Saturday, August 2, 2008

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

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