Understand the relationship
The reasoning behind the result
Upper stages are payload during the lower-stage burn
Uᵢ = payload + Σⱼ>ᵢ(inertⱼ + propellantⱼ)
The first stage accelerates its own structure and propellant, every stage above it, and the final payload. Calculating a first-stage ratio from that stage alone overstates its contribution. The workbench builds carried upper mass backward from the payload, then evaluates burns forward in the entered firing order.
For stage i, initial mass is Uᵢ + inertᵢ + propellantᵢ. Burnout mass is Uᵢ + inertᵢ. The usable propellant is expelled during the burn; the inert mass remains attached until separation.
Burn, discard, then burn again
mf,ᵢ − inertᵢ = m₀,ᵢ₊₁
After an intermediate stage burns out, its inert mass is discarded. This reduces the mass that later stages must accelerate. The ideal model assigns no velocity increment to the separation itself.
The table exposes the continuity equation between adjacent stages. Inert mass on the final stage remains with the payload through the final burn; the mass accounting does not pretend that final structure disappears from its mass ratio.
Adding stage contributions
Δvtotal = Δv₁ + Δv₂ + … + Δvₙ
Each contribution uses that stage's own specific impulse and whole-vehicle burn ratio. The bar lengths show those contributions in the same velocity unit; the total is their arithmetic sum.
Sequential ideal delta-v is a scalar budget. A real trajectory includes directions, losses and changing operating conditions, so this sum does not predict an orbital speed, altitude or mission success.
Represent residual propellant honestly
Usable propellant is the mass actually expelled in the modelled burn. Propellant that remains trapped or reserved belongs with inert mass if it stays aboard until separation. The distinction affects burnout mass and the resulting ratio.
The model supports one to twenty sequential stages. Parallel boosters, cross-feed, overlapping burns and refuelling require different mass histories and are intentionally outside this workbench.
Follow the numbers
Check both mass transitions in a two-stage vehicle
- With payload 500 kg and upper stage 200 kg inert plus 1,800 kg propellant, the booster carries U₁ = 2,500 kg.
- The booster starts at 2,500 + 1,000 + 9,000 = 12,500 kg and burns out at 3,500 kg. Its contribution is 300 × 9.80665 × ln(12,500/3,500).
- After discarding 1,000 kg, the upper stage starts at 2,500 kg and burns out at 700 kg. Its contribution is 350 × 9.80665 × ln(2,500/700).
- Both ratios equal 25/7. Total ideal delta-v is 650 × 9.80665 × ln(25/7), approximately 8,114.293749 m/s.
The upper stage contributes more delta-v here because its specific impulse is higher, even though its propellant mass is smaller.
Quick guide
How to use this calculator
- Enter payload and list stages in firing order, from the first burn to the last.
- For each stage, enter inert mass, usable expelled propellant and constant specific impulse.
- Inspect the mass table: each lower stage carries all unburned upper stages and the payload.
- Compare stage contributions and their sum. No acceleration or speed gain is credited merely for discarding inert mass.
Calculation method
Calculation and interpretation
Model sequential, non-overlapping burns with stage separation, including the upper-stage mass carried by every lower stage.
m₀,i = upper-stage wet mass + payload + inertᵢ + propellantᵢ; Δvtotal = Σ Ispᵢ g₀ ln(m₀,i/mf,i)
Worked example
Check both mass transitions in a two-stage vehicle
The upper stage contributes more delta-v here because its specific impulse is higher, even though its propellant mass is smaller.
m₀,i = upper-stage wet mass + payload + inertᵢ + propellantᵢ; Δvtotal = Σ Ispᵢ g₀ ln(m₀,i/mf,i)
Supported inputs
Precision and limits
Ideal velocity budget
These are ideal, constant-effective-exhaust-velocity burns. Gravity loss, aerodynamic drag, steering loss, finite-burn orbital effects and engine transients are excluded. Delta-v is not an attained ground speed or a launch capability.
Mass definitions
Initial and final mass refer to the entire vehicle accelerated during one burn. Final mass includes structure, payload and any propellant that remains unburned. Expelled propellant is the difference between those masses.
Specific impulse convention
Specific impulse is entered in seconds. Its conversion to effective exhaust velocity uses standard gravity, 9.80665 m/s², even for a burn far from Earth. This constant is a unit convention, not a gravity-loss allowance.
Numerical and practical limits
A mass ratio must be at least one; final mass and exhaust velocity must be positive. A zero-propellant case has zero ideal delta-v. Extremely large required mass ratios may be numerically representable but physically impractical.
Supported staging
Sequential burns with complete intermediate-stage inert-mass discard only. No parallel boosters, cross-feed, overlapping burns, variable Isp or refuelling are modelled.
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