Question

the polar-orbiting environmental satellites (POES) and some military satellites orbit at a much lower level in order to obtain more detailed information. POES complete an Earth orbit 14.1 times per day. What are the orbital speed and the altitude of POES?

Answer 1

period = 24 hours/14.1 = 1.702 hours
times 3600 s/hr = 6128 seconds
v = sqrt(G Me/r)
T = 2 pi r/v = 2 pi r^(1.5) /(G Me)^.5

6128/2pi=r^1.5/(6.67*10^-11*5.98*10^24)^.5

975.2 = r^(1.5)/(3.989*10^14)^.5

975.2 = r^1.5 / 2*10^7

r^1.5 = 1950 *10^7
use log base 10
1.5 log r = 7 + log 1950 = 10.29
log r = 6.86
so
r = 7,244,360 m meters
or 7,244 km
subtract Re =6380km for earth radius = 864 kilometers altitude

v = 2 pi r/T
= 2 pi * 7.24*10^6 /6128
= 7423 m/second or 26724 km/hr

Answer 2

24/14.1 = 1.702hr = 6127sec.

T = 2(Pi)sqrt[r^3/µ]
µ = Earth's gravitational constant = 1.407974x10^16.

T = 2(Pi)sqrt[r^3/1.407974x10^16] from which r = 23,745,774ft. = 4497.3 miles or an altitude of 4497.3 - 3963 = 534.3 miles.

It's orbital velocity derives from
V = sqrt[µ/r] = sqrt[1.407974x10^16/23,745,774] = 24,350fps or 16,602mph.