Equation for Relativistic Exhaust Velocity (Ve): Help Needed

[BitWiz]

Does anyone have an equation that -- given energy applied to a mass over a distance -- will give me an (ideal) final velocity of the mass?

If I direct a gigajoule to accelerate a gram over a distance of a meter, I obviously get into real trouble with c using conventional equations.

I've found equations that apply Lorentz -- but they seem to do it as an adjustment after a raw, conventional velocity has been obtained, using this final, conventional velocity to plug into Lorentz. That can't work in this case. I'm looking for a function Ve( joules, mass, distance ) that works for all non-negative parameter values, and takes relativity into account. Can you help?

Thank you very much!

 

[PeterDonis]

Does anyone have an equation that -- given energy applied to a mass over a distance -- will give me an (ideal) final velocity of the mass?

Take the ratio of total energy to rest energy: that gives you the relativistic $\gamma$ factor. Then invert the formula for $\gamma$ in terms of $v$ to get $v$.

 

[PeterDonis]

If I direct a gigajoule to accelerate a gram over a distance of a meter, I obviously get into real trouble with c using conventional equations.

You should check your math first. Hint: what is the rest energy of one gram of mass, in gigajoules?

 

[PeterDonis]

given energy applied to a mass over a distance

The distance is actually irrelevant to the final velocity. It is only relevant to determining the acceleration required to achieve that velocity. The final velocity only depends on the total energy applied.

 

[BitWiz]

You should check your math first. Hint: what is the rest energy of one gram of mass, in gigajoules?

c² / 1000 ?

 

[PeterDonis]

c2 / 1000 ?

If you calculate this numerically and convert to gigajoules, yes. What do you get?

 

[BitWiz]

The distance is actually irrelevant to the final velocity. It is only relevant to determining the acceleration required to achieve that velocity. The final velocity only depends on the total energy applied.

So, v = v' * rest_mass / (Rest_mass + Jo

If you calculate this numerically and convert to gigajoules, yes. What do you get?

~898756 Gj ?

 

[PeterDonis]

v = v' * rest_mass / (Rest_mass + Jo

No. Try starting from what I said in post #2.

~898756 Gj ?

I think you're too high by an order of magnitude.

 

[BitWiz]

Whoops.
If a joule = 1 (kg) / c², then 1kg = c²j, and a gram is 1000th of that?

If c is ~ 3e8, then c² is 9e16j for a kg. For a gram, 9e13j. For a gigajoule, 9e4?

 

[BitWiz]

No. Try starting from what I said in post #2.

You're saying that gamma is 1 + ( mass-equivalent-of-energy-added / rest mass ) ?

Thanks!

 

[PeterDonis]

You're saying that gamma is 1 + ( mass-equivalent-of-energy-added / rest mass ) ?

That's one way of putting it, yes. Or you could just do everything in energy units; it works out the same either way.