Thursday, October 23, 2008

Orimath Referral and Reselling Business

Lately I had added a new marketing module in my website This new marketing function will enable anybody who have PayPal account to get extra income by referring my products to their friends or resell my products in their website or blog. If someone buy a product in my website, they will get a share. All they have to do are :

After they get the referral code, they can either put the codes in their email and send them to their friends, or they can put referral link in their blogs, websites, or social network profile.

Example of Referral Hyperlink

Example of Referral Banner

Beside referral links, also provide resellers with reseller buttons. By putting these referral buttons in their own websites or blogs, resellers can sell my products direcly from their websites or blogs. Resellers will also get a portion of the sales generated by their reseller buttons. Clicking these buttons will redirect you to PayPal to pay for the products. After purchase is made and paid, you will be sent to my download page to download the products you had paid for.

Example of Reseller Buttons

Oh and last but not least. I just added a new product into orimath. A polynomial solver. This could help you to solve and learn how to solve polynomial using various numerical method. See this link for more into about the product.

Friday, September 26, 2008

Shopping Cart,Multi Download and Discount in

After adding Fractal Gallery into, today I work to upgrade the purchasing and download system. Now any users who wish to buy more than one item wont need to surf around between and PayPal. Just check-up several items that you wish to purchase and put them into your shopping cart. No login is required.

Now, users can grab several items at once shopping cart system

Have you finished your shopping in ? You just need to click that Buy Now button. You will be automatically be brought to PayPal to pay your purchase. By purchasing more than one item in you will get USD 0.15 discount per item, starting from the second item you put into your shopping cart.

This is the Multiple Download page

After finishing the payment procedure using PayPal, user will be redirected back to Meanwhile in, the purchased product will wait for user to download them. Just click once in each picture and user will get the product they bought.

This upgrade have been installed, checked for validity and ready for use. I will have to say thanks for Sonata, who had helped me as the tester and everyone who had visit my site. Thanks to all of you. :)

Wednesday, September 24, 2008

System Update in is the website I used to sell Orimath Quadratic Solver, which is basically a commercial version of the Quadratic Solver I used to answer questions in Yahoo!Answer. The website used to sell only one product, there is practically no other software other than the Orimath Quadratic Solver itself. But that already changed.

Yesterday got a facelift. The index page had been changed to fit more with its role as a webstore. I added a new feature called Fractal Store.

At the moment this post is written 8 fractal patterns are being sold in the web. The pattern is sold as programs capable of rendering the fractal in any resolution. User can also explore the fractals in any range they want to, this would give users freedom to choose which part of the pattern they want to render into high resolution fractal image.

However, since fractal rendering is a time consuming process, it is advised that exploration should be done in low resolution. Should users find certain part of the fractal interesting, they can take high resolution shoot on that particular part.

Tuesday, September 16, 2008

Newton Fractal and Durand Kerner method

Newton Fractal and Durand Kerner method

I had known about Newton Fractal for a long enough time, even before I wrote the code for RTC7681 and RTC7683. But it is impossible to do, as long as I don't know what numerical method people use to get the complex roots of complex polynomial function.

Anyway I just decided to ask the question in Yahoo!Answer recently and several people was kind enough to answer my question. Too bad I can only choose one best answer. Internet is a really helpful thing to have, just imagine how many knowledge do millions of user around the world can have. So I just need to transform Durand-Kerner method into computer code and voila I have my Newton Fractal.

All pictures showed below is a actually Newton Fractal plot of the same complex polynomial function, but different pictures showed different level of detail. The complex polynomial function used is f(x) = x8 + x6 + 5i x5 + ( 3 + 2i )x4 + ( 3 + 5i )x2 -1. The value of a used is 1 + 1i.

The picture above is the Newton Fractal plot of the complex polynomial function mentioned earlier. The plot range is between -300-300i and 300+300i. Even though this is the least detailed plot of the fractal, this one is drawn the latest as that was the last thing I am curious about. If I wasn't too curious about the geometry of the Newton Fractal in big range, I would probably never care to plot it. Anyway the prospect of being able to get the same small detail in big scale win my curiousity.

This particular Newton Fractal was plotted with plot range between -3-3i and 3+3i. So this one have about 100 times more detailed, than the previous Newton Fractal plot result. This is the first plot I told my cute little computer to execute, as that is the range I used to test the program with different polynomial function and a value.

