A satellite is in a circular orbit around the Earth. If its speed is doubled, what will happen to its orbital radius?
Practice Questions
1 question
Q1
A satellite is in a circular orbit around the Earth. If its speed is doubled, what will happen to its orbital radius?
It will remain the same.
It will double.
It will increase by a factor of four.
It will decrease by a factor of four.
If the speed of a satellite is doubled, the orbital radius will decrease by a factor of four, as orbital speed is inversely proportional to the square root of the radius.
Questions & Step-by-step Solutions
1 item
Q
Q: A satellite is in a circular orbit around the Earth. If its speed is doubled, what will happen to its orbital radius?
Solution: If the speed of a satellite is doubled, the orbital radius will decrease by a factor of four, as orbital speed is inversely proportional to the square root of the radius.
Steps: 10
Step 1: Understand that a satellite in orbit has a specific speed that keeps it in a circular path around the Earth.
Step 2: Know that the gravitational force between the Earth and the satellite provides the necessary centripetal force for the orbit.
Step 3: Remember the formula for orbital speed: v = sqrt(G * M / r), where v is the orbital speed, G is the gravitational constant, M is the mass of the Earth, and r is the orbital radius.
Step 4: If the speed (v) is doubled, we can express this as 2v.
Step 5: Substitute 2v into the orbital speed formula: 2v = sqrt(G * M / r_new), where r_new is the new orbital radius.
Step 6: Square both sides of the equation to eliminate the square root: (2v)^2 = G * M / r_new.
Step 7: This simplifies to 4v^2 = G * M / r_new.
Step 8: Rearranging gives r_new = G * M / (4v^2).
Step 9: Compare this with the original radius formula r = G * M / v^2.
Step 10: Notice that r_new is 1/4 of the original radius (r), meaning the orbital radius decreases by a factor of four.
