3,671.96
3.672
3.3333
0.7
7,000
3,671.96
3.672
3.3333
0.7
7,000
The Tsiolkovsky Rocket Equation Calculator computes the delta-v (velocity change) achievable by a rocket given its specific impulse and mass ratio. Derived by Konstantin Tsiolkovsky in 1903, this equation is the fundamental law governing all rocket propulsion — from model rockets to interplanetary spacecraft — and is the cornerstone of orbital mechanics and mission design.
The rocket equation arises from conservation of momentum. As a rocket expels propellant backward at high velocity, it accelerates forward. The relationship is: delta_v = Isp * g0 * ln(m0/mf), where Isp is the specific impulse (a measure of fuel efficiency in seconds), g0 is standard gravity (9.80665 m/s^2), m0 is the initial total mass (rocket plus propellant), and mf is the final dry mass (rocket without propellant). The natural logarithm of the mass ratio is the key quantity: doubling the mass ratio only adds 0.693*Isp*g0 to the delta-v, illustrating the diminishing returns of adding more propellant.
Delta-v is the most important currency of space mission design. Every orbital maneuver — launching to orbit, changing orbital altitude or inclination, traveling to another planet, landing, and returning — costs a certain amount of delta-v. Launching to low Earth orbit requires about 9,400 m/s. Reaching geostationary orbit from LEO requires an additional 4,200 m/s. A Hohmann transfer to Mars requires about 3,600 m/s from LEO. Reaching escape velocity from Earth's surface requires 11,200 m/s.
Specific impulse (Isp) determines how efficiently a propellant produces thrust. Chemical rockets typically achieve Isp of 250-460 seconds (kerosene/oxygen: ~311s, hydrogen/oxygen: ~450s). Ion thrusters achieve 1,000-10,000 seconds but produce very low thrust. Higher Isp means less propellant is needed for the same delta-v, which is why engineers constantly seek higher-Isp propulsion systems.
Tsiolkovsky equation: delta_v = Isp * g0 * ln(m0/mf), where g0 = 9.80665 m/s^2 is standard gravity. Mass ratio R = m0/mf. Propellant mass = m0 - mf. Propellant mass fraction = (m0 - mf)/m0. All units consistent in SI: Isp in seconds, masses in kg, delta_v in m/s.
Delta-v budget: LEO = 9,400 m/s from surface. GTO = +2,500 m/s from LEO. Lunar orbit = +3,200 m/s from LEO. Mars transfer = +3,600 m/s from LEO. A propellant mass fraction above 0.9 (90% propellant) is typical for large launch vehicles. Chemical rockets are mass-ratio limited; high-Isp electric propulsion is power-limited but extremely efficient for deep-space missions.
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With Merlin engines (Isp ~311s) and a mass ratio of about 17, the first stage contributes roughly 8.8 km/s. Gravity losses and atmospheric drag reduce effective delta-v.
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An ion thruster (Isp = 3000s) achieves 15.6 km/s delta-v using only 40% of the initial mass as propellant — highly efficient but slow to accelerate.
Delta-v (delta velocity, or change in velocity) is the total impulse per unit mass available from a rocket. It is measured in m/s or km/s and represents the capability to change the spacecraft's velocity. Every orbital maneuver costs delta-v. Engineers design missions to minimize total delta-v within constraints of launch vehicle capability.
The logarithm arises from integration of the momentum equation: as propellant is burned, the rocket gets lighter and accelerates more for each subsequent unit of propellant burned. The total velocity change is the integral of this varying acceleration, yielding the natural log of the mass ratio. This is why increasing mass ratio has diminishing returns.
Isp is the thrust produced per unit weight of propellant per second, measured in seconds. Higher Isp means more efficient use of propellant. Chemical rockets: kerosene/LOX ~311s, H2/LOX ~450s. Nuclear thermal: ~800-1000s. Ion thrusters: 1000-10000s. Hall effect thrusters: ~1500-3000s. Higher Isp allows smaller propellant loads for the same mission.
The tyranny of the rocket equation: to carry more propellant, you need more rocket structure, which increases dry mass, which requires more propellant. The extra delta-v gained by increasing mass ratio follows a logarithm, which grows very slowly. Doubling the mass ratio adds only Isp*g0*0.693 to delta-v. This is why staging (dropping empty tanks) is essential for reaching orbit.
Staging discards empty propellant tanks during flight. Each stage that is dropped reduces the effective dry mass for subsequent burns. Without staging, the mass ratio needed to reach orbit from Earth's surface would require impossibly large rockets. With two or three stages, mass ratios of 10-20 per stage are achievable with current technology.
g0 = 9.80665 m/s^2 is the standard acceleration of Earth's gravity at sea level. Specific impulse in seconds times g0 gives the exhaust velocity in m/s. Isp in seconds is used because it is mass-independent (the same regardless of the gravitational field where the engine operates), making Isp a universal measure of engine efficiency.
Yes. The same formula applies to ion thrusters, Hall effect thrusters, and other electric propulsion systems. Their very high Isp (1000-10000s) means a small mass ratio achieves large delta-v. However, electric thrusters have very low thrust and require long burn times (months to years), making them suitable for deep-space but not launch from Earth's surface.
Exhaust velocity (ve = Isp * g0) is the speed at which propellant exits the rocket nozzle. It is the fundamental measure of engine performance. For kerosene/LOX engines, ve is about 3050 m/s. For hydrogen/LOX, about 4400 m/s. Ion thrusters can achieve exhaust velocities of 10,000-100,000 m/s.
A complete Earth-Moon-Earth mission requires approximately: 9400 m/s to reach LEO, 3130 m/s for trans-lunar injection, 840 m/s for lunar orbit insertion, 1730 m/s for powered descent to the surface, 1730 m/s for lunar liftoff, 840 m/s to return to trans-Earth trajectory, and 900 m/s for Earth orbit insertion (optional). Total: about 18 km/s without aerobraking.
The rocket equation gives delta-v in vacuum with no gravity. Real launches include gravity losses (extra delta-v wasted fighting gravity during vertical ascent) typically 1000-1500 m/s, and atmospheric drag losses typically 100-400 m/s. The total delta-v needed to reach LEO is about 9400 m/s, roughly 1500 m/s more than orbital velocity of ~7900 m/s.
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