Question about exhaust velocity and terminal velocity

In summary, the conversation discusses a model of a relativistic rocket that accelerates with constant power and variable acceleration and total mass. The surprising result is that a rocket with lower exhaust velocity will have a higher terminal velocity. However, it is pointed out that a rocket in space does not have a terminal velocity and will continue to accelerate as long as it has fuel. The conversation also delves into the equations used for mass energy conservation and linear momentum conservation, as well as clarifying the definition of exhaust velocity. The conversation concludes with the suggestion to provide a more organized and readable version of the equations for further discussion.
  • #1
rhz_prog
17
0
I had made a model assuming a relativistic rocket accelerating with constant power,variable acceleration,variable total mass the result however is somehow counterintuitive. The less exhaust velocity the rocket have, the higher terminal velocity it will have.

Is that means I did something wrong ?
 
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  • #2
Terminal velocity is the final equilibrium velocity of an object against a drag force, typically in freefall. A rocket in space doesn't have a terminal velocity - it will continue to accelerate as long as it has fuel. The closest thing to it you can get is an abitrarily large rocket, regardless of the specifics of the rocket, will have a maximum velocity that approaches C.

Are you trying to calculate an acceleration profile in Relativity? My guess is that your error isn't with the relativity part, it's with the rocket exhaust velocity vs acceleration part. Could you give us the equation you are using...?
 
  • #3
rhz_prog said:
I had made a model assuming a relativistic rocket accelerating with constant power,variable acceleration,variable total mass the result however is somehow counterintuitive. The less exhaust velocity the rocket have, the higher terminal velocity it will have.

Is that means I did something wrong ?

I am assuming you are asking if a rocket with a given total mass and a given fuel payload will reach a higher terminal velocity by the time it runs out of fuel, by using a lower constant exhaust velocity?

You have to specify what you mean by exhaust velocity. Do you mean proper exhaust velocity as measured by the observer on the rocket or exhaust velocity as measured by an inertial observer? If you mean the latter, you should be aware that the exhaust velocity will be getting less over time due to time dilation as a function of the rockets instantaneous relative velocity. In Newtonian calculations in a gravitational field, burning all the fuel as rapidly as possible is the most effecient way to launch a rocket and achieve the maximum terminal velocity for a given fuel load. It can be seen that the opposite extreme, is burning all the fuel so slowly that the rocket is only hovering and its final velocity when the fuel is exhausted is zero. From the Baez FAQ here http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html [Broken] it can be deduced that the final terminal velocity v/c of a photon drive (exhaust velocity=c) rocket is:

v/c = (F+2m)/(F+2m+2m2/F) where F is fuel load of the rocket and m is the payload of the rocket (total mass of the rocket less the fuel load).

Generally speaking the higher the exhaust velocity the greater the efficiency of converting fuel to momentum and the higher the terminal velocity of the rocket. You would have to make it clear about exactly what circumstances you are considering.
 
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  • #4
kev said:
From the Baez FAQ here http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html [Broken] it can be deduced that the final terminal velocity v/c of a photon drive (exhaust velocity=c) rocket is:

v/c = (F+2m)/(F+2m+2m2/F) where F is fuel load of the rocket and m is the payload of the rocket (total mass of the rocket less the fuel load).

Yes I used Baez Relativistic Rocket FAQ to make my model.

I have to modify the equations though, since what I want is a model with :
- constant wattage
- constant exhaust velocity relative to the ship (is this the one called proper exhaust velocity)
- variable proper acceleration
- variable coordinate acceleration

The equations used is attached as 2 jpg images :

Mass Energy Conservation equation :
PBT*EPM*t = (MShip+MFuel-FBT*t)*c^2/(1-v^2/c^2)^(1/2)-(MShip+MFuel-FBT*t)*c^2+EXE
inside massenergyconservation.JPG

Linear Momentum Conservation equation :
(MShip+MFuel-FBT*t)*v/(1-v^2/c^2)^(1/2) = EXE/EXV
inside linearmomentumconservation.JPG

The variables used is as follow :

PBT : Mass of Fuel burned per unit time
EPM : Fuel Potential Energy per unit mass
MShip : Rest Mass of the Ship
MFuel : Rest Fuel Initial Mass
EXE : EXhaust Energy
EXV : EXhaust Velocity

The complete derivations I used is attached as mws file.

Thank you.
 

Attachments

  • Relativity Travel-Momentum.mws
    6 KB · Views: 398
  • linearmomentumconservation.JPG
    linearmomentumconservation.JPG
    4.2 KB · Views: 453
  • massenergyconservation.JPG
    massenergyconservation.JPG
    6.3 KB · Views: 454
Last edited by a moderator:
  • #5
russ_watters said:
Terminal velocity is the final equilibrium velocity of an object against a drag force, typically in freefall. A rocket in space doesn't have a terminal velocity - it will continue to accelerate as long as it has fuel. The closest thing to it you can get is an abitrarily large rocket, regardless of the specifics of the rocket, will have a maximum velocity that approaches C.

Oh I just remember that. Guess I will have to learn more English Physics Vocabularies.

Btw what I mean is the highest velocity the rocket would reach after using some part of its fuel to accelerate itself, provided that the rocket should have enough fuel to decelerate and arrive at its destination.

russ_watters said:
Are you trying to calculate an acceleration profile in Relativity? My guess is that your error isn't with the relativity part, it's with the rocket exhaust velocity vs acceleration part. Could you give us the equation you are using...?

Yes it is given in the mws file I attached on my other reply above.
 
  • #6
I think it is inappropriate for me to give the chaotic mws file like the first one I attached. This is the new mws file which have been divided to section. I hope this one is more readable.

Thank you
 

Attachments

  • Relativity Travel-S2.mws
    16.7 KB · Views: 379

1. What is exhaust velocity?

Exhaust velocity is the speed at which exhaust gases are expelled from a rocket engine. It is typically measured in meters per second (m/s).

2. How is exhaust velocity related to thrust?

Exhaust velocity is directly related to thrust, as it is one of the factors that determines the amount of thrust produced by a rocket engine. A higher exhaust velocity results in a higher thrust.

3. What factors affect exhaust velocity?

The exhaust velocity of a rocket engine is affected by several factors, including the type of propellant, the design of the engine, and the combustion process.

4. What is terminal velocity?

Terminal velocity is the maximum speed that an object can reach when falling through a fluid, such as air. It is determined by the balance between the object's weight and the drag force of the fluid.

5. How is exhaust velocity related to terminal velocity?

Exhaust velocity plays a crucial role in determining the terminal velocity of a rocket or spacecraft. A higher exhaust velocity allows the vehicle to achieve a higher speed and overcome the drag force of the surrounding fluid, resulting in a higher terminal velocity.

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