Sunday 3 March 2013

Rocket Sciences


"No man should understand where his dreams come from, Data." 
~Soong to Data, in Star Trek~  

Roket merupakan salah satu wahana dirgantara yang memiliki makna strategis. Wahana ini mampu digunakan untuk melaksanakan misi perdamaian maupun pertahanan, misalnya sebagai:
Roket Peluncur Satelit (RPS), 
Roket penelitian cuaca, 
Roket kendali, 
Roket balistik dari : darat ke darat, darat ke udara dan udara ke udara.

Dengan kata lain, roket juga bisa berfungsi sebagai peralatan untuk menjaga kedaulatan dan meningkatkan martabat bangsa, baik di darat, laut maupun di udara sampai dengan antariksa.
Oleh karena itu, negara yang menguasai kemandirian teknologi peroketan dengan baik, akan disegani oleh negara-negara lain di seluruh dunia. Indonesia sebagai negara besar dan luas sudah sepatutnya dapat meraih kemandirian yang berkelanjutan dalam penguasaan teknologi roket.
Oleh sebab itu diperlukan upaya yang terus menerus untuk mewujudkan kemandirian ini, salah satunya melalui usaha menumbuh kembangkan rasa cinta teknologi dirgantara, khususnya teknologi peroketan sejak dini.

Rocket Sciences:

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The Tsiolkovsky rocket equation, or ideal rocket equation, describes the motion of vehicles that follow the basic principle of a rocket: a device that can apply acceleration to itself (a thrust) by expelling part of its mass with high speed and move due to the conservation of momentum. The equation relates the delta-v (the maximum change of speed of the rocket if no other external forces act) with the effective exhaust velocity and the initial and final mass of a rocket (or other reaction engine).

For any such maneuver (or journey involving a number of such maneuvers):
\Delta v = v_\text{e} \ln \frac {m_0} {m_1}
where:
m_0 is the initial total mass, including propellant,

m_1 is the final total mass,

v_\text{e} is the effective exhaust velocity (v_\text{e} = I_\text{sp} \cdot g_0 where I_\text{sp} is the specific impulse expressed as a time period and g_0 is Standard Gravity),

\Delta v\ is delta-v - the maximum change of speed of the vehicle (with no external forces acting),

\ln refers to the natural logarithm function.
Units used for mass or velocity do not matter as long as they are consistent.


Derivation

 

 

Consider the following system: Var mass system.PNG




In the following derivation, "the rocket" is taken to mean "the rocket and all of its unburned propellant".
Newton's second law of motion relates external forces (F_i\,) to the change in linear momentum of the system as follows:
\sum F_i  = \lim_{\Delta t \to 0} \frac{P_2-P_1}{\Delta t}

where P_1\, is the momentum of the rocket at time t = 0:

 P_1  = \left( {m + \Delta m} \right)V



and P_2\, is the momentum of the rocket and exhausted mass at time t=\Delta t\,:



P_2  = m\left(V  + \Delta V \right) + \Delta m V_e



and where, with respect to the observer:


V\, is the velocity of the rocket at time t = 0

V+\Delta V\, is the velocity of the rocket at time t=\Delta t\,


V_e\, is the velocity of the mass added to the exhaust (and lost by the rocket) during time \Delta t\,
m+\Delta m\, is the mass of the rocket at time t = 0
m\, is the mass of the rocket at time t=\Delta t\,


The velocity of the exhaust V_e in the observer frame is related to the velocity of the exhaust in the rocket frame v_e by (since exhaust velocity is in the negative direction)
V_e=V-v_e

Solving yields:
P_2-P_1=m\Delta V-v_e\Delta m\,

and, using dm=-\Delta m, since ejecting a positive \Delta m results in a decrease in mass,
\sum F_i=m\frac{dV}{dt}+v_e\frac{dm}{dt}

If there are no external forces then \sum F_i=0 and
m\frac{dV}{dt}=-v_e\frac{dm}{dt}

Assuming v_e\, is constant, this may be integrated to yield:
\Delta V\ = v_e \ln \frac {m_0} {m_1}

or equivalently
m_1=m_0 e^{-\Delta V\ / v_e}      or      m_0=m_1 e^{\Delta V\ / v_e}      or      m_0 - m_1=m_1 (e^{\Delta V\ / v_e} - 1)

where m_0 is the initial total mass including propellant, m_1 the final total mass, and v_e the velocity of the rocket exhaust with respect to the rocket (the specific impulse, or, if measured in time, that multiplied by gravity-on-Earth acceleration).