Curious what would happen if we take a look on those little blob in plot range between -2.3-0.3i and -1.7+0.3i ? I was curious as well. So I decided to spend about 8 hours of my computer time to plot the function in high resolution ( 8000x8000 pixels2 ). I have ordered the ( 70x70 cm2 glossy paper print of the plot result and planned to use it as a poster in my room. Too bad I can't share either the 183 MB BMP file or 20.7 MB JPG File due to the upload speed where I live. But if you click the image, you can still view it in 2000x2000 pixels2 detail.

Extreme Resolution

I was still curious about bigger picture of the Newton Fractal, so I decided to do more experiments. The first extreme experiment result was the picture above, depicting the plot result with range between -3e5-3e5i and 3e5+3e5i. It seems that the greater range we go, the less detail we can expect to get.

And on the range of -3e8-3e8i and 3e8+3e8i, you will get no complex picture at all. See the picture above and you will realize that the pictur consist of only 5 different color, instead of 8. This means that in the big picture of things, the probability to get 5 roots out of 8 from x8 + x6 + 5i x5 + ( 3 + 2i )x4 + ( 3 + 5i )x2 -1 = 0 using Newton-Raphson method is great. To get the other 3 roots however will be almost impossible.

On the other hand, it is not easy to get a plot result with good complexity on the micro scale either. Most of the time you are going to Fall into the Sea, but by using a numerical method called Bisection combined with low resolution exploration, we can expect to arrive in some Island of Complexity. The plot result above for example, was drawn with plot range between -2.2113+0.0002i and -2.2117+0.0006i.

Thursday, September 11, 2008

Who is the god of a simulated reality ?

I have been interested in stories about deities since I was small, and lately I asked lots of questions about this in several internet forums. I am a Christian myself, but I would state that there is no observational evidence about the existence of what we called gods. Some people would say that there is no observational evidence for the inexistence of gods as well, but this argument can be easily countered by the fact that there is no observational evidence about the existence of unicorns and flying sphagetti monster as well. This of course won't mean that gods doesn't exists, but it does mean that the existence of gods is as hypothetical as the existence of either unicorns or flying sphagetti monster.

So what is a possible solution when there are no conclusive evidence ? We should look at how we prove the existence of something hypothetical. A hypothetical object could be proven to be exist, even if they are not visible, by observing whether their predicted effect could be observed in experiments. Of course to do so we have to be clear on the properties of the hypothetical object we are going to look for. In this case I tried to look for possible definitions of gods in case of simulated reality.

So I put my question in a science fiction forum about what they think regarding the gods of a simulated reality. This is a reply from David Jackson, a fellow member of Orions Arm mailing list who happen to run a digital fiction blog. I find his reply to be somewhat insightful, so lets hear how he define a god :

Well ... let's see. What constitutes a god?
  1. IMO, a true god should be the ultimate agent of creation for a thing. He need not craft ever detail of that thing -- it is enough to initiate the creative spark that subsequently develops into the finished product.
  2. Likewise, he need not have absolute authority over the development of the thing once it gets going, but he SHOULD have the power to dictate arbitrary changes and have those changes proceed/develop according to the pre-existing nature of the thing and the overall influence of the change itself.
  3. A god need not be omnipotent, IMO. It is enough that he should be able to obtain a particular piece of information he desires when and if knowing it becomes important.
  4. A god should have power over the future of a thing even after it has left the cradle. So he should be able to recall it for further changes on a whim, terminate the thing outright, suspend its development indefinitely, etc.
So that is David Jackson definition of a god. Then he go on to explain his points whether each objects in a simulated reality could be considered as a god. Since I am the one who make the poll in the Great Big Group, I think I will have to explain why I included each option for my friends there to choose. Green colored text is my reason to include the option as a god candidate, Blue colored text is David Jackson's argument.

So ... how do these criteria match our godly candidates?

The programmer who made the program

It is clear, programmers made the program. Programmers are godlike in the sense that they are the creator deity of the simulated reality. Most people I asked in other forum would choose this option. Programmer is the one who create the environment and the very law that the universe where simulated reality run have to obey. But David Jackson have better knowledge about software development than I am.
Well, he certainly created the program, and has some degree of arbitrary power over its development. But, in the real world, a programmer actually tends to have very little freedom within a project. He is handed a set of goals to accomplish, and sets about accomplishing those goals to the best of his ability. Good managers will give good programmers considerable leeway in designing and implementing their programs ... but programmers without any direction whatsoever are a bad idea.

