Rocket Propulsion in Gravitational Field - II

Earth's gravitational force acts opposite to the rocket's motion, reducing its velocity change. The gravitational acceleration multiplied by the time interval represents the velocity loss due to gravity. A rocket's velocity in a gravitational field is smaller by a factor of gΔt compared to its velocity in space under identical combustion conditions, making gravity a significant opposing force during launch.

Rocket acceleration depends on exhaust velocity of expelled gases, the rate at which mass is ejected, and the rocket's mass without fuel. Higher exhaust velocity increases acceleration, faster fuel burn rate increases thrust and acceleration, and lower rocket mass increases acceleration. Conventional propulsion systems achieve exhaust velocities around 2.5 × 10³ m/s, while rocket mass decreases continuously during flight as fuel burns.

Faster combustion allows rockets to overcome gravitational deceleration more effectively. Since gravity continuously opposes motion with acceleration gΔt, burning more fuel in shorter time intervals generates greater thrust and acceleration. This rapid mass ejection compensates for gravitational losses and enables the rocket to achieve significant velocity changes during launch from Earth's surface.

A rocket's thrust is greater in outer space than in the atmosphere or on a launch pad because gases are easier to expel in a vacuum. Thrust is defined as the rate of change of the rocket's mass multiplied by exhaust velocity. In space, without atmospheric resistance and gravitational opposition, the same combustion rate produces more effective thrust and acceleration.

The ratio of initial mass to mass after fuel combustion directly influences the rocket's velocity change. This mass ratio, combined with exhaust velocity and time interval, determines how much velocity the rocket gains. Larger mass ratios indicate more fuel burned relative to remaining rocket mass, producing greater velocity changes during the same time period.

Rocket acceleration increases continuously because the rocket's mass decreases dramatically as fuel burns. Since most of the rocket is fuel initially, acceleration increases throughout flight, reaching maximum just before fuel exhaustion. With constant thrust and decreasing mass, the acceleration a = F/m increases proportionally, making the rocket progressively faster as it loses mass.

The velocity change equation is derived by integrating the instantaneous velocity change expression with initial and final limits. This integration accounts for the negative ratio of reduced mass to initial mass, exhaust velocity, and gravitational acceleration over time. The resulting expression shows that velocity change depends on exhaust velocity, mass ratio, and gravitational deceleration factor gΔt.