APOLLO MOON LANDING CALCULATIONS			 8 Feb 1996

Steven S. Pietrobon, Small World Communications, 6 First Avenue
Payneham South SA 5070, Australia           fax +61 8 7117 1416
steven@sworld.com.au                  http://www.sworld.com.au/

Here are various delta-v's I have calculated for the Apollo Moon landing. I 
used the Apollo 14 flight report for most of my data:

The masses of each of the stages are (converted from imperial units)

      m_p+m_s (t)   m_s (t)   m_p (t)
LES         4.1          ?         ?  (Launch Escape System)
SC         46.3          ?         ?  (Spacecraft)
S-IVB     119.9       11.8     106.1  (Third Stage plus Instrument Unit)
S-II      491.5       39.1     452.4  (Second Stage)
ISA         4.6        4.0       0.6  (Interstage Adaptor)
S-IC     2284.9      132.6    2152.2  (First Stage)

where m_p is the propellant mass and m_s is the structure mass.

The S-II was assembled with the interstage adaptor (ISA) between S-IC and S-II. 
The ISA is in fact made from two parts. A lower thin part bolts permantly to 
the S-IC in the vehicle assembly building (VAB). The S-IC with this lower 
section then separates from the large upper section of the ISA. Solid 
propellant separation rockets on the large ISA then fire to settle the 
propellants in the S-II. The large ISA is then seperated from S-II. Similarly, 
the S-IVB is assembled with the ISA between the S-II and S-IVB (the large 
conical section). The ISA is permantly bolted to the S-II in the VAB, with the 
S-II and ISA seperating from the S-IVB on staging.

The F-1 is has a vacuum v_e = 3002 m/s. The first stage v_d is then

v_d_1 = 3002 ln(1 + 2152.2/(132.6+4.6+491.5+119.9+46.3+4.1)) = 3922 m/s

This compares with the actual 2713 m/s from [1] where gravity and air friction 
and pressure losses are taken into acount. The delta-V from the ISA is

v_d_2 = 2491 ln(1 + 0.6/(4.0+491.5+119.9+46.3+4.1)) = 2 m/s

The ISA is jettisoned 28.3 seconds after S-II ignition. Each J-2 engine 
generates 1038.6 kN thrust with a v_e = 4153 m/s at a mixture ratio of 5.5:1. 
For a total thrust of 5193 kN and a constant propellant usage rate this gives 
the total propellant usage of 35.4 t. The v_d before the ISA is jettisoned is 

v_d_3 = 4153 ln(1 + 35.4/(4.0+456.1+119.9+46.3+4.1)) = 227 m/s

At 34 seconds after S-II ignition the LES is jettisoned. The propellant mass
used after ISA separation is 7.1 t. The v_d before the ISA is jettisoned is

v_d_4 = 4153 ln(1+ 7.1/(449.0+119.9+46.3+4.1)) = 47 m/s

The remaining S-II propellant is 452.4-(35.4+7.1) = 409.9 t. The v_d after LES 
jettison is

v_d_5 = 4153 ln(1 + 409.9/(39.1+119.9+46.3)) = 4558 m/s

The total v_d from the second stage is then 2+227+47+4558 = 4834 m/s (compared 
with 4108 m/s actual). The S-IVB is accelerated by two solids with 15.1 kN 
thrust each. The total propellant mass is 0.05 t. The v_d from the S-IVB ullage 
motors is then

v_d_6 = 2491 ln(1 + 0.05/(119.85+46.3)) = 1 m/s

The S-IVB then fires for 150 s to go into low Earth orbit (LEO). The propellant
ratio is 5.0:1 with F = 888.7 kN and v_e = 4176 m/s. For a single J-2 engine 
this gives an m_p = 31.9 t. The v_d is

v_d_7 = 4176 ln(1 + 31.9/(87.95+46.3)) = 890 m/s

The total v_d into LEO is then 3922+4834+1+890 = 9647 m/s. The remaining S-IVB 
propellant is 106.1-31.9 = 74.2 t giving the trans lunar injection (TLI) v_d as

v_d_8 = 4176 ln(1 + 74.2/(13.75+46.3)) = 3360 m/s

The total Saturn V v_d is then 9647+3360 = 13,007 m/s, about 2000 m/s larger
than the escape speed.