## 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.

The equation is named after Konstantin Tsiolkovsky who independently derived it and published it in his 1903 work.

The Physics of Rocket

## Derivation

Consider the following system:

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
$\Delta v = c \cdot \tanh \left(\frac {v_e}{c} \ln \frac{m_0}{m_1} \right)$

## 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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