How can a space shuttle pilot determine how fast the ship is going in space?



Answer:
Speed can be determined by spatial relationship with any two objects of known position versus time.

The accuracy of GPS is only as good as the accuracy of the satellite providing the fix. The planets and stars are also reliable for reference.

They have to know their three dimensional position in space so by comparing this position over different times gives speed of motion.
From his speedometer you FOOL !
The distance above the earth lets them know their radius of orbit, and the duration it takes them for a complete orbit... combine these and you have the velocity.
They look at the speedometer they wouldn't want to get a ticket.
From the rate of change of their relative position with earth.
He looks at his display
LOL.

The shuttle doesn't even move. In it's own frame of reference, it's the everything else that is moving around it.

Motion is relative.

And the space shuttle is the center of the universe.
I assume that radar from ground stations measures its speed, and NASA radios it back to them.

It seems that astronauts also can use with the Earth, or background stars as a reference by sighting them through optical telescopes, and measuring the rate of change.

Good question, though. NASA has a website for e-mails from children. You could ask them how it is done.
space shuttles use the very sophisticated inertial systems to detect their speed. Once aligned to a star, the exact position and speed can be calculated by the on board computers. The readouts are typically in feet/sec or meters/sec.
it sends the signal continuously to earth. and from base station it gets its relative speed wrt earth's distance.

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