Why is there a discrepancy between rocket velocity calculations in orbit?

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SUMMARY

The discussion centers on the discrepancy in rocket velocity calculations in orbit, specifically between 5609 m/s and the answer key's 3950 m/s. The participant asserts that velocity should be calculated relative to the center of mass of the Earth rather than the Earth's surface. The correct gravitational force exerted on the rocket is confirmed as F=12268.1 N. The calculations involve understanding centripetal acceleration and gravitational forces at an altitude equal to the Earth's radius.

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JM2107
In the following problem, I got that the velocity is 5609m/s but the answer key says that the velocity of the rocket is 3950m/s. I know both answers can be calculated but I don't understand why the radial distance would be the distance from the rocket to the Earth's surface. I personally think that the velocity of the rocket should be calculated in respect to the center of the mass that is is orbiting around. Could some please tell me if I am right, it would be greatly appreciated. Oh yeah, and I got the correct force which is F=12268.1N. Once again, thanks for any help provided in advance


Here's part of the problem:

1. A 5000 kg rocket orbits the Earth at an altitude above the earth’s surface equal to the radius of the earth[R_earth=6.4*10^6m].
a. Calculate the force exerted on the rocket by the earth.
b. Calculate:
i. the velocity of the rocket
 
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Did you try to equate needed for orbiting centripetal acceleration v2/r to the acceleration of gravity g/4 at that distance? It comes right, v~4 km/sec
 

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