🚀 Two-Stage Rocket — Tsiolkovsky Equation & Staging

Configure two rocket stages — thrust, specific impulse, propellant and dry mass — then launch. Watch stage separation, observe the ΔV budget in real time, and see why staging is key to reaching orbit.

🟠 Stage 1

Thrust 2 000 kN Isp 310 s Propellant mass 200 t Dry mass 20 t

🔵 Stage 2

Thrust 100 kN Isp 420 s Propellant mass 40 t Dry mass + payload 5 t

⏸ READY

Telemetry

Altitude—

Velocity—

Acceleration—

Mass now—

ΔV remaining—

Time—

Stage—

Tsiolkovsky:
ΔV = Isp·g₀·ln(m₀/m₁)
F_net = Thrust − mg − D
ρ(h) = ρ₀·e^−h/8500
Orbit ≈ 7800 m/s at 200 km

Why Two Stages?

The Tsiolkovsky rocket equation ΔV = I_sp·g₀·ln(m₀/m₁) shows that ΔV depends on the mass ratio m₀/m₁. Once Stage 1 exhausts its propellant, its heavy empty tanks are dead weight. Dropping them (staging) dramatically improves Stage 2's mass ratio and its ΔV contribution. A single stage carrying the same propellant and payload would need a huge structural fraction, reducing the mass ratio and making orbit nearly impossible. Real launch vehicles like the Falcon 9, Saturn V and Soyuz all use this principle. Reaching low Earth orbit requires about 9–10 km/s of ΔV including gravity and drag losses.

About this simulation

This simulator applies the Tsiolkovsky rocket equation — ΔV = I_sp·g₀·ln(m₀/m₁) — to a real two-stage launch. Each stage has its own thrust, specific impulse (Isp), propellant mass and dry mass, so you can see exactly how staging turns dead weight (empty Stage 1 tanks) into extra delta-v for Stage 2. A live physics loop integrates thrust, gravity that weakens with altitude, and exponential atmospheric drag to track altitude, velocity and remaining ΔV budget in real time, right up to the 200 km low-Earth-orbit target.

🔬 What it shows

An animated rocket climbs from the pad, burning Stage 1's propellant while thrust, gravity (which weakens with the inverse square of distance from Earth's centre) and drag act on it. Once Stage 1's propellant hits zero, the sim triggers a staging event — dropping the empty Stage 1 structure — after a 1.5-second separation coast, then ignites Stage 2. A side graph plots altitude and velocity history against the Kármán line (100 km) and orbital velocity (7,800 m/s at 200 km).

🎮 How to use

Drag the Thrust, Isp, Propellant mass and Dry mass sliders for Stage 1 (orange) and Stage 2 (blue) to design your own rocket — changing any slider while idle recalculates the total ΔV budget instantly. Hit "🚀 Launch!" to run the simulation, watch the status badge move through STAGE 1 BURN → STAGE SEP → STAGE 2 BURN → ORBIT REACHED (or FUEL DEPLETED/CRASH), and use "↺ Reset" to try a new configuration.

💡 Did you know?

The mass ratio m₀/m₁ inside the Tsiolkovsky equation is logarithmic, so a rocket needs to be mostly propellant just to gain a modest amount of ΔV — this is why real vehicles like the Falcon 9, Saturn V and Soyuz all discard empty stages rather than hauling their dead tanks all the way to orbit.

Frequently asked questions

What does the Tsiolkovsky rocket equation actually calculate?

It calculates the maximum velocity change (ΔV) a rocket stage can achieve from burning all its propellant: ΔV = Isp·g₀·ln(m₀/m₁), where m₀ is the stage's starting mass (propellant plus dry mass plus whatever it carries above it) and m₁ is its mass once the propellant is gone. In the sim, this formula runs separately for Stage 1 and Stage 2, and the two ΔV values are added to get the total budget shown in the telemetry panel.

Why does dropping Stage 1 improve performance instead of just carrying it along?

Because ΔV depends on the ratio of starting mass to ending mass, not on absolute mass. Once Stage 1's tanks are empty they are pure dead weight, so Stage 2's own mass ratio (and therefore its ΔV) is much better if it doesn't have to keep accelerating that empty structure. The simulator models this directly: at burnout, Stage 1's dry mass is subtracted from the total mass right when the "✂️ STAGE SEP" event fires.

What do the Thrust, Isp, Propellant mass and Dry mass sliders control?

Thrust (kN) sets how much force the engine produces; Isp (specific impulse, in seconds) sets exhaust efficiency, feeding into effective exhaust velocity ve = Isp·g₀; Propellant mass is the fuel burned away over the stage's burn; Dry mass is the leftover structure (tanks, engine) once propellant is spent. For Stage 2, dry mass also includes the payload. Each parameter can be tuned independently for Stage 1 and Stage 2.

Why does the rocket sometimes run out of fuel before reaching orbit?

Reaching the simulator's 200 km / 7,800 m/s orbit target needs the combined ΔV of both stages to overcome gravity losses and atmospheric drag, which the physics loop applies continuously as thrust minus weight minus drag. If your slider settings give a low total ΔV budget (shown in the ΔV bar) relative to what altitude and drag demand, the status badge will read "💨 FUEL DEPLETED" once Stage 2's propellant is exhausted below orbital velocity.

Why does drag matter less as the rocket climbs?

The simulator models air density with an exponential atmosphere, ρ(h) = ρ₀·e^(−h/8500), so density (and therefore drag, which scales with ½·ρ·C_d·A·v²) falls off rapidly with altitude. By the time the rocket nears the Kármán line at 100 km, drag has become negligible, which is why real launch vehicles spend their early, slow, low-altitude flight fighting drag the hardest.