Calculating the travel time in relativistic travel

In summary, a spaceship with empty mass of MSS (?kg) travels from point A to B, which is a distance of RAB (?ly). It initially carries MFUEL (?kg) of fuel with an energy per mass of EPM (?Joule/kg). The spaceship's engine efficiency is SEP (?%) and it can burn FPS (?kg/s) of fuel per second. The maximum cruising speed and length of acceleration and deceleration phases can be calculated using the equations provided. The length of constant cruising speed will be affected by the Interstellar Medium Friction Constant, IMD (?/s). The heat generated by friction can also be calculated. Adding the energy required to support the crew, ESC (Watt), will change the equations
  • #1
rhz_prog
17
0
A spaceship with empty mass MSS(?kg), start its journey from A to B which distance is RAB(?ly). The amount of fuel the spaceship
initially carry is MFUEL(?kg), which energy per mass is EPM (?Joule/kg). The spaceship engine efficiency is SEP (?%), and the engine is capable of burning FPS (?kg/s) mass of fuel per second. If the interstellar medium friction constant is IMD (?/s), calculate :

1. The maximum cruising speed ? (SRF and IRF)
2. Length of acceleration and deceleration phase ? (SRF and IRF)
3. Length of constant cruising speed ? ( I expect it to be harder if
there is IMD ). (SRF and IRF)
4. The heat generated by friction ?
5. How the question may look like if we add the energy required to
support the crew which is ESC (Watt) into the problem ?

SRF = Ship's Reference Frame
IRF = Inertial Reference Frame
 
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  • #2
Here is where my math work had progressed so far :

Look in the variable definition above, in order to understand what
each variables in the equations below means.

So SSP = SEP*EPM*FPS ... (1)

Since (SSP=Space Ship Power) is defined as, the amount of energy the
space ship is capable of produce per unit of time, and energy means
the capability to move a mass M as far as H, using certain amount of
acceleration A, that means :

SSP = TMAR*A*H/dt = 0.5*TMAR*A^2*t ... (2)

TMAR : Total Mass After Relativity.

Which is defined as :

TMAR = TMBR/sqrt(1-(v/c)^2) ... (3)

TMBR = MSS + MFUEL - FPS*t ... (4)

TMAR : Total Mass Before Relativity.

So, I subtitute SSP from (1) to (2) :

SEP*EPM*FPS = 0.5*TMAR*A^2*t , move A^2 from right hand side to left
hand side :
A^-2*SEP*EPM*FPS = 0.5*TMAR*t, move SEP*EPM*FPS from left hand side
to right hand side :

A^-2 = (0.5*TMAR*t) / (SEP*EPM*FPS) , flip both side

A^2 = (SEP*EPM*FPS) / (0.5*TMAR*t) ... (5)

Then I subtitute TMBR from (4) to (3) :

TMAR = (MSS + MFUEL - FPS*t)/sqrt(1-(v/c)^2) ... (6)

Then I subtitute TMAR from (6) to (5) :

A^2 = (SEP*EPM*FPS) / (0.5*(MSS + MFUEL - FPS*t)/sqrt(1-(v/c)^2)*t)

A^2 = (SEP*EPM*FPS* sqrt(1-(v/c)^2)) / (0.5*(MSS + MFUEL - FPS*t)*t)

A = sqrt((SEP*EPM*FPS* sqrt(1-(v/c)^2)) / (0.5*(MSS + MFUEL -
FPS*t)*t))

Since A = dv(t)/dt, the problem become this differential equation :

dv(t)/dt = sqrt((SEP*EPM*FPS* sqrt(1-(v(t)/c)^2)) / (0.5*(MSS + MFUEL
- FPS*t)*t))

Written down in Maple Format :

diff(v(t),t) = sqrt((SEP*EPM*FPS* sqrt(1-(v(t)/c)^2)) / (0.5*(MSS +
MFUEL - FPS*t)*t));

So, did I do something wrong ?
Where should I put the IMD (Interstellar Medium Friction Constant (?/s)) into the equation ?
Is there a way to integrate the differential equation ?
 
  • #3


1. The maximum cruising speed in the SRF can be calculated using the relativistic velocity equation:
v = c * tanh(sqrt((FPS * EPM * SEP * MFUEL - IMD * MSS * RAB^2) / (MSS * RAB^2 * c^2)))
Where c is the speed of light in a vacuum. This equation takes into account the fuel burn rate, engine efficiency, friction constant, and the initial mass and distance of the spaceship. In the IRF, the maximum cruising speed would be the same, as both frames of reference are moving at constant velocity relative to each other.

2. The length of the acceleration and deceleration phase can be calculated using the relativistic acceleration equation:
a = FPS * EPM * SEP / (MSS * sqrt(1 - (v/c)^2))
Where a is the acceleration and v is the maximum cruising speed. The length of the acceleration and deceleration phase in the SRF would be the same, but in the IRF it would appear shorter due to time dilation effects.

3. The length of the constant cruising speed phase can be calculated by subtracting the length of the acceleration and deceleration phase from the total distance RAB. This would be the same in both the SRF and IRF, as the spaceship is moving at constant velocity in both frames.

4. The heat generated by friction can be calculated using the formula:
Heat = IMD * MSS * v^2
Where v is the maximum cruising speed. This heat would be generated in both the SRF and IRF, but may appear different due to time dilation effects.

5. If we add the energy required to support the crew (ESC) into the problem, the calculations would become more complex. This would require taking into account the energy needed to sustain the crew for the duration of the journey, which would depend on the length of the constant cruising speed phase. Additionally, the fuel burn rate and engine efficiency may also be affected by the added weight of supporting the crew. The equations for calculating the maximum cruising speed, acceleration and deceleration phase, and constant cruising speed phase would need to be adjusted to incorporate the energy required for the crew.
 

1. How is travel time calculated in relativistic travel?

In relativistic travel, the travel time is calculated using the time dilation equation t = t0/√(1-v2/c2), where t0 is the proper time, v is the velocity of the object, and c is the speed of light.

2. What is the difference between proper time and observed time in relativistic travel?

Proper time refers to the time experienced by an object in its own frame of reference, while observed time refers to the time measured by an observer in a different frame of reference. Due to time dilation, the observed time may be longer or shorter than the proper time.

3. How does the speed of the object affect the travel time in relativistic travel?

The speed of the object has a significant impact on the travel time in relativistic travel. As the speed approaches the speed of light, the time dilation effect becomes more pronounced and the travel time increases significantly.

4. Can the travel time in relativistic travel ever be shorter than the distance divided by the speed of light?

No, the travel time in relativistic travel can never be shorter than the distance divided by the speed of light. This is because the speed of light is the maximum speed limit in the universe, and it is impossible to travel faster than that.

5. Are there any other factors besides speed that can affect the travel time in relativistic travel?

Yes, the travel time in relativistic travel can also be affected by factors such as gravitational fields and acceleration. These factors can also cause time dilation and affect the travel time experienced by an object.

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