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How hight up? In a vaccuum, there's no areodynamic drag to deal with, just 1/6g. And the CSM they were joining with was orbiting at about 60 miles up. It didn't take a lot of fuel to get up there, compared to launching into Earth orbit.
Quoting: Anonymous Coward 22634579 can you say "shill"?
Lets do the analysis:
Weight: 10,300 lbs (we'll call this Mo)
Thrust: 3,500 lbs (we'll call this Th)
APS specific impulse: 311 sec
APS propellant mass: 5,187 pounds
Gravity moon: 1.624 m/s² (we'll call this Gl
CSM speed = 3649.3 mph
Orbit assumed to be 100 km (numbers from 100km to 160km are listed and CSM could come to within 20 km of the moon)
Simple test is V = AT
Given the starting acceleration of 1G, the time is around 170 seconds.
The fuel consumption of 11.3 lb/s. 11.3*170 = 1921 lbs.
So they had more fuel than they needed - 2.5 times as much. Some of this was used to correct the orbit etc. etc. This is one reason Apollo 13 fared so well, they had fuel to burn.This isn't quite correct see below
yer cooking the books.
When I recomputed the problem in metric units I realized a number of things.
1. The original was a simple trajectory - perhaps too simple.
2 . The force of gravity is reduced by the 11.25 lbs/s fuel consumption. This effect is linear
3. The force of gravity is reduced by centripetal force - which varies with speed and is exponential (1/2 speed = 3/4 weight, orbital speed = 0 weight).
We really should be solving for a two dimensional path and perhaps using polar coordinates, however for a simple scalar solution:
Vc is current speed.
Rm is radius of the moon
T is time
A = (Th - Mo*LG( 1 - (Vc*Vc)/Rm -5.1 T))/Mo
Integrate with respect to time and orbital velocity. Solve for Velocity = Command Module speed.
If you try approximate this you get a number in the range of 441 seconds the same as they did. Fuel exhaustion occurs at 461 seconds so they have 20 seconds to spare.
No adjustments 900 seconds.
Adjustment for fuel burn reduces burn time to 600 seconds.
Adding adjustment for centripetal force makes the burn time right around 440.
As to how they flew it - they had cheat sheets for angles and burn times that were precomputed by the engineering staff.
