# Rocket Motion – Variable Mass Problem (Mechanical Physics)

Rocket Motion – Variable Mass Problem (Mechanical Physics Part 2)

• v is upward vertical rocket velocity related to Stationary Earth.
• vo is the initial rocket upward velocity of propellant Fuel relative to stationary earth. also initial exhaust velocity relative to stationary earth.
• D is downward atmospheric force, this is neglect this time for simplicity.
• T – Thrust force applying Newton’s 3rd Law.
• u – exhaust velocity relative to the rocket.
• Fext – Net external force expended in accelerating rocket and also variable mass propellant fuel.

## Defining Ideal Rocket

• There is no twisting or turning moment force acting on the rocket.
• The thrust T acts precisely at the rocket’s center of mass.
• The rocket and it’s propellent fuel mass solely involved in upward translational rectilinear motion.
• There is a realistic assumption of constant fuel burn rate implying constant thrust T.
• The gravity acceleration is assumed to be constant.

In the beginning part of the analysis, the rocket does not escape the earth’s gravity field.

## Generalized Equation (Rocket Motion – Variable Mass Problem)

This generalizes the rocket equation considers rocket weight by factoring out external gravity force of rocket and fuel weight from Fa but keeping drag D as part of Fa .

Rocket Motion – Variable Mass Problem (Mechanical Physics Part 2)

## Rocket Equation 2 (Rocket Motion – Variable Mass Problem)

This equation is really / corollary to the above equation.

• g is constant gravitational field acting upon rocket and propellant masses and also thrust force T.
• m – Total mass including propellant fuel.

[katex]T\quad =\quad \frac { ud(m) }{ dt }[/katex], thrust excreted by exhaust velocity of propellent.

g is constant gravitation field then,

Since it vaeies inversly thrue rocket length.

• vo – Initial rocket velocity.
• vbo – Burn out velocity, rocket velocity moment at complete.
• dv = vbo – vo, maximum burn out velocity.
• u = – ( v – vo ), effective exposed velocity.

Initial total rocket and propellant

## Rocket Equation 3 (Rocket Motion – Variable Mass Problem)

Also, this gives maximum rocket velocity at time t. Rocket Motion – Variable Mass Problem (Mechanical Physics Part 2)

## Rocket Equation 4 (Rocket Motion – Variable Mass Problem)

Propellent mass function,

Which is the fraction of initial mass that is reaction mass.

## Thrust to weight ratio

In other words the specific impulse can be defined as the ratio of the thrust produce to the rate at which the rocket consium its fuel.

## Question: (Rocket Motion – Variable Mass Problem (Mechanical Physics Part 2))

Consider a rocket moving in a space suppose that 2.1×106 kg of fuel a consiume during a burn last in 1.5×102 s. Give that there is a constant force on the rocket of 3.4×107 N. During this burn calculate the velocity of the exposting fuel. Hence calculate the increasing speed resulting some their burn. If the initial mass is 2.8×106 kg.

What is the initial vertical acceleration that can be important to this rocket when it is launched from the earth?