MY SIMULATOR

Staged Rocket: Split the Delta-v

Design the stages, then fly them to orbit.

INTERACTIVE 3D

Educational model of generic launch vehicles (point mass, non-rotating Earth) — not a design tool; masses and engine data are illustrative.

Ideal Δv

Stage 1 share

Structure

Pitch-over

Lab & science

Drag to orbit, wheel / pinch to zoom. Space = launch, R = reset.

Graphs (same as the panel)

Setup

Number of stages

Propellants (vacuum Isp, stage 1 / upper; sea-level Isp is lower)

Payload mass

Lift-off thrust-to-weight

Time warp

Presets

Camera

Graphics (High on every device; Low = smaller shadows, lower pixel ratio)

Design & flight readout (model values)

Experiments

1. Sweep the stage 1 share from 20 % to 80 % in 2.5 % steps: payload fraction and a full simulated ascent for each. 2. Compare 1 to 4 stages at the same delta-v and structure.

Missions

Mission 5: a stage with Isp 300 s burns until its mass has fallen to one tenth of the starting mass. What delta-v (km/s) does the rocket equation give? Enter a number.

The science

What you see

A rocket cannot reach orbit with one stage of ordinary propellants because most of the lift-off mass would be tanks that are still carried after they are empty. Dropping each empty stage lets the next one accelerate less mass. The bars and curves show how much payload survives for a given total delta-v, structure fraction and split between the stages. The flight uses the same numbers: first stage steered along a pitch programme, upper stage steered to flatten out near 180 km. Attitude is drawn along the velocity.

Equations

Rocket equation       dv = Isp g0 ln(m0 / mf)                    (g0 = 9.80665 m/s^2)
Stage with payload    m_p above, stage mass m = m_s + m_prop, eps = m_s / m
                      mass ratio r = (m_p + m)/(m_p + eps m)  =>  m = m_p (r - 1)/(1 - r eps)   (needs r eps < 1)
Equal stages          r = exp(dv_total / (N ve)),  payload fraction = [ (1 - r eps) / (r (1 - eps)) ]^N
Optimal split         identical stages: equal dv per stage (a higher Isp upper stage takes a larger share)
Flight (2-D)          dv/dt = T/m - D/m + g(r)        g = -mu r / |r|^3,  mu = 3.986004e14 m^3/s^2
                      D = 1/2 rho v^2 Cd A,   rho = 1.225 exp(-h/8500 m),   Cd = 0.3
Thrust                Isp(h) = Isp_vac - (Isp_vac - Isp_sl) exp(-h/H),   T = mdot Isp(h) g0
Velocity budget       v_final = dv_ideal - gravity loss - drag loss - pressure loss - steering loss
                      gravity loss = integral g sin(gamma) dt,  drag loss = integral D/m dt
Orbit                 e = specific energy 1/2 v^2 - mu/r,  a = -mu/(2 e),  perigee/apogee = a (1 -/+ e) - R_E

Assumptions & honest simplifications

Point-mass vehicle in a vertical plane, spherical non-rotating Earth (a real launch gains up to about 0.46 km/s from Earth's rotation), exponential atmosphere with a single scale height, constant drag coefficient, no wind, no structural loads or max-q throttling.
Steering is a simple programme, not an optimal guidance law: the first stage follows a pitch schedule from vertical to the chosen angle by 60 km; the last stage flattens linearly to horizontal at 180 km. It reaches orbit for sensible designs but is not a minimum-loss trajectory; some designs that could reach orbit with better steering show as sub-orbital here.
Isp values (310/350 s kerosene, 330/365 s methane, 310/440 s hydrogen upper stage) and structure fractions are typical order-of-magnitude values, not data of any real vehicle. Sea-level Isp is taken as 90 % (stage 1) and 80 % (upper stages) of vacuum. Upper-stage thrust-to-weight at ignition is 1.0.
Stage length is drawn from propellant mass with a bulk tank density of 800 kg/m^3 and a diameter that grows with the cube root of lift-off mass; shapes are generic and not any real rocket. Fairing jettison at 110 km is visual only. Stage separation adds a 2 s coast.

Independent numerical checks

Also in node: node check.js.

Sources

K. E. Tsiolkovsky (1903) rocket equation; standard multistage-rocket optimisation (equal-ratio / Lagrange-multiplier staging) as in Sutton & Biblarz, Rocket Propulsion Elements, and Wertz & Larson, Space Mission Analysis and Design; typical gravity and drag losses of 1.3-1.8 km/s for launches to low Earth orbit are common textbook figures. Generic models only; no company or vehicle is represented.

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