The problem is not the atmosphere. Getting to space is relatively easy. Getting to orbital velocity is the hard part. The crux of the problem is the exponential nature of the rocket equation.
The way I see it there are three hurdles to getting a self sufficient colony going. Any solution will probably require combination of advancement in several of the following areas.
1. Funding. Anyone who has seen "The right stuff" knows it's funding that makes the rockets go up. In this category there are two options for this.
1a. Government funding. This requires political will. The only way I see this happening is if two large powers are competing like in the cold war. Until that happens people would rather devote there production possibilities frontier to penis enhancing sports cars and episodes of jersey shore. But if we really wanted to we could probably make a colony in the next 15 years for about 10% of an Iraqi invasion.
1b. Philanthropy. Lets face it the funds of the private sector pale in comparison to government funding. This might work however with substantial improvements in the other two categories.
2. Lowering the mass necessary for self sufficiency. According to a NASA study "Advanced Automation for Space Missions" A semi-self sufficient moon colony could be started with 100 tons. (This would still require supply missions of hard to manufacture items like microprocessors and tool heads.)
3. Lowering the cost to orbit. There seems to be several solutions to this.
3a. Conventional chemical rockets: The big improvement that needs to be made here is re-usability. Rocket fuel accounts for less than 1% of the cost per launch.
Space X is making some headway in this department. There current quoted price is in the $3.7million/ton to LEO range. With reusable designs an improvement in cost by a factor of 10 seems plausible. This puts a colony in the $10billion range within reach.
3b. Other more exotic orbital launch systems(nuclear light-bulb, launch loop, space elevator
The competing agency FARPA is developing competitor to the LS3 technology. The name for this project is the Military Utility Logistics Engine. The stats are about the same except:
MULE has a payload of only 200lbs
MULE is quieter
MULE is capable of in situ resource utilization simplifying logistics
MULE is capable of doubling as a food source.
MULE's per unit cost is
FARPA is also working on a more advanced project known as DONKEY, that will have self replicating abilities. Unfortunately this project is still in the early development stages.
When asked about the cost discrepancy between the $5,000,000,000 LS3 project vs the much more cost effective $500 MULE project, Congressmen Porkbarrel, R, MA replied: "I'm sorry I cant hear you over the sound of all this bribe money"
Here is a link to an early prototype of LS3:
And starvation is one hell of a motivator.
Unfortunately starvation is not a motivator for the people who most influence the global economic system. Profit is the motivator.
Baitin' is what I do for entertainment too.
Take the case of throwing a baseball. This is case 2 from parent
Assume that the magnitude of the drag on the ball is proportional to the square of its velocity. Also, assume that the magnitude of the gravitational force is constant. You get the following set of differential equations:
x''(t)*m=A * (x'(t)^2+y'(t)^2) * cos(theta)
y''(t)*m=A * (x'(t)^2+y'(t)^2) * sin(theta) + g * m
x(t) is the horizontal position of the ball at time t
x'(t) is the horizontal velocity of the ball
x''(t) is the acceleration
y(t) is the vertical position of the ball
(x'(t)^2+y'(t)^2) is the square of the velocity
theta is the angle of travel above the horizontal
m, A and g are constant over time*
A=-1/2*drag coefficient*cross sectional area of ball*air density
m is the mass of the ball
g is acceleration due to gravity