Anyone who run the program and play

The program have left its cradle and people who may not know anything about how the program is written, puchased the program and play it. The player have full control on the lives of any creature in the simulated reality, and the programmer will not interfere. Players in this sense are gods, just like Odin, Osiris and Zeus are gods.
He is constrained by the nature of the program, warts and all. Since he has no authority over the nature of his experience regarding the program and no ability to dictate arbitrary changes, I'd say he's out as a god.

The computer where the simulation is run

My reason to include this option is they are the one that make sure the law the programmer had made is always obeyed in simulated reality universe.
Basically this is the same as the person who runs and plays the program. They experience different aspects of the same process. A user is just as much a part of a program's process flow as the CPU is ... so by the same argument, no, the computer is not a god. It's merely an environment.

The simulation software

This is the very law of the simulated reality universe itself. If the simulated universe can evolve, this is the very equation that govern how the evolution are being done.
Am I a god of myself? Considering how a bad cold can knock me down ... I'd say no. Likewise, the first time a segfault brings the simulation to its knees, it pretty much loses all claim on godhood.

The harddisk

My reason is the harddisk store the deeds of all creature in the simulated reality universe. The harddisk know everything and is closer to omniscience (relative to the simulated reality) than even the programmers themselves.
Part of the environment. Not even a directive part of the environment, so, no.

The server admin

Well, some of them say they are gods, so I included them here :)
Have you met many server admins? They only think they're gods.

The memory (RAM)

This is the very environment where creatures inside the simulated reality lived and evolved.
Same basic principle as the hard disk, so no.


When all is said and done, I'd have to say that the project manager or designer is the closest thing to a god I can think of when it comes to a software project. He is tasked with identifying a need, formulating a plan to address that needs, and delegating the implementation of that plan to his programmer peons. Depending on his position in the company, he may be able to initiate holds and recalls of the product. If he is forward-thinking, he can direct his programmers to include the kinds of "kill switches" Microsoft builds into all its products to obsolete them after some amount of time in-market.

Sure ... every now and then some hacker comes along with a crack to disable your kill switch ... but, then, Adam bit that apple. So a PM still has at least as much claim to godhood as Yahweh.

Tuesday, August 19, 2008

Example of Chaos in Rule 23-3

The evolution property of Cellular Automata Rule 23-3 known as Conway's Life have been known to be chaotic. This means a small change in initial condition, will have a big effect on the evolution of said world.

This Morning (August 10th 2008,1 AM (GMT+8)) I had been lucky enough to observe this particular property of Rule 23-3 while doing some experiments involving 21 Conway's Worlds.

The experiments involved 7 worlds with worldsize of 30 x 30 pixels, 7 worlds with worldsize of 60 x 60 pixels and 7 worlds with worldsize of 120 x 120 pixels. The main objective of the experiments was to determine the average longevity of Conway's world in relation to the worldsize of said Conway's world.

What interest me the most was an unusually high longevity observed in one world with worldsize of 120 x 120 pixels. This particular initial condition below, was shown to have longevity of 9150 cycles.

Initial Condition
Death Condition

This unusually high longevity (about 2.135 Standard Deviation from the mean value) had made me wonder about the rarity or abundance of Conway's world with such long lifetime. So I alter the initial condition by killing one live cell from the original initial condition above. This had decreased the lifetime of the world down to 3059 cycles.

Initial Condition
Death Condition

Since killing one live cell in the initial condition made the longevity of the world being near to the mean longevity value, this might means that initial conditions that produce world with high longevity value to be rare. But more experiments must be done before taking any further conclusions.

If you are interested to help calculating the average lifetime of a Conway's world with certain worldsize, just download the program and do some experiments by yourself. Submit your experiment results to this particular maling list Yahoo!Groups : Cellular Automata Research.

Saturday, August 2, 2008

My website :

July 2008, is a month full of web programming work. My partner in web busines, Lucy Rahayu had asked me to make 2 new websites. One of them is This website is basically selling frozen foods like dumplings and meatballs, produced by an Indonesian food company named TMVFood, owned by one of our friends.

Lucy is the one doing the web design and marketing work, while I do the programming and webmaster work. We are new in this business and this is the first time we make a complex website for e-commerce purpose, so it is natural that we got some difficulty during the process of making this website. Both of us don't know from the start what facility an e-commerce website should have.

