9.17:
Rocket Propulsion In Empty Space – II
In space, a rocket accelerates by burning its fuel and ejecting the burned gases. The operation of the rocket follows the conservation of linear momentum. The ejected gases from the rocket have momentum in the opposite direction to that of the rocket's motion.
Here, the mass of the expelled gases is equal to the mass lost by the rocket. Dividing both the sides by dt and rearranging the equation, the expression for the acceleration of the rocket is obtained.
The acceleration of the rocket continuously increases as the mass of the rocket decreases for a constant velocity. The product of mass and acceleration is the force for the flight, which depends on the velocity of the expelled gases and the combustion rate.
Rearranging the equation for force and integrating the equation for the rocket's velocity with initial and final limits, the ideal rocket equation is derived.
The rocket achieves maximum velocity when the ratio mi over m is made as large as possible such that the fuel primarily constitutes the initial mass of the rocket.
9.17:
Rocket Propulsion In Empty Space – II
The motion of a rocket is governed by the conservation of momentum principle. A rocket's momentum changes by the same amount (with the opposite sign) as the ejected gases. As time goes by, the rocket's mass (which includes the mass of the remaining fuel) continuously decreases, and its velocity increases. Therefore, the principle of conservation of momentum is used to explain the dynamics of a rocket's motion. The ideal rocket equation gives the change in velocity that a rocket experiences by burning off a certain mass of fuel, which decreases the total rocket mass. This equation was originally derived by the Soviet physicist Konstantin Tsiolkovsky in 1897.
The total change in a rocket's velocity depends on the mass of the fuel that is being burned during the flight, which is not linear. Furthermore, the rocket's acceleration depends on the speed of the exhaust gases. Therefore, the speed of the exhaust gas should be as high as possible to achieve the maximum velocity. Also, for a given speed of the exhaust gas, the maximum speed for the rocket is achieved when the ratio of the initial mass to the final mass of the rocket is as high as possible; that is, the mass of the rocket without fuel should be as low as possible, and it should carry a maximum amount of the fuel. The ideal rocket equation only accounts for the reaction force exerted by the exhaust gases on the rocket. It does not account for any other forces acting on the rocket.
This text is adapted from Openstax, University Physics Volume 1, Section 9.7: Rocket Propulsion.