Rocket Propulsion in Gravitational Field - I

When a rocket is launched, its initial motion is influenced by the Earth's gravitational field. As the rocket travels upward, the external gravitational force acts on it opposite to its motion.

This force acting for a time dt applies an impulse equal to the change in momentum of the rocket and the fuel system.

Here, the initial momentum of the rocket of mass m moving with a velocity v is mv. Furthermore, the final momentum is the sum of the momentum of the rocket and the expelled gases.

Due to combustion, the momentum of the expelled gases traveling in the negative y-direction is a product of its mass and relative velocity.

Subsequently, the rocket's momentum is a product of mass decreased by dm_ex and velocity increased by dv.

Here, dm_ex corresponds to the decrease in the rocket's mass, minus dm. For any time dt, the rocket's motion is one-dimensional; therefore, considering the magnitude of the above equation and simplifying it further, neglecting the smaller terms, an expression for dv is derived.