I have to think not only about the code, but also how to integrate the links to the design, so the website esthetic value is preserved. I have to admit that Lucy is a good web designer anyway.

Shown above is the product menu page of At the time this entry is being written, there are only three products there. The owner of TMVFood plans to add more products later.

Click any picture in the product menu page and you will be redirected into a page containing the full detail of said product. Shown here is Chicken-Shrimp Dumplings, or "Siomay Ayam Udang" in Bahasa Indonesia.

This page contain details about the product and the price of said product. Anyway the more you buy a kind of product, the less each unit of them will cost. This page also have price list stating how much a product will cost, given the amount of product you buy.

This website also contain some delicious recipes. The one shown above is "Sup Bola-bola".

And finally if you are interested to ask us some questions, we have a contact page. Just write your name, email address (we will need it to send our replies) and your message. We will reply your message as soon as possible.

Mathematical Basis of RTC7683

Download RTC7683 here

Problem Description

I had made an attempt to do the math for relativistic rocket in my previous blog post. However it turns out that the calculations in said post turns out to be wrong. I got to know why it is wrong when I looked for information from Yahoo!Answer, Physic Forum and Adam Getchell. I initially planned to used the information for upgrading RTC7681 to RTC7682. It turns out that in the previous post, I ignored conservation of linear momentum and thus the whole mathematic attempt is wrong. I will have to redo the calculation, this time including Conservation of Linear Momentum.

Anyway, this is the problem description :

A Spaceship with empty ress mass of MShip have fuel capacity of MFuel. The Ship's engine is capable of burning some amount of the fuel per unit time ( FBT ), turning them into kinetic energy per mass unit ( EPM ). The Spaceship is assumed to use Photon Drive as suggested in Baez Relativistic Rocket FAQ.

Actually I want to make SRF Exhaust Velocity variable, but it turns out to be more complex than I had initially thought. It is easier to assume that the SRF Exhaust Velocity to be c, as lightspeed is frame invariant. Anyway if anyone who read this post know how, I will be more than thankful if you want to tell me.

Conservation of Mass-Energy and Conservation of Linear Momentum

To calculate anything about rocket, we have to consider both Conservation of Mass-Energy and Conservation of Linear Momentum. Anything else about the rocket can be derived, if we can model how Mass-Energy and Linear Momentum is conserved in our model.

Equation RTC7683-1.Conservation of Mass-Energy

Equation RTC7683-2.Conservation of Linear Momentum

To get rid of EXE ( Exhaust Energy ), we would use equation RTC7683-2 to derive what EXE is equal to.

Equation RTC7683-3.Exhaust Energy Equation

Equation RTC7683-4.Equations derived from both Conservation

Function v(t) and t(v)

To get the IRF Velocity of the ship over time, we have to turn Equation RTC7683-4 into Quadratic Equation over v.

Equation RTC7683-5A.Quadratic Equation on v

Equation RTC7683-5B.The a,b,c part of
Quadratic Equation in Equation RTC7683-5A

Equation RTC7683-5C.Velocity as Function of Time

For the program RTC7683, we are going to need calculating the time required to reach certain velocity in order to calculate the time required for deceleration.

Equation RTC7683-6A.Quadratic Equation on t

Equation RTC7683-6B.The a,b,c part of
Quadratic Equation in Equation RTC7683-6A

Equation RTC7683-6C.Time as Function of Velocity

Coordinate and Proper Acceleration

Coordinate Acceleration is the second derivative of displacement as observed in a reference frame over time.

Equation RTC7683-7.Coordinate Acceleration

Proper Acceleration is the amount of acceleration an accelerometer inside the ship would measure.

Equation RTC7683-8.Proper Acceleration

Download RTC7683 here

Wednesday, July 2, 2008

Update on Cellular Automata Programs

I had updated my cellular automata programs. The previous version seems to have problem with the width of the screen in computers without some Japanese fonts installed.

The new links is here :

The Cellular Automata v1 is a program to simulate single state cellular automata. Thus the color of each cell could only be black or red (or other color, if you choose so.

The Cellular Automata v2 is a program to simulate multiple state cellular automata. The rule required are a bit more complex than the one for Cellular Automata v1. But once you find a good rule, you could have more fun. The Cellular Automata v2 package have three rules inside in (*.rul) files.

