RTC7681 Graph Plotting

Download RTC7681 here.

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%)

Download RTC7681 here.

Mathematical Basis of Relativistic Travel Calculator RTC7681

Problem Description

Download Relativistic Travel Calculator RTC7681 here

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; begin    FBT2:=FBTA*FBTA;    EPM2:=EPMA*EPMA;    t2:=t*t;    PL:=((2*EPMA*FBTA*t*c2*(MF+MS))-(2*FBT2*EPMA*t2*c2)+(FBT2*EPM2*t2));    P1:=sqrt(abs(PL));    P2:=c2*(MF+MS)+FBTA*(EPMA-c2)*t;    CalculateVelo:=P1*c/P2; end;

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; begin    v2:=v*v;    EPM2:=EPMA*EPMA;    P2:=c2*EPM2*(c2-v2);    P1:=EPMA*(c2-v2)+v2*c2-sqrt(P2);    P2:=c2*(MS+MF);    P3:=((v2*EPM2)+(v2*c2*c2)-(c2*EPM2)+(2*c2*c2*EPMA)-(2*v2*EPMA*c2))*FBT;    CalculateTime:=P1*P2/P3; end;

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; begin    DT:=(t1-t0)/400;    lt:=t0;    Y0:=CalculateVelo(EPMA,FBTA,MS,MF,t0);    Acc:=0;    for hi:=1 to 400 do begin       lt:=lt+DT;       Y1:=CalculateVelo(EPMA,FBTA,MS,MF,lt);       Acc:=Acc+0.5*(Y1+Y0);       Y0:=Y1;    end;    Acc:=Acc*DT;    CalculateDist:=Acc; end;

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

Download Relativistic Travel Calculator RTC7681 here

Orimath Quadratic Equation and Function Solver

Orimath Quadratic Equation and Function Solver is a software capable of solving basic mathematic problems related to quadratic equation or quadratic function. In this sense, Orimath Quadratic Equation and Function Solver can act as both a Quadratic Equation Calculator and a Quadratic Function Calculator.

The picture depicted above is the main part of the Quadratic Equation and Function Solver. The main interface is designed so user can define their problem easily and checks out what questions about the quadratic equations or quadratic functions, the Quadratic Equation Calculator have to solve for them.

Unlike its freeware counterpart, this new Quadratic Equation Calculator is capable of telling the steps required to get the answer in user friendly HTML interface. This make the program good and useful for students who have problem with their mathematic homework, students who want to learn more about quadratic equations and quadratic functions, and teachers who want to make some problems for their students to solve.

The program is also capable of plotting the quadratic function neatly. The plotting interface also allow students to looks for the value of y(x) for a given value of x, making it easier to plot or graph a quadratic function.

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Plotting Quadratic Function

A picture can speak thousands words, that is exactly the reason why plotting a function is a good way to understand the properties of said function. To understand some properties of quadratic function, we have to see how the quadratic curve is shaped or located if each constants in y(x) = ax² + bx + c is modified. The program used to plot the quadratic functions is Orimath Quadratic Equation and Function Solver.

The picture above shows how the shape of the curve change if the value of a is modified. Modifying the value of a will dilatate the curve relative to the x-axis.

The picture above shows how the position of the curve change if the value of b is modified. Modifying the value of b will move the position of the curve diagonally.

The picture above shows how the position of the curve change if the value of c is modified. Modifying the value of c will move the position of the curve vertically along the y-axis.

Take a look at our Orimath Quadratic Equation and Function Solver.