stopping-force multiplier 1.5; study 1–2. These are assumed catch parameters, not published tower capabilities.
• Upright vertical motion, constant Earth gravity, no aerodynamic force, no lateral or rotational energy. Engines stop at first arm contact. Two arms share vertical loads equally. Shape dimensions and vertical travel in the schematic are illustrative.
• Ranges above are sensitivity-study suggestions, not statistical confidence intervals. Input limits are numerical study bounds, not hardware qualification limits.

Use for preliminary final-approach sensitivity studies under the stated assumptions. Selected braking thrust must exceed initial weight, contact speed must be lower than command speed, and propellant use must not exceed the usable supply. The solver may calculate a hypothetical burn beyond available propellant to explain the shortage; error conditions withhold result cards for that case.

Aerodynamic drag is deliberately omitted. It can be significant for Earth landings, as Blackmore discusses, so the ignition result is an idealized vertical requirement rather than a flight instruction. There is no trajectory optimization, engine switching, throttle schedule, thrust-vector control, lateral capture envelope, ship flip, tower dynamics, load rating, stress analysis, rebound, or structural certification.

Peak force depends on an assumed multiplier, not a solved force history. Catch duration and animation follow mean constant deceleration; unequal sharing and contact shock can give higher arm loads. With zero contact speed, support is static and no catch stroke is traversed. The constant-mass comparison is an initial-mass braking benchmark, not the main solution. Published reference checks validate initial propulsion and that comparison; properties and a separate numerical integration audit test the variable-mass solution. Flight schedules and mission-specific masses are not asserted.