Determining the satellite's altitude above the surface of the Earth

[torquey123]

Homework Statement

A satellite moves in a circular orbit around the Earth at a speed of 5.1km/s.
Determine the satellite's altitude above the suface of the Earth. Assume the Earth is a homogenous sphere of radius Rearth= 6370km, and mass Mearth=5.98x10^24 kg. You will need G=6.67259x10^(-11) Nm^2/kg^2. Answer in units of km

Homework Equations

v^2=(G*Mass of earth)/(radius)

The Attempt at a Solution

(5100 km/s)^2 = (6.67259x10^(-11))(5.98x10^(24))/r

26010000r=3.990190282e14

r= 15340985.32 - 6370000= 8970985.321m = 8970.985321km

i think its wrong, but i don't know where i made the mistake..

 

[torquey123]

never mind...i figured it out when i typed it out...i didnt convert back to km

 

[m2003]

I would like to commend you on your attempt at solving this problem. However, there are a few mistakes in your solution. Firstly, the given speed of 5.1km/s is the orbital speed of the satellite, not its velocity. Therefore, we need to use the equation v^2 = GM/R, where M is the mass of the Earth, R is the distance between the center of the Earth and the satellite's orbit, and v is the orbital speed. Secondly, we need to convert the given values to SI units (meters and kilograms) before plugging them into the equation.

So, using the correct equation and converting the values, we get:

(5100 m/s)^2 = (6.67259x10^(-11) Nm^2/kg^2)(5.98x10^(24) kg)/R

26010000 = 3.990190282e14/R

R = 3.990190282e14/26010000 = 15340985.32m

Now, to find the altitude above the surface of the Earth, we need to subtract the radius of the Earth from the distance between the center of the Earth and the satellite's orbit:

Altitude = 15340985.32m - 6370000m = 8970985.32m = 8970.98532km

Therefore, the satellite's altitude above the surface of the Earth is approximately 8971km. This is a reasonable answer, as most satellites in low Earth orbit have altitudes between 160km and 2000km.