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After Apollos 3rd stage burn to get to the moon the Command Module (CM) turned around and pulled the Lunar Module out of the 3rd stage. Both the CM and the LM turned around and flew to the moon.

But how exactly did the CM/LM separate from the 3rd stage? I mean all parts where traveling into the same direction with the same speed.

Was there a burn of CM to gain more speed and to get distance between to the 3rd stage?

Did the CM/LM somehow changed their direction to get out of the way of the 3rd stage?

Was the 3rd stage able to maneuver to get away from the CM/LM?

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This paper Earth Departure Trajectory Reconstruction of Apollo Program Components Undergoing Disposal in Interplanetary Space states that the 3rd stage was placed on a differing trajectory by dumping residual propellant.

For example

Using residual propellant dumps following TLI and CSM/LM ejection, the Apollo 10 S-IVB approached the Moon over its trailing hemisphere and obtained a gravity assist leading to disposal in interplanetary space.

And confirmed in the TECHNICAL INFORMATION SUMMARY APOLLO-10 (AS- 505)

enter image description here

Following LM extraction, the S-IVB stage will undergo a residual propellant, retrograde dump and safeing sequence. Thrust from available propellants in the launch vehicle auxiliary propulsion system and from main propulsion system venting is used to "propel" the expended S-IVB/IU to pass behind the moon and into a solar orbit.

See this question: How was the S-IVB stage configured for the post-extraction retrograde dump? for details about how the dumps were performed. The answer explains that not only residual propellant was dumped, but the Auxiliary Propulsion System (thrusters used for propellant settling) was fired, and residual helium pressurant was also dumped.

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  1. A "CSM/LV Sep" button on the control panel was pressed, which ignited detonating cord and separated the CSM from the Spacecraft–lunar module adapter (SLA), and the four adapter panels from each other and the S-IVB upper stage. This exposed the LM.
  2. The CSM's translation thrusters were used to move it a safe distance away. Rotation thrusters were then used to pitch up the CSM 180° and roll it to the proper alignment angle for docking. Translation thrusters were then used to move it back to the LM. A T-shaped docking target on the top of the LM aligned optically with a reticle pattern on the CMP's left-hand docking window to ensure proper spacecraft alignment.
  3. A soft dock was achieved when a probe at the top of the CSM was inserted into a hole in the center of a cone-shaped drogue at the top of the LM and three small capture latches closed. Hard dock was achieved by activating a mechanism which retracted the probe and caused twelve more capture latches to close around the command module's docking flange.
  4. A pressure equalization valve in the CM forward hatch was opened to allow oxygen to fill the LM through a similar valve in its hatch that was left open at launch. When the pressure equalized, the pilot removed the CM hatch, removed the probe and drogue, inspected the capture latches, and connected two umbilical cables which electrically connected the CM and LM. He then replaced the CM hatch.
  5. The LM hold-down attachments and umbilical connection to the S-IVB Instrument Unit were released, and the CSM's translation thrusters were used to pull the CSM/LM stack a safe distance away from the S-IVB, which would then be steered by ground control either to a heliocentric orbit, or to a deliberate crash landing on the Moon.

Source: https://en.wikipedia.org/wiki/Transposition,_docking,_and_extraction

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    $\begingroup$ Hi, unfortunately, apart from a part of point 5, most of what you've copied from Wikipedia isn't relevant to the question asked here. The remaining bit doesn't seem to add anything not already covered by the other answer. $\endgroup$ – TooTea Dec 18 '19 at 13:19
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    $\begingroup$ @TooTea, apparently we understood the question differently. According to my understanding of the question, this answer is much better than the other one. $\endgroup$ – prl Dec 19 '19 at 3:19
  • $\begingroup$ You forgot a step 0 - according to what I learned they did the maneuver in "parking" position aka they came to a complete stop. To make the manual steering as easy as possible. $\endgroup$ – eagle275 Dec 19 '19 at 15:43
  • $\begingroup$ If two umbilical cables which electrically connected the CM and LM were laid through the open hatch, how was it possible to close the hatch again without damaging the cables? $\endgroup$ – Uwe Dec 22 '19 at 23:21

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