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If a missile is launched from Yemen, and destined for Israel which is "nearly 1,000" (less than) miles away, is it likely for that missile to be intercepted at 62 miles of altitude? Where on the trajectory would that place the interception?

interception

Details from https://www.dailymail.co.uk/sciencetech/article-12716503/Israel-space-war-Arrow-missile-defense-system.html

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    $\begingroup$ 62 miles (converts to 99.779 km) is suspiciously also the Kármán line, the semi-official and completely arti-ficial altitude where "space begins". Above it, aerodynamics plays very little role and the trajectory is most ballistic (i.e. most predictable) My guess is that a reporter just inserted this specific number because it's the Kármán line, not because the interceptor waits patiently ("59... 60... 61... ok 62 miles, shoot") $\endgroup$
    – uhoh
    Commented Nov 25, 2023 at 21:18
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    $\begingroup$ 62 miles is also suspicously close to 100 km, which is a round number. And Israel of course uses metric - this might be just a journalist using too many significant digits. $\endgroup$
    – MSalters
    Commented Nov 27, 2023 at 11:02

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The numbers are reasonable. Georgia State's astrophysics department has several online ballistics calculators; the "will it clear the fence" and "where will it land" calculators tell me that with an initial velocity of 4000 m/s and a launch angle of 50 degrees, a missile will travel about 1000 miles, and at about 943 miles downrange it'll be a little over 62 miles up, implying an intercept about 57 horizontal miles from the target area. This isn't considering the powered ascent time, atmospheric drag, or earth curvature, but the peak altitude for this ballistic trajectory is pretty close to the reported peak altitude of 500km for the Iranian-designed Ghadr missile that was recently intercepted by Israel, so it should be roughly in the ballpark.

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    $\begingroup$ So that means at 943 miles it's 62 miles up? And the remaining range (100-943) 57 miles it's descending at a slope greater than 1. I didn't realize missiles descended like that if so. $\endgroup$
    – Evan Carroll
    Commented Nov 25, 2023 at 21:54
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    $\begingroup$ Yeah, in the absence of atmospheric drag, if you launch at 50 degree elevation, you land at 50 degree elevation. A 45-degree launch angle gets you the greatest range in the no-atmosphere, flat world case, but a higher launch elevation gets you into thinner air sooner, and if you have performance to spare, steeper launches get you higher-velocity, harder-to-intercept terminal trajectories. $\endgroup$
    – Russell Borogove
    Commented Nov 25, 2023 at 21:55
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    $\begingroup$ @EvanCarroll Perhaps you were thinking of cruise missiles? Those approach from quite shallow angles (they basically fly like an airplane, but typically at quite low altitudes in attempt to avoid radar.) Ballistic missiles are a different matter entirely and tend to approach from quite steep angles, generally launching well into outer space and spending only their last few seconds of flight in atmosphere above the target - at extremely high velocities. This is why intercepting ICBM reentry vehicles required crazy stuff like Sprint missiles. $\endgroup$
    – reirab
    Commented Nov 27, 2023 at 6:30

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