9.19:
Rocket Propulsion in Gravitational Field – II
When a rocket is launched, its motion is affected by the Earth's gravitational force, which is directed opposite to the rocket's motion.
The change in instantaneous velocity of the rocket is expressed as the negative ratio of reduced mass of the rocket to its initial mass for a given constant velocity minus the acceleration due to gravity times the time interval.
Integrating the above equation with initial and final limits, the mathematical expression for the change in velocity under the influence of the Earth's gravity is obtained.
Thus, the velocity of expelled gases and the ratio of initial mass to the mass after fuel combustion during time Δt influences the change in the rocket's velocity.
The rocket's velocity is smaller by a factor gΔt compared to the rocket's velocity in space for the same combustion rate.
Thus, it is essential to have quicker combustion to achieve a significant change in velocity during the launch.
9.19:
Rocket Propulsion in Gravitational Field – II
A rocket's velocity in the presence of a gravitational field is decreased by the amount of force exerted by Earth's gravitational field, which opposes the motion of the rocket. If we consider thrust, that is, the force exerted on a rocket by the exhaust gases, then a rocket's thrust is greater in outer space than in the atmosphere or on a launch pad. In fact, gases are easier to expel in a vacuum.
A rocket's acceleration depends on three major factors, consistent with the equation for the acceleration of a rocket: First, the greater the exhaust velocity of the gases relative to the rocket, the greater the acceleration is. The practical limit for the exhaust velocity is roughly 2.5 x 103 m/s for conventional (non-nuclear) hot-gas propulsion systems. The second factor is the rate at which mass is ejected from the rocket. The thrust of a rocket is defined as the rate of change of the rocket's mass multiplied by the velocity of exhaust gases with units of newtons. The faster the rocket burns its fuel, the greater its thrust and the greater its acceleration. In the presence of gravitational force, it is required to burn more mass in a very short time interval to achieve a higher acceleration. The third factor is the mass of the rocket without fuel. The smaller the mass is (all other factors being the same), the greater the acceleration.
Rocket mass decreases dramatically during flight because most of the rocket is fuel, to begin with, so that acceleration increases continuously, reaching a maximum just before the fuel is exhausted.
This text is adapted from Openstax, College Physics, Section 8.7: Introduction to Rocket Propulsion and Openstax, University Physics Volume 1, Section 9.7: Rocket Propulsion.