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I am really interested in the Apollo mission and I was wondering how fast did the astronaut pass by relative to the moon. I wanted to compare it with the speed of the car. I have been able to recalculate their trajectory but I can't seem to be able to convert the velocity of the spacecraft relative to the moon instead of the earth.

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Converting between different frames of reference is pretty straight forward.

You subtract the velocities. That's it.

If you have the spacecraft's velocity relative to the Earth, you can just subtract the Moon's velocity relative to Earth. Then you have the spacecraft's velocity relative to the Moon.

In the case of a lunar transfer orbit, the spacecraft is moving at about 180 m/s in the outermost part.
The moon is moving in the same direction, at 1,080 m/s.
Their relative velocity is thus 840 m/s.

From there, the spacecraft starts falling into the gravity well of the Moon, gaining speed. We can see how much by using the following formula:

$$v^2 = v_e^2 + v_{\infty}^2$$

Where $v$ is the current velocity, $v_e$ the local escape velocity, and $v_{\infty}$ the velocity we had when entering the system.

Using the values for 110km above the Moon's surface, the Apollo was moving at approximately 2,500 m/s before slowing down using the service module's engine to enter orbit.

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