Tsiolkovsky rocket equation

The Rocket Equation of Space Physics describes a fundamental law of rocket propulsion by continuous ejection of mass support. The equation was first later erected in 1903 by Konstantin Tsiolkovsky and independent of him by Hermann Oberth and Robert Goddard.

Equation

Consider a single-stage rocket with initial mass and initial velocity zero. The engine stumble in the supporting mass of infinitesimally small portions and with constant speed. This assumption and the restriction to non-relativistic velocities is justified for chemical drives, see Specific Impulse. Other forces, such as gravity or friction will not be considered. Under these conditions, the Rocket Equation applies to the velocity of the rocket as a function of the rest mass ( the reduced to the spent fuel initial mass ):

Derivation

The mass of the rocket had already taken on and change now to a small viewing unit. The support material is in the reference system of the missile with the speed in the system of the observer with so ejected and thus transmits the pulse. Since no external forces act, the total momentum of the rocket and supporting mass is obtained:

And thus

This differential equation is now integrated indeterminate. Integration of the left-hand side yields ( an antiderivative of ). On the right side only has to be integrated, since it was assumed to be constant. It is calculated as the integral. The minus sign is replaced by the reciprocal of the logarithm of the additive constant C by a factor of C ', thus:

The initial condition is satisfied by, that

Consequence

The terminal velocity when the total fuel mass is expelled is,

So the greater the greater the discharge velocity and the smaller the residual mass, consisting of the load, the engine and structural material.

It is noteworthy that terminal velocities are larger than achievable. However, in order to reach speeds far beyond parts of the structure ( empty tanks ) or the engine ( booster) will go left, see multi-stage rocket. Clearly, the case of successive levels set, wherein the upper stage represent the payload of the lower stages.

Assume a two-stage rocket, the steps of which have a mass of 100 and 20 are ( in arbitrary units) and 90 % of fuel, ie structural masses of 10 and 2 have. The payload also amounts to 2 units. The Rocket Equation is applied twice, with the contributions of both levels add ( you can see that if you change the burnout of the first stage in the reference system in which the second stage is initially at rest ):

For comparison, the single-stage missile with the same fuel and structural mass:

Limitations

The influence of gravity is not considered in the Rocket Equation. For vertical missile launches, low heights of rise and neglecting air resistance

With the acceleration of gravity and the burning time. This formula is unsuitable to optimize the achievement of the orbit, because this is the thrust vector changes continuously.