Design a fictional two-stage rocket, pass the pre-launch go / no-go check, fly it through the atmosphere layers and put a small science satellite on a circular orbit.
Specific impulse +-1.5 % (1 sigma), drag coefficient +-10 %, upper wind +-4 m/s around the slider value.
The science
Rocket equation. Δv = Isp g₀ ln(m₀/mf) for each stage; the stage Δv add. Staging pays because the empty tank mass of stage 1 is thrown away before stage 2 has to accelerate it. This design carries about 10.2 km/s of ideal Δv.
Why 10 km/s when orbit needs 7.7 km/s? Circular speed at radius r is vc = √(μ/r), 7.73 km/s at 300 km. Gravity loss ∫ g sinγ dt (about 1.7 km/s here), drag loss and steering loss make up the difference; they are counted in the budget panel.
Thrust depends on altitude. F = ṁ Isp,vac g₀ − pa(h) Ae; the sea-level Isp (285 s) is lower than the vacuum value (315 s) because ambient pressure acts on the nozzle exit area.
Atmosphere and loads. ISA 0 to 86 km (piecewise-linear temperature, barometric formula), exponential with a 5.9 km scale height above. Drag D = ½ ρ v² CD(M) A with a transonic drag rise; dynamic pressure q = ½ ρ v² peaks at max-Q near 64 s. The engine is throttled back when q approaches its limit.
Equations of motion. Planar, spherical non-rotating Earth: r̈ = −μ r̂/r² + (T − D)/m, RK4 with dt = 0.05 s while the engines burn and velocity Verlet (leapfrog, symplectic) in the vacuum coast. Mass flows at ṁ = F/(Isp g₀).
Guidance. Stage 1: vertical rise, a kick, then thrust along the velocity (a gravity turn) but never steeper than a linear pitch programme. Stage 2: closed-loop terminal guidance that commands the vertical acceleration a = 6Δh/t² − 4vr/t (double integrator) so the stage arrives at the target height with zero vertical speed; cut-off when the specific orbital energy reaches that of the target circle. The orbit is then a = −μ/(2ε), e = √(1 + 2εh²/μ²).
Simplifications. Fictional vehicle, not a copy of any real rocket; 2-D flight plane, no Earth rotation or J2, no structural or aero-stability model, no stage-1 recovery, wind modelled only as drag on the relative airspeed, round-number drag, mass fractions and thrust values. Not a design tool. Peaceful science mission only. Models built with Blender, scene by Three.js.