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This is from the Apollo 11 mission report. It says the vehicle was going at 7.6 km/s when the CSM separated from the third stage. It was not in orbit at this point, it was thousands of KM from earth.
Is this accurate? Am I missing something? That seems incredibly far below other sites saying that you need 10km/s or more to leave the earth.
As @Rikki-Tikki-Tavi points out escape velocity is the velocity you would need at (or near) the surface of Earth to make out out of Earth orbit. Of course, just like anything thrown up into the air, the spacecraft decelerates as it moves away from the Earth.
Having escape velocity means that your total energy (relative to the Earth in this case) is greater than or equal to zero.
Using the equations from this answer
$$\mathscr{E}_{tot} = \frac{1}{2}v^2 - \frac{GM}{r}$$
where $\mathscr{E}_{tot}$ is energy per kilogram. Since both terms use the same $m$ you can just calculate energy per kilogram by dropping it.
Let's do the calculation for the "actual" numbers in the table.
altitude 7066 km
Earth radius 6378 km
r 13444 km
GM 3.986E+5 km^3/s^2
v 7.609 km/s
$\begin{align}\frac{1}{2}v^2 & =28.948\\ -\frac{GM}{r} & = -29.649\\ \mathscr{E}_{tot} & = -0.701 \ km^2/s^2 \end{align}$
So the spacecraft has plenty of energy, but not quite enough to completely escape Earth's gravity. That means if it didn't approach the moon, it would keep going until all of the kinetic energy was used up by being stored as potential energy.
$-\frac{GM}{r_{max}} = -0.701$ gives
$r_{max} = 569,000$ km. or a little past the orbital radius of the Moon if the Moon's gravity wasn't there.
So the numbers work out just fine.
If the spacecraft returned to LEO at say 400 km altitude or $r = 6778$, the velocity would then be given by
$-0.701 = \frac{1}{2}v^2 - \frac{GM}{6778 km}$
$v = \sqrt{2(-0.701 + GM/6778 km)} $
and that gives 10.78 km/s
It is normal for a spacecraft to slow down as it coasts towards the Moon. The spacecraft is still in Earth orbit and is being pulled by Earth’s gravity during its entire trip to the Moon. During the powered thrust phase known as TLI (trans lunar injection) the maximum speed reached by the Apollo 11 spacecraft was 10.8 km/sec (24,200 mph) when it was at an altitude of about 300 km (180 miles). By the time of Command Module separation from the S-IVB stage 26 minutes later it had reached an altitude of 7,000 km as you noticed in the table, and by then it had slowed to 7.6 km/sec (17,000 mph). During the approximately three day coast to the Moon it continued to slow, and by the time it reached the Moon it had slowed to 0.9 km/sec (2,000 mph). It then began to speed up again as the Moon’s gravity pulled it towards the Moon.
As explained in other answers the Apollo spacecraft did not reach escape velocity. At the 300 km altitude of TLI escape velocity was 10.9 km/sec. The velocity reached during TLI was just short of this at 10.8 km/sec. It did not have to reach escape velocity because the Apollo spacecraft never left Earth’s orbit. In fact even while orbiting the Moon the spacecraft was still in Earth orbit, as of course the Moon is.
Even if a spacecraft accelerates to escape velocity, as soon as the engines cut off and it begins coasting it will immediately begin slowing down. This slowing will occur as long as the spacecraft is close enough to Earth to be influenced by its gravity, which effectively extends several million miles out into space. After getting far away enough from Earth the spacecraft will stop slowing and remain at whatever velocity it has remaining, and will be influenced from that point forward only by the Sun’s gravity, or any planets or other objects that it comes within vicinity of.
