Given that
G = gravitational constant = 6.67408E-11 m^3/kg s^2
M = mass of earth = 5.97E+24 kg
r = radius of earth 6,371,000 m
we can calculate escape velocity as
1. Vesc = sqrt(2GM/r) = 11.18597 km/s
If we use g = Acceleration due to gravity at earth's surface = 9.80665 m/s^2
r = radius of earth 6,371,000 m
and use the formula
2. Vesc = sqrt(2gr) = 11.178387 km/s
If we divide formula (1) by formula (2), we get
11.178387 km/s / 11.18597 km/s
= 0.99932 or 99.93% accuracy.
So why not use sqrt(2gr) instead of sqrt(2GM/r) to calculate escape velocity, since calculating GM involves so many numbers and is so cumbersome?