Rocket Equation Calculator (Tsiolkovsky)

What is the Tsiolkovsky rocket equation?

The Tsiolkovsky rocket equation describes how much velocity change (delta‑v) an ideal rocket can achieve from a given amount of propellant and exhaust velocity. It assumes:

• Constant exhaust velocity relative to the rocket
• No external forces (no drag, no gravity losses)
• Instantaneous or very long, low‑thrust burns in free space

Specific impulse and exhaust velocity

Rocket engines are often specified by specific impulse \(I_{sp}\) instead of exhaust velocity. Specific impulse is the thrust per unit weight flow of propellant and has units of seconds.

Mass ratio and propellant fraction

Two useful derived quantities are the mass ratio and the propellant fraction:

For a fixed exhaust velocity, delta‑v grows only with the logarithm of the mass ratio. This is the famous “tyranny of the rocket equation”: to gain a bit more delta‑v, you need exponentially more propellant.

How this rocket equation calculator works

This tool is more flexible than a simple one‑direction calculator. It can:

• Solve for delta‑v from masses and engine performance
• Solve for required mass ratio or propellant mass for a target delta‑v
• Convert between exhaust velocity and specific impulse
• Estimate burn time from thrust or mass flow rate

Core computation steps

1. Convert all inputs to SI units (kg, m/s, N, s).
2. Infer missing masses using \(m_p = m_0 - m_f\) when possible.
3. Infer \(v_e\) from \(I_{sp}\) and \(g_0\), or vice versa.
4. Apply the rocket equation to solve for the unknown among \(\Delta v\), \(v_e\), and \(R = m_0/m_f\).
5. Compute propellant fraction \(\phi\), mass flow rate \(\dot{m}\), and burn time \(t\) if thrust is provided.

Example: LEO launch stage

Suppose you have a stage with:

• Initial mass \(m_0 = 500{,}000\ \text{kg}\)
• Final mass \(m_f = 100{,}000\ \text{kg}\)
• Specific impulse \(I_{sp} = 300\ \text{s}\)

Then:

• Mass ratio \(R = m_0/m_f = 5\)
• Exhaust velocity \(v_e = 300 \cdot 9.80665 \approx 2942\ \text{m/s}\)
• Delta‑v \(\Delta v = 2942 \ln(5) \approx 4{,}735\ \text{m/s}\)

This is not enough for orbit by itself, but it could be one of multiple stages. Use the presets and adjust masses to explore realistic scenarios.

Limitations and real‑world considerations

• Gravity and drag losses: Real launches lose 1–3 km/s of effective delta‑v to gravity and atmospheric drag.
• Throttling and staging: Engines do not always run at constant thrust; staging changes mass abruptly.
• Non‑collinear thrust: Steering losses occur when thrust is not perfectly aligned with velocity.

Despite these simplifications, the Tsiolkovsky rocket equation remains the fundamental first‑order tool for sizing rockets and understanding why high‑Isp engines and staging are so valuable.

FAQ

Can I use this for ion or electric propulsion?

Yes. Ion engines have very high specific impulse (often 2,000–4,000 s), which you can enter directly. Just remember that real ion missions are low‑thrust and require trajectory optimization beyond the simple rocket equation.

How do I model a multi‑stage rocket?

Treat each stage separately: for each stage, enter its initial and final mass and engine performance, compute its delta‑v, then sum the delta‑v values of all stages to get the total.

Why does my required mass ratio explode at high delta‑v?

Because delta‑v grows with the logarithm of mass ratio, the inverse relationship is exponential:

For missions like interstellar travel, even tiny increases in target delta‑v can require enormous increases in propellant mass, which is why alternative concepts (staging, refueling, beamed propulsion, sails) are explored.