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On the SpaceX mission I hear periodic "Downrange distance: N kilometers" for various values of N (initially 1, then 3, then 383, 920, 1500...).

What is the meaning of this value?

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This is simply the distance across the ground from the launch site.

Once the mission has reached orbit these distances are less relevant so they typically are not announced.

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To get to orbit there are two issues. Height, gotta get high enough. Speed, gotta get going fast enough.

However the lower atmosphere is thick and a pain, so rockets generally launch almost straight up at first, to get out of the atmosphere as fast as possible, then tilt over to accelerate to orbital speeds. There is almost always an initial component of horizontal velocity, and the distance they travel is what is being called out as a downrange distance.

This is important since once they clear particular boundaries, the exclusions needed to keep shipping, boaters, and airplanes out of that area can be relaxed or released.

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