LABORATORIES / TIME-DEPENDENT FLIGHT MODEL
See burn, separation and free flight
on the same timeline.
Calculate vertical motion of an example one- or two-stage vehicle with changing mass, thrust, gravity and drag. Pause or rewind time and investigate charts alongside vehicle state.
You start playback yourself. Leaving the view pauses it. The rocket and atmosphere are schematic; profile scale follows the highest calculated altitude.
Interpret results within the numerical model’s limits.
Altitude km
A time chart appears after calculation.
Vertical velocity m/s
A time chart appears after calculation.
Net vertical acceleration m/s²
A time chart appears after calculation.
Total carried mass kg
A time chart appears after calculation.
Dynamic pressure q kPa
A time chart appears after calculation.
Measurements at the selected time
The chart’s vertical line and values below refer to the same model time. Positive velocity is upward; negative is downward.
- Net acceleration
- —
- Dynamic pressure
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- Thrust
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- Stage 1 remaining propellant
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- Stage 2 remaining propellant
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- Air density
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Open burn and flight events
| Event | Time (s) | Altitude (km) | Vertical velocity (m/s) | Mass (kg) |
|---|---|---|---|---|
| No calculation yet. | ||||
Open the sampled time-series table
The table shows 10-second samples and event times. Integration uses smaller steps; charts sample that solution. Summary peaks are scanned across all accepted integration points and events, not just this table; they are not exact analytical extrema of a continuous curve.
| Time (s) | Altitude (km) | Vertical velocity (m/s) | Acceleration (m/s²) | Mass (kg) | q (kPa) |
|---|---|---|---|---|---|
| No calculation yet. | |||||
Explore the same inputs together.
The link contains only numeric vehicle inputs from the last calculation. The recipient starts calculation on their own device.
02 / READ THE EQUATIONS AND LIMITS
Changing mass, forces calculated together.
dh/dt = v
dv/dt = [F − ½ρCdA v|v|] / m − μ / (R + h)²
dmpropellant/dt = −F / (Isp · g₀)
ρ(h) = 1,225 · exp(−h / 8.500) kg/m³ · q = ½ρv²
Upward is positive. Drag opposes motion and therefore acts upward during descent. Thrust and effective specific impulse are constant within each stage. Engines stop when propellant ends; in the two-stage model, first-stage dry mass is discarded instantly and the second engine starts without delay. If liftoff has not occurred before first-stage burnout, calculation ends on the pad without upper-stage ignition. Payload and the unused second stage are carried throughout the first burn.
Motion is solved with fourth-order Runge–Kutta using steps no larger than 0.05 seconds. Comparing one full step with two half steps controls local numerical error; steps shrink when needed, especially at high thrust and drag. Steps also shorten at propellant exhaustion and sampling boundaries. Trial states failing accuracy checks are excluded from charts and results. Apogee and ground intersection are located within smaller intervals. The model stops after 1,800 seconds or at 20,000 km altitude. If a time or altitude limit stops it, reached values are shown without claiming apogee was found.
The atmosphere uses sea-level density 1.225 kg/m³ and constant scale height 8,500 m in a simplified exponential model, not an ISA atmosphere. It does not represent actual high-altitude variations in temperature, composition and density. Constant Cd excludes Mach effects, heating, shape changes, breakup and re-entry aerothermodynamics. Descent towards the ground is therefore a continuation of the educational model, not a landing-system demonstration.
Earth is spherical and non-rotating, with μ = 3.986004418 × 10¹⁴ m³/s², R = 6,371,000 m and g₀ = 9.80665 m/s². Wind, horizontal velocity, lift, attitude, flight control, pressure-dependent engine performance, startup transients, separation impulses, parachutes and post-contact motion are excluded. If initial thrust does not exceed weight, the vehicle remains on the pad; propellant consumption can change that balance.
How should results be interpreted?
- Altitude is vertical height above the ground, not evidence of orbital access.
- Vertical speed is zero at apogee. This establishes neither an orbit nor a weightlessness achievement.
- q is not a structural strength limit. Ascent and descent may have different peaks; the chart displays the entire calculated flight.
- Mass changes discontinuously at separation; speed remains continuous because no separation impulse is assumed.