The value m_0 - m_1 is the total mass of propellant expended, and hence:
M_f = 1-\frac {m_1} {m_0}=1-e^{-\Delta V\ / v_\text{e}}


where M_f is the propellant mass fraction (the part of the initial total mass that is spent as reaction mass).


\Delta V\ (delta v) is the integration over time of the magnitude of the acceleration produced by using the rocket engine (what would be the actual acceleration if external forces were absent). In free space, for the case of acceleration in the direction of the velocity, this is the increase of the speed. In the case of an acceleration in opposite direction (deceleration) it is the decrease of the speed.


Of course gravity and drag also accelerate the vehicle, and they can add or subtract to the change in velocity experienced by the vehicle. Hence delta-v is not usually the actual change in speed or velocity of the vehicle.


If special relativity is taken into account, the following equation can be derived for a relativistic rocket, with \Delta v again standing for the rocket's final velocity (after burning off all its fuel and being reduced to a rest mass of m_1) in the inertial frame of reference where the rocket started at rest (with the rest mass including fuel being m_0 initially), and c standing for the speed of light in a vacuum:

\frac{m_0}{m_1} = \left[\frac{1 + {\frac{\Delta v}{c}}}{1 - {\frac{\Delta v}{c}}}\right]^{\frac{c}{2v_e}}



Writing \frac{m_0}{m_1} as R, a little algebra allows this equation to be rearranged as
\frac{\Delta v}{c} = \frac{R^{\frac{2v_e}{c}} - 1}{R^{\frac{2v_e}{c}} + 1}

Then, using the identity R^{\frac{2v_e}{c}} = \exp \left[ \frac{2v_e}{c} \ln R \right] (here "exp" denotes the exponential function;  

see also Natural logarithm as well as the "power" identity at Logarithm#Logarithmic identities) and the identity



\tanh x = \frac{e^{2x} - 1} {e^{2x} + 1} (see Hyperbolic function), this is equivalent to
\Delta v = c \cdot \tanh \left(\frac {v_e}{c} \ln \frac{m_0}{m_1} \right)

 

 

Applicability

 

 

  Pendidikan, Riset dan Kompetisi Pembuatan Iptek Roket di Tanah Air

 

Uses


The rocket equation captures the essentials of rocket flight physics in a single short equation. It also holds true for rocket-like reaction vehicles whenever the effective exhaust velocity is constant; and can be summed or integrated when the effective exhaust velocity varies. It does not apply to non-rocket systems, such as aerobraking, gun launches, space elevators, launch loops, tether propulsion.

Delta-v is of fundamental importance in orbital mechanics. It quantifies how difficult it is to perform a given orbital maneuver. To achieve a large delta-v, either m_0 must be huge (growing exponentially as delta-v rises), or m_1 must be tiny, or v_\text{e} must be very high, or some combination of all of these.

In practice, very-high delta-v has been achieved by a combination of 1) very large rockets (increasing m_0), 2) staging (decreasing m_1), and 3) very high exhaust velocities.

The Saturn V rocket used in the Apollo space program is an example of a large, serially staged rocket. The Space Shuttle is an example of parallel staging where all of its engines are ignited on the ground and some (the solid rocket boosters) are jettisoned to lose weight before reaching orbit.

The ion thruster is an example of a high exhaust velocity rocket. Instead of storing energy in the propellant itself as in a chemical rocket, ion and other electric rockets separate energy storage from the reaction (propellant) mass storage.

Not only does this allow very large (and in principle unlimited) amounts of energy to be applied to small amounts of ejected mass to achieve very high exhaust velocities, but energy sources far more compact than chemical fuels can be used, such as nuclear reactors. In the inner solar system solar power can be used, entirely eliminating the need for a large internal primary energy storage system.

Sources:

1. Tsiolkovsky Rocket Equation
2. NASA Jet Propulsion Laboratory
3. Lembaga Penerbangan dan Antariksa Nasional (LAPAN)



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