Answer: 544.818 km
The velocity [tex]V[/tex] of the space shuttle orbiting around the Earth can be determined by the following equation:
[tex]V=\sqrt{\frac{GM_{E}}{a}}[/tex] (1)
Where:
[tex]V=7589.288\frac{m}{s}[/tex] is the Velocity of the space shuttle
[tex]G=6.67(10)^{-11}\frac{Nm^{2}}{kg^{2}}[/tex] is the Gravitational constant
[tex]M_{E}=5.972(10)^{24}kg[/tex] is the Mass of the Earth
[tex]a[/tex] is the distance from the center of the Earth to the space shuttle
Now, firstly we need to find [tex]a[/tex] fromo equation (1):
[tex]a=\frac{GM_{E}}{V^{2}}[/tex] (2)
Substituting the known values:
[tex]a=\frac{(6.67(10)^{-11}\frac{Nm^{2}}{kg^{2}})(5.972(10)^{24}kg)}{(7589.288\frac{m}{s})^{2}}[/tex] (3)
[tex]a=6915818.153m=6915.818km[/tex] (4)>>>This is the distance from the center of the Earth to the space shuttle
But, we are asked to find the distance from the surface of the Earth.
If we want to find it, we have to use the following equation:
[tex]a-r_{E}=r[/tex] (5)
Where:
[tex]r_{E}\approx 6371km[/tex] is the radius of the Earth
[tex]r[/tex] is the distance from the surface of the Earth to the space shuttle
So, we are going to substract the radius of the Earth to the total distance from the earth's center to find this value:
[tex]6915.818km-6371km=544.818km[/tex] (6)
Finally, the distance is:
[tex]544.818km[/tex]
