Rocket Equation Calculator

Tsiolkovsky Rocket Equation Calculator

Standard chemical rockets usually range from 250 to 450s.

The total mass of the rocket including fuel (kg or tons).

The mass of the rocket after fuel is consumed (kg or tons).

Calculation Result

Δv = 0 m/s

(Equivalent to 0 km/s)

Understanding the Rocket Equation

The Tsiolkovsky Rocket Equation is the fundamental principle behind spaceflight and orbital mechanics. Developed by Konstantin Tsiolkovsky in 1903, it describes the maximum change in velocity (Delta-v) a vehicle can achieve based on its exhaust velocity and the ratio of its initial and final mass.

The Mathematical Formula

Δv = v_e * ln(m₀ / m_f)

Where:

• Δv (Delta-v): The total change in velocity. This is the "budget" used to reach orbit or travel between planets.
• v_e (Exhaust Velocity): How fast gas leaves the nozzle. Calculated as I_sp × g₀ (where g₀ ≈ 9.80665 m/s²).
• m₀ (Initial Mass): Total mass, including propellant.
• m_f (Final Mass): Mass after the propellant has been burned (structure, engines, payload).
• ln: The natural logarithm.

Why Delta-v Matters

Delta-v is the most critical metric in mission planning. Unlike cars that measure travel in distance (kilometers or miles), spacecraft measure travel in Δv. For example, to reach Low Earth Orbit (LEO), a rocket needs approximately 9,400 m/s of Δv. Because of the logarithmic nature of the equation, increasing your Δv requires an exponential increase in fuel, which is often referred to as "the tyranny of the rocket equation."

Practical Example

Suppose you have a small orbital stage with the following specs:

• Specific Impulse (I_sp): 320 seconds
• Initial Mass (m₀): 5,000 kg
• Final Mass (m_f): 1,500 kg

Step 1: Calculate effective exhaust velocity: 320s × 9.80665 m/s² = 3,138.128 m/s.

Step 2: Calculate the mass ratio: 5,000 / 1,500 = 3.333.

Step 3: Apply the natural log: ln(3.333) ≈ 1.204.

Step 4: Final Δv: 3,138.128 × 1.204 = 3,778 m/s.

function calculateDeltaV() { var isp = parseFloat(document.getElementById("ispValue").value); var m0 = parseFloat(document.getElementById("initialMass").value); var mf = parseFloat(document.getElementById("finalMass").value); var g0 = 9.80665; // Validation if (isNaN(isp) || isNaN(m0) || isNaN(mf)) { alert("Please enter valid numeric values for all fields."); return; } if (mf <= 0) { alert("Final mass must be greater than zero."); return; } if (m0 < mf) { alert("Initial mass (wet mass) must be greater than or equal to final mass (dry mass)."); return; } // Calculations var ve = isp * g0; var massRatio = m0 / mf; var deltaV = ve * Math.log(massRatio); var deltaVKm = deltaV / 1000; // Display Results document.getElementById("deltaVResult").innerText = deltaV.toLocaleString(undefined, {minimumFractionDigits: 2, maximumFractionDigits: 2}); document.getElementById("deltaVKmResult").innerText = deltaVKm.toLocaleString(undefined, {minimumFractionDigits: 4, maximumFractionDigits: 4}); document.getElementById("exhaustVelocityText").innerText = "Effective Exhaust Velocity (ve): " + ve.toFixed(2) + " m/s"; document.getElementById("resultArea").style.display = "block"; // Scroll to result smoothly document.getElementById("resultArea").scrollIntoView({behavior: 'smooth', block: 'nearest'}); }