## 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*.***v**is the initial rocket upward velocity of propellant Fuel relative to stationary earth. also initial exhaust velocity relative to stationary earth._{o}**D**is downward atmospheric force, this is neglect this time for simplicity.**T**– Thrust force applying Newton’s 3^{rd}Law.**u**– exhaust velocity relative to the rocket.**F**– Net external force expended in accelerating rocket and also variable mass propellant fuel._{ext}

## 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.

**Derivation of Rocket Equation**

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

[katex]{ F }_{ a }\quad +\quad T\quad -\quad mg\quad =\quad \frac { md(v) }{ dt }[/katex]

This generalizes the rocket equation considers rocket weight by factoring out external gravity force of rocket and fuel weight from **F _{a}**

_{ }but keeping drag

**D**as part of

**F**.

_{a}Rocket Motion – Variable Mass Problem (Mechanical Physics Part 2)

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

[katex]\frac { dv }{ dt } \quad =\quad \frac { T }{ m } \quad -\quad g[/katex]

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.

**v**– Initial rocket velocity._{o}**v**– Burn out velocity, rocket velocity moment at complete._{bo}- dv = v
_{bo}– v_{o}, maximum burn out velocity. - u = – ( v – v
_{o}), 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.

## Relationship between total impulse and specific impulse

## Determination of burn out time

## Rocket Equation 5

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

Consider a rocket moving in a space suppose that 2.1×10^{6} *kg* of fuel a consiume during a burn last in 1.5×10^{2} *s*. Give that there is a constant force on the rocket of 3.4×10^{7} *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 2.8×10^{6} *kg*.

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

**Answers,**

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

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