Question

a geosynchronous satellite has an orbital period of 24 hours so that it stays fixed at a point above earths surface. how far away does a geosynchronous satellite need to be from the surface of earth, in kilometers? (1 point) 42,300 km can be at any distance 35,900 km 29,500 km

Explanation:

Step1: Recall the formula for the orbital radius of a geosynchronous satellite

The formula for the orbital radius \( r \) of a satellite is given by \( r=\sqrt[3]{\frac{GMT^{2}}{4\pi^{2}}} \), where \( G = 6.67\times10^{-11}\space Nm^{2}/kg^{2} \) (gravitational constant), \( M = 5.97\times 10^{24}\space kg \) (mass of the Earth), and \( T=24\times3600\space s \) (orbital period).

Step2: Calculate the orbital radius

First, substitute the values into the formula:

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After calculation, \( r\approx4.23\times 10^{7}\space m = 42300\space km \) (this is the distance from the center of the Earth).

Step3: Subtract the Earth's radius

The radius of the Earth \( R_{E}\approx6400\space km \). The distance from the Earth's surface \( d=r - R_{E} \).

$$ d=42300 - 6400=35900\space km $$

Answer:

35,900 km