The two artificial satellites, 1 and 2 orbit the Eart in circular orbits having radii R1 and R2, respectively. If R2 = 2R1, the accelerations a2 and a1 of the two satellites are related by which of the following:
1) a2=4a1
2) a2= 2a1
3) a2= a1/4
4) a2= a1
5) a2= a1/2
The acceleration of a satellite in a circular orbit is given by the formula a = v^2/r, where v is the velocity of the satellite and r is the radius of its orbit.
Since the two satellites are in circular orbits, their velocities v1 and v2 are related to their radii R1 and R2 as follows:
v1 = √(GM/R1)
v2 = √(GM/R2)
Substituting these expressions into the formula for acceleration, we have:
a1 = v1^2/R1 = (GM/R1)/(R1) = GM/R1^2
a2 = v2^2/R2 = (GM/R2)/(R2) = GM/R2^2
Given that R2 = 2R1, we can substitute this into the expression for a2 to get:
a2 = GM/(2R1)^2 = GM/(4R1^2) = GM/R1^2/4 = (GM/R1^2)/4 = a1/4
Therefore, the correct answer is option 3) a2 = a1/4.