Mass of fuel would ALSO increase as spaceship comes near the speed of light

[ZapperZ]

something else you might want to think about is that the ship and fuel mass increasing close to the speed of light is going to put you closer and closer to collapsing into a black hole so the extra fuel mass will get you nowhere past a certain point. i do not remember where or how much but somebody somewhere has calculated how much energy can be in a local area before collapsing and it's not infinity. if you rearrange e=mc^2 you get m=e/c^2. the mass of a particular amount of energy is equal to it's energy in joules divided by the speed of light in meters per second squared. once this reaches 3.4 times the mass of sol it will collapse into a black hole.

Er.. please be careful with your response. The "m" in that equation is the invariant mass. Relativistic mass is not an appropriate term anymore. Please read the FAQ in the General Physics forum.

Your application of Special Relativity is faulty here. You are also responding to a thread that is more than a year old.

Zz.

 

[schonovic]

sorry about responding to an older thread. i didn't think i used the term relativistic mass in that response but o.k., what does invariant mass mean? do you mean that a heavy mass close to the speed of light will not collapse into a black hole even if it is massive as a black hole?

 

[ZapperZ]

Did you even attempted to read the FAQ that was suggested?

You did refer implicitly to relativistic mass when you said:

...the ship and fuel mass increasing close to the speed of light...

If that isn't a reference to relativistic mass, then you've just violated conservation of energy.

Zz.

 

[schonovic]

i seem to be missing the point, even if the fuel mass increases with the ship mass: so what, nothing can exceed the speed of light anyways so why do we care if the fuel mass increases...does it make a difference?

 

[ZapperZ]

i seem to be missing the point, even if the fuel mass increases with the ship mass: so what, nothing can exceed the speed of light anyways so why do we care if the fuel mass increases...does it make a difference?

Yes, because you are using that faulty "mass increase" to cause a "... collapse into a black hole.."

I'm assuming that you now know what "invariant mass".

Zz.

 

[schonovic]

o.k. Zz i stand corrected. it is invarient mass. i simply use m=e/c^2 to calculate how much recoil the weapon systems on my science fiction starships experience. if i fire a 10^17 joule laser on a 450,000 M.T. starship how much kick will the ship experience? however it seems i can't do it that way. maybe i'll have to learn a new way, thanks.

 

[JesseM]

o.k. Zz i stand corrected. it is invarient mass. i simply use m=e/c^2 to calculate how much recoil the weapon systems on my science fiction starships experience. if i fire a 10^17 joule laser on a 450,000 M.T. starship how much kick will the ship experience? however it seems i can't do it that way. maybe i'll have to learn a new way, thanks.

If m is the rest mass rather than the relativistic mass, then E=mc^2 is the energy of an object at rest (so 0 kinetic energy), for a moving object the total energy is given by E^2 = m^2 c^4 + p^2 c^2, where p is the relativistic momentum given by p = mv/sqrt(1 - v^2/c^2) for massive objects (for photons this formula for relativistic momentum doesn't work, but since a photon has m=0 you can reduce the previous formula to E=pc for photons, so p=E/c for photons). If you let M equal the relativistic mass, so M=m/sqrt(1 - v^2/c^2), it turns out that the formula for the total energy of a moving object reduces to E=Mc^2.

So if all the energy of the laser goes into increasing the linear kinetic energy of the starship (not a totally realistic assumption but never mind), and if the starship was initially at rest in the frame where the oncoming laser had an energy of 10^17 joules, then before being accelerated the ship must have had a rest energy of E = (450,000,000 kg)*(299792458 m/s)^2 = 4.04439830431568 * 10^25 kg m^2 / s^2 = 4.04439830431568 * 10^25 joules. Then after it's accelerated, the new total energy of the ship must be 4.04439830431568 * 10^25 + 10^17 joules = 4.04439831431568 * 10^25 joules. So this energy must be the one that comes out of E=Mc^2=mc^2/sqrt(1 - v^2/c^2), which means we have:

4.04439831431568 * 10^25 joules = 4.04439830431568 * 10^25 joules / sqrt(1 - v^2/c^2)

This indicates that the gamma-factor 1/sqrt(1 - v^2/c^2) must be given by:

(4.04439831431568 * 10^25 joules) / (4.04439830431568 * 10^25 joules) = 1.00000000247256

Solving 1/sqrt(1 - v^2/c^2) = 1.00000000247256 for v gives a velocity of about 7*10^-5 * c, or about 21,000 meters/second, in about the same range as the Voyager 1 spacecraft (around 17,000 meter/second).

