# Tsiolkovsky rocket equation The Tsiolkovsky rocket equation is a mathematical equation that describes the motion of vehicles that follow the basic principle of a rocket. A device that can apply acceleration to itself using thrust by expelling part of its mass with high velocity can thereby move due to the conservation of momentum. The equation is named after Russian scientist Konstantin Tsiolovsky who independently derived it and published it in his 1903 work.

### About Tsiolkovsky rocket equation in brief The Tsiolkovsky rocket equation is a mathematical equation that describes the motion of vehicles that follow the basic principle of a rocket. A device that can apply acceleration to itself using thrust by expelling part of its mass with high velocity can thereby move due to the conservation of momentum. The equation is named after Russian scientist Konstantin Tsiolovsky who independently derived it and published it in his 1903 work. It had been derived earlier by the British mathematician William Moore in 1810, and later published in a separate book in 1813. Robert Goddard in America independently developed the equation in 1912 when he began his research to improve rocket engines for possible space flight. Hermann Oberth in Europe independently derived the equation about 1920 as he studied the feasibility of space travel. It can also be derived from the basic integral of acceleration in the form of force over mass. According to Newton’s Second Law of Motion, the propelling force on a rocket is equal to its current acceleration at any time. The rocket is subject to a constant force but its total mass is decreasing steadily because it is expelling gas. This creates a force propelling the rocket that is constant but decreasing in speed. The value m0−m1 is the total mass of propellant expended, and hence: where Mf{displaystyle Mf}} is the propellant mass fraction.

If thrust is the only force involved, the integral is found to be: Realising that impulse over the change in mass is equivalent to force mass, which is itself equivalent to exhaust velocity,the integral can be equated to a rocket at rest with no forces exerted on it. From its engine is started the rocket expels gas mass at a constant flow rate. Now, the rocket velocity relative to the exhaust is relative to p × ve. The rocket has initially been started and at the time of launch the mass of the fuel on board is equal to m0 – mf – ve – the speed at which the fuel is expelled from the rocket. This is the rate at which it is expelled into space. In free space, for the case of acceleration in the direction of the velocity, this is the increase of the speed. Of course gravity and drag also accelerate the vehicle, and they can add or subtract to thechange in velocity experienced by the vehicle. Hence delta-v is not usually the actual change in speed or velocity of the vehicle and it is usually not known how fast the rocket is moving. In the following derivation, ‘the rocket’ is taken to mean “the rocket and all of its unburned propellant’