Question

a satellite is in orbit about earth. its orbital radius is (7.9\times10^{7}) m. the mass of the satellite is 2523 kg and the mass of earth is (5.974\times10^{24}) kg. determine the orbital speed of the satellite in mi/s. 1 mi/s = 1609 m/s. 565 × mi/s

Explanation:

Step1: Use the formula for orbital speed

The formula for the orbital speed \(v\) of a satellite around a planet is \(v = \sqrt{\frac{GM}{r}}\), where \(G = 6.67\times10^{- 11}\space N\cdot m^{2}/kg^{2}\) is the gravitational constant, \(M\) is the mass of the planet (Earth in this case), and \(r\) is the orbital radius of the satellite.
Substitute \(G = 6.67\times10^{-11}\space N\cdot m^{2}/kg^{2}\), \(M = 5.974\times 10^{24}\space kg\), and \(r=7.9\times 10^{7}\space m\) into the formula:

$$v=\sqrt{\frac{6.67\times 10^{-11}\times5.974\times 10^{24}}{7.9\times 10^{7}}}$$

First, calculate the numerator: \(6.67\times 10^{-11}\times5.974\times 10^{24}=6.67\times5.974\times10^{-11 + 24}=39.84658\times10^{13}\)
Then, divide by \(r\): \(\frac{39.84658\times 10^{13}}{7.9\times 10^{7}}=\frac{39.84658}{7.9}\times10^{13 - 7}\approx5.044\times10^{6}\)
Take the square - root: \(v=\sqrt{5.044\times 10^{6}}\approx2246\space m/s\)

Step2: Convert the speed from m/s to mi/s

We know that \(1\space mi/s = 1609\space m/s\). To convert \(v\) from \(m/s\) to \(mi/s\), use the conversion factor \(v_{mi/s}=\frac{v_{m/s}}{1609}\)
Substitute \(v = 2246\space m/s\) into the conversion formula: \(v_{mi/s}=\frac{2246}{1609}\approx1.4\space mi/s\)

Answer:

\(1.4\space mi/s\)