Mechanical Physics

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

• v is upward vertical rocket velocity related to Stationary Earth.
• v_o 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 3^rd Law.
• u – exhaust velocity relative to the rocket.
• F_ext – Net external force expended in accelerating rocket and also variable mass propellant fuel.

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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.

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Derivation of Rocket Equation

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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)

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

Initial total rocket and propellant

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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)

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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

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Rocket Equation 5

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?

Answers,

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

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