9.16:
Rocket Propulsion in Empty Space – I
Rocket propulsion is a classic example of Newton's third law of motion. Fuel combustion leads to the ejection of gases at a high velocity, which provides the necessary force for flight as the rocket gets propelled in outer space.
The principle of momentum conservation holds true for a system of rocket and fuel in space. The initial momentum of the rocket containing the fuel is equal to the final momentum, which is the sum of the rocket's momentum and the momentum of the expelled gases.
If the direction of the rocket is along the positive x-axis, the expelled gases with mass dmex travel in the negative x-direction with a velocity vex.
Here, the initial momentum of the rocket is mv. Due to fuel combustion for time dt, the momentum of expelled gases and the momentum for the rocket is calculated.
Expressing the equation in terms of the magnitude, the equation is simplified further and neglecting the smaller terms from the equation, an equation for conservation of linear momentum for the rocket is obtained.
9.16:
Rocket Propulsion in Empty Space – I
The driving force for the motion of any vehicle is friction, but in the case of rocket propulsion in space, the friction force is not present. The motion of a rocket changes its velocity (and hence its momentum) by ejecting burned fuel gases, thus causing it to accelerate in the direction opposite to the velocity of the ejected fuel. In this situation, the mass and velocity of the rocket constantly change along with the total mass of ejected gases. Due to conservation of momentum, the rocket's momentum changes by the same amount (with the opposite sign) as the ejected gases. However, 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 instrumental in explaining the dynamics of a rocket's motion.
Using the conservation of momentum principle, the velocity of the rocket at any given instant can be calculated using the ideal rocket equation. Similarly, the thrust acting on the rocket and its instantaneous acceleration can be estimated.
This text is adapted from Openstax, University Physics Volume 1, Section 9.7: Rocket Propulsion.