A satellite is in a circular orbit around the Earth. What is the relationship between its orbital speed v and the radius r of the orbit?

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Q: A satellite is in a circular orbit around the Earth. What is the relationship between its orbital speed v and the radius r of the orbit?

Solution: The orbital speed of a satellite is given by v = sqrt(G * M / r), where M is the mass of the Earth.

Steps: 10

Step 1: Understand that a satellite moves in a circular path around the Earth.
Step 2: Recognize that the force keeping the satellite in orbit is gravity.
Step 3: Know that the gravitational force depends on the mass of the Earth (M) and the distance from the center of the Earth (r).
Step 4: Remember that the formula for gravitational force is F = G * (M * m) / r^2, where G is the gravitational constant and m is the mass of the satellite.
Step 5: Realize that for the satellite to stay in orbit, the gravitational force must equal the centripetal force needed to keep it moving in a circle.
Step 6: The centripetal force is given by F = (m * v^2) / r, where v is the orbital speed of the satellite.
Step 7: Set the gravitational force equal to the centripetal force: G * (M * m) / r^2 = (m * v^2) / r.
Step 8: Cancel the mass of the satellite (m) from both sides of the equation since it appears in both terms.
Step 9: Rearrange the equation to solve for v: v^2 = G * M / r.
Step 10: Take the square root of both sides to find the orbital speed: v = sqrt(G * M / r).