Suppose we have a spaceship with empty mass of 1000 tonnes. On board is a stored-antimatter energy source capable of propelling the ship by ejecting reaction mass at 0.99 c - and it can do this with 100% energy efficiency. We plan to accelerate at 1G to a velocity of 0.9 c.
How much reaction mass must be carried? How much antimatter “fuel” will be consumed?
This sounds a bit like a homework problem (the numbers are kind of specific), so I’ll just leave this link here. If it’s not a homework problem, then let me know and I’ll show you how to use the equations in question.
It’s definitely not homework.
The link you have provided looks quite useful - I’ll try to digest it.
In that case: the derivation of the ∆v equation is a little complicated, but if you’re fine with just using the result, you have:
∆v/c = tanh ( I[sub]sp[/sub]/c * ln (m[sub]0[/sub]/m[sub]1[/sub] ) )
where ∆v is the final speed (0.9 c); I[sub]sp[/sub] is the specific impulse (0.99 c in your case); m[sub]0[/sub] is the initial mass of the spaceship plus that of the fuel; and m[sub]1[/sub]is the final mass. You want to find m[sub]0[/sub], which (rearranging the above equation) is given by
m[sub]0[/sub] = m[sub]1[/sub] exp( arctanh( ∆v/c ) / (I[sub]sp[/sub]/c) ),
which in your case works out to a initial spaceship mass (fuel + payload) of 4424 tonnes.
The rate of acceleration, by the way, is immaterial if all you’re concerned about is the final velocity. It does come into play, however, if you want to figure out how long it takes to get up to speed (which works out in your case to 1.43 years of shipboard time, or almost exactly two years of Earth time.)