Final velocity is the velocity of the spacecraft just as the last of the fuel is used. The final velocity of a typical pressure fed rocket stage operating in vaccum with no gravity is five thousand meters per second. A pressure fed hydrogen peroxide rocket will have a final velocity of four thousand meters per second. A top of the line graphite pumped rocket fueled with oxygen and hydrogen will have a final velocity of nine thousand meters per second.

In spacecraft in vacuum, bipropellant rocket, tripropellant rocket, pumped rocket and rocket cost final velocity is calculated from the empty mass ratio, exhaust velocity and payload ratio.

final velocity = exhaust velocity * log( empty mass ratio / ( 1.0 + payload ratio ) )
 

In astropolis, atmospheric spacecraft, multi stage spacecraft and spacecraft cost final velocity is numerically integrated as the craft rises up through the atmosphere. The craft faces drag which is proportional to the air pressure and the cube root of its mass. The trajectory is chosen to enter low earth orbit with a low energy expenditure. To that end the angle is calculated at each point in the integration.

function getAngle(
    inAcceleration,
    inAltitude,
    inDeceleration,
    inFinalAcceleration,
    inGravity,
    inLowDragAltitude,
    inLowDragOrbital,
    inXVelocity,
    inYVelocity )
{
    velocity = sqrt( inXVelocity * inXVelocity + inYVelocity * inYVelocity )
    relativeOrbital = velocity / inLowDragOrbital
    relativeGravity = 1.05 * inGravity * ( 1.0 - relativeOrbital * relativeOrbital )
    timeTill = 2.5 * ( 1.01 * inLowDragOrbital - inXVelocity ) / ( inAcceleration + inFinalAcceleration )
    extraTill = inLowDragAltitude - inAltitude - inYVelocity * timeTill
    upwardsTill = ( extraTill / timeTill / timeTill ) + relativeGravity
    upwards = 0.03 / inG * inDecel * inLowDragOrbital * inLowDragOrbital
    upwards += 1.2 * upwardsTill / inAcceleration
    if ( velocity > inLowDragOrbital ) {
        upwards = 0.0
    }
    if ( upwards < 0.0 ) {
        upwards = 0.0
    }
    return atan( upwards )
}
 
 

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