Orbital Velocity Problems: Problem 1: A satellite is orbiting Mars at an altitude of 400 km above the surface. The radius of Mars is approximately 3,390 km and its mass is 6.42times 10^wedge 23kg Calculate the orbital velocity of the satellite. Problem 2: The International Space Station orbits Earth at an altitude of 408 km Given the radius of Earth is approximately 6,371 km and its mass is 5.97times 10^wedge 24kg determine the orbital velocity of the ISS. Problem 3: A moon of Jupiter orbits at a distance of 1,070,000 km from the planet's center. The mass of Jupiter is 1.90times 10^wedge 27kg Find the orbital velocity of the moon. Problem 4: A communication satellite is in geostationary orbit around Earth at an altitude of 35,786 km. Calculate its orbital velocity assuming the radius of Earth is 6,37 I km and its mass is 5.97times 10^wedge 24kg

To solve these problems, we will use the formula for orbital velocity:<br /><br />\[ v = \sqrt{\frac{GM}{r}} \]<br /><br />where:<br />- \( v \) is the orbital velocity,<br />- \( G \) is the gravitational constant (\(6.67430 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2}\)),<br />- \( M \) is the mass of the central body (in kg),<br />- \( r \) is the distance from the center of the central body to the satellite (in meters).<br /><br />Let's solve each problem step by step.<br /><br />### Problem 1: Satellite Orbiting Mars<br /><br />Given:<br />- Altitude = 400 km = 400,000 meters<br />- Radius of Mars = 3,390 km = 3,390,000 meters<br />- Mass of Mars = \(6.42 \times 10^{23} \, \text{kg}\)<br /><br />First, calculate the total distance from the center of Mars to the satellite:<br />\[ r = 3,390,000 \, \text{m} + 400,000 \, \text{m} = 3,790,000 \, \text{m} \]<br /><br />Now, plug the values into the formula:<br />\[ v = \sqrt{\frac{(6.67430 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2}) \times (6.42 \times 10^{23} \, \text{kg})}{3,790,000 \, \text{m}}} \]<br /><br />\[ v = \sqrt{\frac{4.285 \times 10^{13} \, \text{m}^3 \text{s}^{-2}}{3,790,000 \, \text{m}}} \]<br /><br />\[ v = \sqrt{1.135 \times 10^6 \, \text{m}^2 \text{s}^{-2}} \]<br /><br />\[ v \approx 1,060 \, \text{m/s} \]<br /><br />### Problem 2: International Space Station (ISS) Orbiting Earth<br /><br />Given:<br />- Altitude = 408 km = 408,000 meters<br />- Radius of Earth = 6,371 km = 6,371,000 meters<br />- Mass of Earth = \(5.97 \times 10^{24} \, \text{kg}\)<br /><br />Total distance from the center of Earth to the ISS:<br />\[ r = 6,371,000 \, \text{m} + 408,000 \, \text{m} = 6,779,000 \, \text{m} \]<br /><br />Now, plug the values into the formula:<br />\[ v = \sqrt{\frac{(6.67430 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2}) \times (5.97 \times 10^{24} \, \text{kg})}{6,779,000 \, \text{m}}} \]<br /><br />\[ v = \sqrt{\frac{3.986 \times 10^{14} \, \text{m}^3 \text{s}^{-2}}{6,779,000 \, \text{m}}} \]<br /><br />\[ v = \sqrt{5.88 \times 10^6 \, \text{m}^2 \text{s}^{-2}} \]<br /><br />\[ v \approx 7,720 \, \text{m/s} \]<br /><br />### Problem 3: Moon of Jupiter<br /><br />Given:<br />- Distance = 1,070,000 km = 1,070,000,000 meters<br />- Mass of Jupiter = \(1.times 10^{27} \, \text{kg}\)<br /><br />Now, plug the values into the formula:<br />\[ v = \sqrt{\frac{(6.67430 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2}) \times (1.90 \times 10^{27} \, \text{kg})}{1,070,000,000 \, \text{m}}} \]<br /><br />\[ v = \sqrt{\frac{1.268 \times 10^{17} \, \text{m}^3 \text{s}^{-2}}{1,070,000,000 \, \text{m}}} \]<br /><br />\[ v = \sqrt{1.183 \times 10^4 \, \text{m}^2 \text{s}^{-2}} \]<br /><br />\[ v \approx 108.8 \, \text{m/s} \]<br /><br />### Problem 4: Communication Satellite in Geostationary