Unfortunately I am really busy with my web programming work, that I have no time left to do any deeper research on it. Anyway I am going to have some good surprise for programmers in the near future. Have fun!

Sunday, June 29, 2008

My Enterpreneurship Attempt

Since I am currently starting my new business, I thought it should be useful for me to read more about investing. Well, someone like Warren Buffet started investing in financial terms from very young age, most of us don't. So I run into this particular link, and that reminds me about my own enterpreneurship attempts.

Personally I had loved math since I was 7 ,learned programming since I was 14 and start being serious learning my religion since I was 13. Not really an advantage for me during high schools, since we was not taught programming and won't get any points for it. But after being memed by my father since I was five about he having a system working for him, and he only had to command the system to fulfill his wishes, I know that programming is the path I should go.

It turned out that I was right. During university my programming skill help me greatly to understand most of the math taught ( and not taught ) to me. I don't need to derive mathematic equations myself, since the computer can do it for me, but the most helpful part is the computer's calculation speed and graphing capability. I don't know if anyone can turn hundreds 3D points into a graph in their mind, but I know I can't, that is where programming can be a really useful thing to know.

I think that was an investment in my part. But my friend told me something that I can't deny, "Fendy I am amazed how you can make hundreds of programs, but still make no money from your skill". Well, it is not really that I am not trying to. But some months before that, I just lost my first entrepreneurship attempt, after I worked the whole 6 months to make the full system without receiving any payment, during the beginning of the commercialization.

That was really a shock to me. That system was like my own child, I weave every part of its DNA, hoping that one day the system would grow and become sentient on its own right ( I designed the system so it could grow and evolve, evolution sometimes is the only way to answer some mathematic problems, extrasolar planet finders being one of its application, you will need a GRAPE to do accurately it however ). I had read OA during those times and some ideas leading to the creation of the system was based on things written there ( Have to thanks Todd a lot for some discussion about sexuality in the OA main list, that was really helpful ). Having nothing to do anymore in Jakarta, I have pack up and go back to my hometown Surabaya, and here I am until now.

In Surabaya, I spent several months writing things in my blogs, answering questions in Yahoo!Answer and taking some Japanese courses. It is funny though that during these downtime, I learned lots of new things from the internet, made several simple programs every month. One day when I was answering questions, I notice that there are certain questions that I can answer very quickly in detail, if only I have the right program to do it. I spent two days to do the code and start answering questions in detail in short time.

Then my friend come and asked me, "Why dont you sell it for $2 ? I can market it for you.", that surely reminded me on the thread a net friend of mine, Chris had started in Great Big Group some months before, my answer at that time was "lets start immediately." So I spent some weeks learning how PayPal works and start selling programs in my website, while my friend take care about the web design and marketing part. The result was not great yet, but one ideas done make more ideas come out. Robert Kiyosaki said in his books, that most people lost two or three companies before they own the one that made them filthy rich. And information age, provide a way to own a company at the cost of less than $50, the only thing required is idea alone and in my case I got more idea after I execute an idea.

So here's some lesson I learned and wish to share with everyone who read this post :
  • Starting a business is more about people than about money.
  • Don't be afraid to fail. Had I didn't do my first entrepreneurship attempt, I won't know how to do web-programming at all.
  • Work even when you are not being paid. Someone who never want to work without being paid, are not going to be paid when they are not working.
  • Don't waste your breath to curse or blame someone. Each breath worth a lesson to learn.

I really hope that everyone who read this particular entry could start being creative and found their own business. There are countless number of web-based business that already helped billions of people in the planet to live their lives. Some of them like PayPal, Google, Yahoo!, GNU Project and FSF had spent billions of dollars and millions work hours to place the foundation and infrastructure required, to ensure that all creative, talented and brave entrepreneurs in the information age to start their business for less than the cost of a bag of peanuts. So what hold you down ? Feel free to contact us, in case you need some help setting up your web-based business.

Tuesday, May 13, 2008

RTC7681 Graph Plotting

The mathematic detail of calculation provided by RTC7681 is provided here.

This post will focus on what kind of graphs RTC7681 is capable to produce. In short, RTC7681 is a calculator that calculate in detail the value of various variables involved in a relativistic travel. Instead of simply relating the maximum speed to various variables, RTC7681 also take the fact that the mass of fuel used to provide acceleration decrease over time during both acceleration and deceleration phase into consideration. The consideration about fuel mass depletion during both acceleration and deceleration phase is intended to provide more detail about the nature of relativistic travel, since there seems to be statistically significant percentage of people ( including me in the past ), who have wrong impression about the mass dilation as mass increase from nowhere. The mass increase during relativistic travel, actually did not come from nowhere, it come from the potential energy stored initially in the ship as fuel.

