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If not, what are the joules required to do so? And what would the peak wattage be if I am accelerating said ship over 1 year?

Thanks.

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If not, what are the joules required to do so? And what would the peak wattage be if I am accelerating said ship over 1 year?

Thanks.

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jtbell

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Show us the details of how you worked it out, and someone will be glad to tell you whether your basic method is correct and whether you carried out the arithmetic properly.Am I correct in my calculations that this requires app. 1x10^30 joules?

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Filip Larsen

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The relativistic kinetic energy [1] of an object with (rest-) mass m is E_{k} = (γ-1)mc^{2}, where γ is the Lorentz factor [2], so you should be able to estimate the minimum amount of energy you need to accelerate that ship of yours and from there the average power needed if this has to take a year. If the energy is stored in a fuel and brought along the ship you may want to take a peek at the relativistic rocket [3] to see what high speeds means for the fuel mass to payload mass ratio.

[1] http://en.wikipedia.org/wiki/Kinetic_energy#Relativistic_kinetic_energy_of_rigid_bodies

[2] http://en.wikipedia.org/wiki/Lorentz_factor

[3] http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html [Broken]

[1] http://en.wikipedia.org/wiki/Kinetic_energy#Relativistic_kinetic_energy_of_rigid_bodies

[2] http://en.wikipedia.org/wiki/Lorentz_factor

[3] http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html [Broken]

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http://www.circlon-theory.com/images/starshiptitanic.gif

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Borek

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I assume that you're doing this as a relativistic rocketry problem. I'll call the rapidity of the rocket θ. This is related to its velocity by [itex]v=ctanh\left(\theta \right)[/itex]. Lets say the ship has an initial mass including fuel of [itex]m_{i}[/itex] and a final mass [itex]m[/itex]. It will have a ship frame exaust port exaust velocity of [itex]v_{ex}[/itex] which relates to the specific impulse [itex]I_{sp}[/itex] of the fuel by [itex]v_{ex}=gI_{sp}[/itex].

If not, what are the joules required to do so? And what would the peak wattage be if I am accelerating said ship over 1 year?

Thanks.

A change in the rapidity of the ship will then be given by

[itex]\Delta \theta =\frac{v_{ex}}{c}ln\left(\frac{m_{i}}{m}\right)[/itex]

You find the change in the rapidity from this formula and calculate the final velocity from the rapidity according to the first equation. In the case of zero initial velocity you can compose these as

[itex]v=ctanh\left(\frac{v_{ex}}{c}ln\left(\frac{m_{i}}{m}\right) \right)[/itex]

I like this version as its easy to crunch numbers on a scientific calculator, but if you want you can use log properties and the hyperbolic trig version of Euler's identities to write this without the hyperbolic trig function like at a site someone else linked.

In the case of an ideal matter anti-matter rocket [itex]v_{ex}=c[/itex].

As for the question of power, this final velocity - mass ratio relation for the rocket is independent of the flow rate at which you choose to burn it off. Burn it off how you like. However, it is common to analyse the case that you are burning off the fuel at a time dependent rate such that it accelerates at a constant "proper acceleration". This is where the occupant feel like they undergo a constant "g-force" or in otherwords, if one stood on a weight scale it would read a constant weight for the cosmonaught for the whole burn. If the proper acceleration is [itex]\alpha [/itex] then the ship time rate of the mass burn relates to the proper acceleration generally by

[itex]\alpha =\frac{v_{ex}}{m}\frac{dm}{dt'}[/itex], and for constant proper acceleration this integrates to result in

[itex]\alpha \frac{\Delta t'}{c} =\frac{v_{ex}}{c}ln\left(\frac{m_{i}}{m}\right)[/itex]

You can then replace this into the equation for the velocity to get the velocity as a function of ship time for constant proper acceleration. If you want more information about this, just let me know.

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