edit: sorry, I thought you wanted to know how much kick the ship would gain if it absorbed the energy of a laser, but rereading I see you were asking about the recoil when it fires the laser. This is a bit more complicated, since in order to shoot out a laser the ship must convert some internal potential energy into photons, and thus its rest mass will actually change slightly (the rest mass of a multiparticle system includes internal potential energy and heat along with the rest mass of all the particles individually), along with a change in its kinetic energy. I'm not sure how to solve this problem exactly, although if m is its rest mass before firing the laser and m' is the rest mass after firing, then since energy is conserved the following should be true in the frame where the ship was initially at rest:

mc^2 = m'c^2/(1 - v^2/c^2) + 10^17 joules

...since m'c^2/(1 - v^2/c^2) incorporates both the ship's rest energy and its kinetic energy after firing the laser, and the total energy after firing the laser is (ship's rest energy) + (ship's kinetic energy) + (laser's energy).

 

[Dale]

sorry about responding to an older thread. i didn't think i used the term relativistic mass in that response but o.k., what does invariant mass mean? do you mean that a heavy mass close to the speed of light will not collapse into a black hole even if it is massive as a black hole?

If it is not a black hole in the frame where it is at rest then it will not be a black hole in any other frame regardless of its total energy (which can be arbitrarily high) in that other frame.

 

[schonovic]

JesseM thanks for taking all the effort over a simple science fiction spaceship. I'm sorry i did not actually completely elaborate on how i calculate the kick my laser gives my ship. i first use m=e/c^2 to calculate the mass of the laser energy, then i multiply the lasers mass times its velocity to figure how many Newtons of force it exerts durring the fring. then i apply that force to the mass of my ship to figure the kick of the laser. if this is wrong i simply figure it is for science fiction as long as i always follow the same rules for all ships I'm o.k. I'm just a simple sci-fi creator and do not have a degree in physics.

 

[JesseM]

JesseM thanks for taking all the effort over a simple science fiction spaceship. I'm sorry i did not actually completely elaborate on how i calculate the kick my laser gives my ship. i first use m=e/c^2 to calculate the mass of the laser energy, then i multiply the lasers mass times its velocity to figure how many Newtons of force it exerts durring the fring.

That won't work at all--mass times velocity gives momentum, not force! (force is mass * acceleration) But figuring out the momentum (p) of the laser should actually work, since conservation of momentum implies the ship should gain an equal and opposite momentum (in the frame where the ship was originally at rest before firing the laser). For photons with have zero rest mass, the equation E^2 = m^2 c^4 + p^2 c^2 reduces to E=pc, so the momentum of the laser (and the equal and opposite momentum of the ship after firing) would be given by p=E/c. For a laser with energy 10^17 joules = 10^17 kg meters^2 / second^2, that works out to a momentum of (10^17 kg meters^2 / second^2 ) / (299792458 meters/second) = 333564095 kg meters/second. So that should be the momentum change of the ship, and although relativistic momentum is given by m*v/sqrt(1 - v^2/c^2), for velocities that are small compared to the speed of light it's reasonably accurate to just use the classical momentum m*v. So:

m*v = 333564095 kg meters/second
v = (333564095 kg meters/second) / m

And with m = 450000000 kg, this gives the ship a velocity of 0.74 meters/second after firing the laser. But caution, this is a very different answer from the one I got using energy--since this calculation is a lot simpler, I think the previous one probably contained an error somewhere.

Anyway, to sum up, if a ship of mass m fires a laser with energy E, then the ship's change in velocity v (in the frame where it was previously at rest) can be calculated with this simple equation:

v = E/mc

(make sure your units are consistent though--for example, if you use joules for energy, then since 1 joule = 1 kilogram*meter^2/second^2, you must use kilograms for the mass m of the ship and c = 299792458 meters/second)

 

[schonovic]

i know you say my way will not work but i'v tried it your way and my way and they get the exact same answer. it's my presentation that is faulty. I'm not describing my method in clear detail. i'll try again. first i use m=e/c^2. then i multiply m*light speed to get Newtons. then i divide Newtons by ships mass to get velocity after firing the laser. maybe it works because you could be accelerating @ 300000000m/sec^2 for one second. it's wrong i guess i just got lucky somehow it gets the same answer. anyways thanks for the work and the nice handy dandy equation. I'm glad we have guys like you to steer us fumbling laymen in the right direction again thanks and let me get off this year ond thread. bye.

 

[qraal]

Hi jonnyk
If you look closer at the discussion of relativistic rockets at Baez's site you'll notice total energy is conserved. Read it a bit closer and you'll see what I mean.

 

[qraal]

Hi jonnyk
If you look closer at the discussion of relativistic rockets at Baez's site you'll notice total energy is conserved. Read it a bit closer and you'll see what I mean.

Egads! I just noticed how old the post I am replying to was. Oops.

 

[schonovic]

mass times velocity gives momentum but that momentum and mass applied to an inert object has an effect.