Rocket Propulsion in Gravitational Field - I

Gravitational force acts opposite to the rocket's upward motion, applying a negative impulse that decreases the system's total momentum. This external force acting over time dt changes the rocket's momentum according to the impulse-momentum theorem. The gravitational force reduces the momentum gain that would occur from fuel ejection alone, requiring the rocket to expel more fuel to achieve desired velocity changes.

The rocket's momentum change equals the expelled gases' momentum in the opposite direction, following Newton's third law. Expelled fuel traveling in the negative y-direction carries momentum equal to its mass times relative velocity. This reaction produces an equal and opposite force on the rocket, causing it to accelerate upward while the system's total momentum is conserved.

Rocket mass decreases because combustion burns fuel, converting it into expelled gases. The mass decrease (dmex) represents the difference between initial and final rocket mass as fuel is consumed and ejected. This mass reduction is critical to the rocket equation, as the same thrust produces greater acceleration in a lighter rocket.

The ideal rocket equation calculates rocket velocity by applying conservation of momentum while accounting for the gravitational force's negative impulse. In a gravitational field, the momentum of the entire system decreases by the gravitational force multiplied by the time interval. This principle allows determination of the rocket's velocity at any instant despite external gravitational forces acting on the system.

The rocket's momentum increases from fuel ejection but simultaneously decreases due to gravitational force acting downward. The net momentum change equals the expelled fuel's momentum minus the gravitational impulse. Conservation of momentum applies to the entire system, with the rocket gaining upward momentum while gravity applies a downward impulse over the time interval.

Impulse equals the net external force (gravitational force) multiplied by the time interval dt. For a rocket, this gravitational impulse is negative, opposing the rocket's upward motion and reducing its momentum gain. The impulse-momentum relationship shows that this external force directly changes the rocket's momentum, requiring fuel ejection calculations to compensate for gravity's effect.

Relative velocity is the speed at which expelled gases move relative to the rocket. The momentum of expelled fuel equals its mass times this relative velocity. Higher relative velocities produce greater momentum transfer to the rocket, enabling more efficient propulsion and greater velocity changes despite gravitational forces acting on the system.