The examples of the graph below depicts the change in several variables value during relativistic travel for a spaceship massing 1 metric tons, carrying 10 metric tons of fuel, with engines capable of burning 0.08 kilograms of fuel per second and convert the fuel mass into kinetic energy at 0.1 efficiency ( 8.987551 1015 Joule / kg = 0.1 c2 ).

Changes in Velocity over time

The spaceship's velocity during a relativistic travel is not always constant. During both the acceleration and deceleration phases, the velocity changes as the spaceship accelerate or decelerate. The mass dilation effect increase the total mass, while burning the fuel for acceleration decrease the mass. Since the amount of fuel burned during both acceleration and deceleration phase is assumed to be constant, the change in mass affect the change in the amount of acceleration the engines are capable to provide.

Changes in Velocity over time
( Fuel Burned during Acceleration Phase 50%)

Changes in Velocity over time
( Fuel Burned during Acceleration Phase 76.833%)


Changes in Acceleration over time

The amount of fuel burned during both acceleration and deceleration phase are assumed to be constant. However since the total mass of the spaceship changes due to loss of mass by fuel and dilation of mass due to relativistic effect, the spaceship's acceleration provided by the engine do change over time during both acceleration.

Changes in Acceleration over time
( Fuel Burned during Acceleration Phase 50%)

Changes in Acceleration over time
( Fuel Burned during Acceleration Phase 76.833%)

IRF Distance Travelled over time

IRF Distance Travelled, is the distance travelled by the spaceship over time, as observed by an observer in an Inertial Reference Frame (IRF). Said observer in Inertial Reference Frame could be in the origin or in the destination point. The distance travelled by the spaceship is the result of finite integration of its velocity function over time.

Distance Travelled over time
( Fuel Burned during Acceleration Phase 50%)

Distance Travelled over time
( Fuel Burned during Acceleration Phase 76.833%)

IRF Ship's Mass over time

IRF Ship's Mass is the mass of spaceship as observed by an observer in an Inertial Reference Frame (IRF). Said observer in Inertial Reference Frame could be in the origin or in the destination point. Since the velocity of the ship change during both acceleration and deceleration phase, observers in Inertial Reference Frame are going to observe change in the amount of mass dilation during these both phases.

IRF Ship's Mass over time
( Fuel Burned during Acceleration Phase 50%)

IRF Ship's Mass over time
( Fuel Burned during Acceleration Phase 76.833%)

SRF Fuel Mass over time

SRF Fuel Mass is the amount of fuel available in the spaceship over time, as observed by observers inside the Ship's Reference Frame (SRF). Since the amount of fuel burned per unit time is assumed to be constant in this model, we should not expect anything other than two linear graphs with equal gradient, separated by a horizontal line.

SRF Fuel Mass over time
( Fuel Burned during Acceleration Phase 50%)

SRF Fuel Mass over time
( Fuel Burned during Acceleration Phase 76.833%)

IRF Total Mass over time

IRF Total Mass is the mass of the ship added with the mass of fuel still available in the ship, as observed by observers in an Inertial Reference Frame. Said observer in Inertial Reference Frame could be in the origin or in the destination point. Preservation of energy will make it impossible for the spaceship to gain mass without losing its fuel mass.

IRF Total Mass over time
( Fuel Burned during Acceleration Phase 50%)

IRF Total Mass over time
( Fuel Burned during Acceleration Phase 76.833%)

Monday, May 12, 2008

Mathematical Basis of Relativistic Travel Calculator RTC7681

Problem Description

Relativistic Travel Calculator RTC7681 is a software intended to calculate various variables and graph various function over time related to relativistic travel. The variables used in this blog entry are :

- MShip :The rest empty mass of the spaceship intended to be used.
- MFuel :The rest mass of the fuel provided for the spaceship.
- FBT :Amount (mass) of fuel burned per unit time.
- EPM :The amount of energy provided per unit mass of fuel burned.
- DistanceOfDestination :The distance between spaceship's destination and its origin.

