How Do NASA and other organisations make sure they are at the right spot to pick them up and that the astronauts don’t land in some war zone or remote hard to reach area?
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The blunt cone lifting body shape employed by the Apollo capsules and adopted by SpaceX, Artemis etc. can be flown with remarkable accuracy.
They have an asymmetric centre of gravity and thrusters that can rotate the capsule about that centre of gravity. This rotation allows the angle of attack of the blunt base to be varied to steer left or right, up or down. The Apollo capsules actually did two re-entries on return to Earth: an initial one to scrub off speed and then they would climb again, losing even more speed, before descending more steeply into the designated landing zone.
So it wasn’t so much a question of calculating where the capsule would land but one of having the Command Module Pilot fly the capsule to the required spot.
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