russ_watters said:
I think what you are saying here is wrong (and is your basis for use of the word "squandered"). Could you please explain in more detail why you think this is true...perhaps by responding to post #82 or providing some actual vector math.
Post #82 mentions three launch scenarios but then only shows a launch profile for one of them. That launch profile is a failure -- the result would be a crash and burn. Here is that profile:
russ_watters said:
Starting: 900 mph at 90 degrees (east)
Needed: 17500 mph at 135 degrees (angle guessed)
Assist: 636 mph
Wasted: 900-636=264mph
Delta-V: 16,863 @ 137 degrees
As I read it, that's an impulsive burn with a delta v of 16,863 mph on a ramp angled 47 degrees above the horizontal. Plus the Earth's rotation gives you a launch angle of 45 degrees true. That gives you an elliptical orbit that intersects with the surface of the Earth.
You have enough energy to get into a circular low Earth orbit. But you do not have the right angle to do so. With this launch profile, you are going have ##\frac{\sqrt{2}}{2}v## too much vertical velocity and ##(1-\frac{\sqrt{2}}{2})v## too little horizontal velocity. You would need a circularizing burn somewhere to make up the difference. More delta v to spend.
By contrast, if you had simply thrust horizontally, an initial delta v of only 17500-900 = 16600 mph would have done the job and put you into a circular low Earth orbit. Less delta v. And no need to spend more to avoid crashing.
To some extent this is an apples and oranges comparison. Your launch profile plus circularizing burn puts the craft into a high Earth orbit. The horizontal launch profile puts the craft into a low Earth orbit. The two are not the same. [To say nothing of the pesky atmosphere]
But here is the thing. Mechanical energy is conserved. It does not matter what direction the craft is moving at a particular speed.
Its energy at a given altitude is a function of speed alone. An impulsive horizontal burn of 16,863 mph added to a starting velocity of 900 mph gives you an orbital speed of 17,763 mph. An impulsive burn of 16,863 mph at a 47 degree angle skyward added to a starting velocity of 900 mph horizontal only gets you to 17,500 mph.
You've squandered energy and you're not getting it back.
If I were doing it right, I'd save that couple of hundred mph of delta v and go straight into a Hohmann transfer from a circular low Earth orbit. My Hohmann transfer is more efficient than yours due to the Oberth effect. I get to start my burn moving low and fast. You start your burn moving high and slow.
