I did the gravity calculation, using rough numbers.
A one lb rocket traveling 4000 fps (roughly mach 4 anyway) would have
249,000 ft-lb of energy.
Which is what you'd need to throw it 249,000 feet up a 1G gravity well
in a perfect vacuum.
So yes, a rocket traveling at Mach 4.1 can indeed coast more than
200,000 feet straight up.
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