what is the altitude of the satellite?

Question

At what altitude will a satellite complete a circular orbit of the Earth in 1.7 h ? put in two significant figures

Answer 1

When a satellite or any other object is in circular orbit, the equation for the time the object takes to complete one orbit is given by:

T = √(4π²r³)/(G⋅M)

where T = time the object takes to complete one complete orbit (must be in seconds), r = radius of the circular orbit from the center of the earth (in meters), G = 6.67x10^-11, and M = mass of the earth in kg (or if it was another planet, it would be the mass of that central body).

If you were to rearrange the equation shown above and solve it for r, you would get the following:

r = [(T²⋅G⋅M)/(4π²)]^1/3

r = [ ((6120 s)²(6.67x10^-11)(5.97x10²⁴))/(4π²) ]^1/3

r = 7229048 m or 7.23x10⁶ m

The radius of the orbit, r, is also given by:

r = R + h

where R = radius of the earth (or if it was another planetary body, it would be the radius of that planet) and h = the altitude above the earth's surface. The radius of the earth is 6.38x10⁶. When you solve the equation for h, you get:

h = r - R

h = 7.23x10⁶ m - 6.38x10⁶ m

h = 850 km

So now you can say that if a satellite takes 1.7 hours to complete one circular orbit around the earth, the altitude of the satellite is 850 kilometers above the earth's surface.

Answer 2

Use Kepler's 3rd law to get the radius "a" of the orbit, then subtract the radius of the earth R_e from the radius of the orbit to get the height h.

a = [GMT² / (4π²)]^1/3

Look up in your physics book the values of G (constant of universal gravitation) and M (the mass of the earth). Convert the orbital period T to seconds. Plug all this into the equation for "a", then get

h = a - R_e