The calculation done in RTC7681, assumes that :
- The spaceship travel in linear trajectory.
- The spaceship always use the same engine capacity for both acceleration and deceleration.
- Friction with interstellar medium is ignored.
- The total rest mass of the spaceship is decreasing as the spaceship engine burn the fuel.

Below are the definition of several terms used in this blog entry :

- IRF :Inertial Reference Frame, any IRF values are measured by observers in the origin.
- SRF :Ship's Reference Frame, any IRF values are measured by observers inside the ship.
- Acceleration Time :The time required by the spaceship to accelerate to an intended maximum IRF velocity.
- Acceleration Distance :Distance travelled by the ship during Acceleration Time.
- Deceleration Time :The time required by the spaceship to decelerate to zero IRF velocity.
- Deceleration Distance :Distance travelled by the ship during Deceleration Time.

Function of Velocity, Time, Distance and Acceleration

In order to find the function that relate these variables, we have to remember a concept called energy preservation. To gain kinetic energy, something have to lose its potential energy. The potential energy of the ship is stored as mass of fuel, while the kinetic energy of the ship as a whole depend on the total mass of the ship and fuel, multiplied by the square of its velocity, dilated by the mass dilation caused by relativistic velocity. From the equivalence between ship's potential and kinetic energy, we got we got :

Equation RTC7681-0. Energy Preservation

Equation RTC7681-1. Energy Preservation

To Solve this for v and t, we have to do the following steps :

Equation RTC7681-2. Energy Preservation

Equation RTC7681-3. Energy Preservation

Equation RTC7681-4. Quadratic Equation relative to v

Since equation RTC7681-4 is a Quadratic Equation relative to v, we can use any technique used to solve a Quadratic Equation , to solve it for v. Assuming that v is positive, we got the function of v over t below :

Equation RTC7681-5. Function of velocity over time

function TForm1.CalculateVelo(const EPMA,FBTA,MS,MF,t : double ) : double;
var P1,P2,PL,t2,FBT2,EPM2 : double;

Equation RTC7681-5 in Delphi Code

By turning equation RTC7681-4 to it's alternative form below, we can easily see that equation RTC7681-4 is a Quadratic Equation relative to t.

Equation RTC7681-4B. Quadratic Equation relative to t

So assuming that t is positive, we got the function of t over v below :

Equation RTC7681-6. Function of time over velocity

function TForm1.CalculateTime(const EPMA,FBTA,MS,MF,v : double ) : double;
var P1,P2,P3,v2,EPM2 : double;

Equation RTC7681-6 in Delphi Code

Acceleration and Deceleration Distance over time can be calculated using numerical integration below of the equation below :

Equation RTC7681-7. Function of Acceleration Distance over Time

function Torm1.CalculateDist(const EPMA,FBTA,MS,MF,t0,t1 : double ) : double;
var Acc,DT,Y0,Y1,lt : double;
    hi : cardinal;
   for hi:=1 to 400 do begin

Equation RTC7681-7 in Delphi Code

While the amount of acceleration and deceleration over time is the differentiation result of v over t.

Equation RTC7681-8. Function of Acceleration over time

Maximum Amount of Fuel Expended during Acceleration Time

The amount of fuel expended during acceleration time should never exceed a certain value or the ship will not be have enough fuel later during deceleration time. The maximum mass of fuel expendeable during acceleration will be called MPF here on. To find the value of MPF, remember that the proportion between the amount of fuel required to accelerate an object and the mass of the object, is always the same for the same velocity. Which means :

So, if the spaceship is intended to stop in its destination, the amount of fuel expended during acceleration should never be greater than MPF.

Exception in Acceleration Distance

Sometimes after some calculation, amount of fuel required to be expended during both acceleration and deceleration phase are not as great as we first intended them to be. In such situation, the amount of fuel required to be expended must be recalculated. The recalculation will involve solving equation RTC7681-9 for t0 and t1 :

Equation RTC7681-9. Exception Handling Formula

The variables in the equations above are :

- t0 :Length of Acceleration Phase.
- t1 :Length of Deceleration Phase.
- MFuel0:Initial amount of fuel when the spaceship start accelerating.
- MFuel1:Initial amount of fuel when the spaceship start decelerating.

We also have to mind the fact that the Initial amount of fuel when the spaceship start decelerating is equal with the Initial amount of fuel when the spaceship start accelerating subtracted with the amount of fuel burned during Acceleration Phase.

MFuel1 = MFuel0 - FBT t0