Apollo 8 delta-v calculation

The Apollo 8

Apollo 8 mission launched on December 21, 1968, and it was the second crewed spaceflight mission flown in the Apollo space program. The first mission, Apollo 7, stayed in Earth orbit. Apollo 8 mission was the first ever mission to leave the Earth's gravitational sphere, and to reach the Moon. The crew orbited the Moon ten times and returned home. It went without the landing. However, the mission was the first mission to see and take a photo of the far side of the Moon. 

The mission consisted of three astronauts: Frank Borman, James Lovell, and William Anders. 

Apollo 8 was the third flight and the first crewed launch of the Saturn V rocket. 

Earthrise: Taken from Apollo 8; source: Wiki

When building a rocket, it is a good thing to do staging. Staging basically means throwing away big, expensive parts of your rocket while the rest continues on the way up. The main advantage is not to carry dead weight. 

When one has multiple powered stages, the question arises as to how much Δv (a measure of the rocket's ability to push, and a function of the amount of propellant) to allocate to each stage. 

Let's make some calculation on very interesting example Apollo Saturn V, Apollo 8, related to my previous article Tsiolkovsky rocket equation. 

On all the missions after Apollo 8, there was also a Lunar Module, which flew with the Command and Service Module until the moon. However, because it is difficult to work out exactly how much delta-v is spent before the Lunar Module is jettisoned, let’s analyze the Apollo 8 mission, which did not carry a Lunar Module. Instead, Apollo 8 carried it. 

A Lunar Test Article (LTA), mass 9 000 kg, which is jettisoned along with the third stage,.

What is the total delta-v available to Apollo 8?

We can work backwards through this list:

CSM (empty): 11 900 kg

CSM (full): 28 800 kk

S-IVB (empty) + LTA + CSM (full): 51 300 kg

S-IVB (full) + LTA + CSM (full): 160 800 kg

S-II (empty) + S-IVB (full) + LTA + CSM (full): 200 900 kg

S-II (full) + S-IVB (full) + LTA + CSM (full): 657 000 kg

S-IC (empty) + S-II (full) + S-IVB (full) + LTA + CSM (full): 797 000 kg

full Apollo 8 rocket: 2 947 000 kg

With these, we can calculate the delta-v produced in each set of burns:

So, all together the delta-v budget is 15.73 km/s.