The orbital radius of a satellite around Earth is 7,000 km. Using the formula \( v = \sqrt{\frac{GM}{r}} \) (where \( GM = 3.986 \times 10^{14} \, \text{m}^3/\text{s}^2 \)), find the orbital speed in m/s.

["Interesting Deep Dive: Calculating the Orbital Speed of a Satellite at 7,000 km from Earth’s Center", "When exploring satellite motion, one fundamental question arises: What is the speed of a satellite orbiting Earth at a distance of 7,000 km from its center? Understanding satellite dynamics is not only crucial for space missions but also a fascinating application of physics and orbital mechanics. In this article, we’ll explore how to compute the orbital speed using Einstein and Newton’s principles—specifically with the formula ( v = \sqrt{\frac{GM}{r}} )—where ( v ) is orbital speed, ( G ) is the gravitational constant, ( M ) is Earth’s mass-related parameter, and ( r ) is the orbital radius.", "### Satellite Orbit: A Delicate Balance of Gravity and Momentum", "For a satellite in a stable circular orbit, gravity provides the centripetal force that keeps it moving in a closed path. The balance of forces leads directly to the orbital speed formula we’ll use. Given Earth’s standard gravitational parameter ( GM = 3.986 \ imes 10^{14} , \ ext{m}^3/\ ext{s}^2 ), and knowing the orbital radius ( r = 7,000 , \ ext{km} = 7,000,000 , \ ext{m} ), we can plug values into the equation:", "[\nv = \sqrt{\frac{GM}{r}} = \sqrt{\frac{3.986 \ imes 10^{14}}{7,000,000}}\n]", "### Step-by-Step Calculation", "Start by dividing:", "[\n\frac{3.986 \ imes 10^{14}}{7,000,000} = \frac{3.986 \ imes 10^{14}}{7 \ imes 10^6} \approx 5.694 \ imes 10^7\n]", "Now take the square root:", "[\nv = \sqrt{5.694 \ imes 10^7} \approx 7,546 , \ ext{m/s}\n]", "### Why 7,500 m/s Matters", "The result—about 7,546 meters per second—means a satellite orbiting at 7,000 km from Earth’s center travels at a speed of roughly 7.5 km/s. This speed is typical for low Earth orbits, enabling satellites to overcome gravity and maintain a stable path. For context, the International Space Station orbits closer (~400 km), moving faster (~7.7 km/s), while higher orbits slow the satellite accordingly.", "### Real-World Implications", "This calculation isn’t just academic. Engineers rely on such formulas to design satellite missions—telecom, weather, and GPS satellites all depend on precise orbital speeds. Understanding how distance affects speed helps ensure orbital stability, fuel efficiency, and mission success.", "---", "In summary, the orbital speed of a satellite around Earth with an orbital radius of 7,000 km is approximately 7,546 m/s. This elegant application of ( v = \sqrt{GM/r} ) demonstrates how physics governs space travel and illustrates the precision needed in modern astronomy and aerospace engineering.", "If you’re fascinated by how satellites navigate Earth’s gravity, exploring this formula deepens your appreciation for the science behind modern connectivity and space exploration."]