Two artificial satellites, 1 and 2, orbit the Earth in circular orbits having radii R1 and R2, respectively, as shown above. If R2 = 2R1, the accelerations A2 and A1 of the two satellites are related by which of the following?
1)A2 = 4*A1
2)A2 = 2*A1
3)A1 = (A1)/4
4)A2= A1
5)A2= (A1)/2
The acceleration of an object moving in a circular orbit is given by the equation a = v^2/r, where v is the velocity of the object and r is the radius of the orbit.
The velocity of an object in a circular orbit can be found using the equation v = √(GM/r), where G is the gravitational constant and M is the mass of the object being orbited (in this case, the Earth).
Since the two satellites are orbiting the same central mass (the Earth), they have the same mass M. Therefore, we can calculate their velocities as follows:
v1 = √(GM/R1)
v2 = √(GM/R2)
Substituting R2 = 2R1 into the equation for v2, we get:
v2 = √(GM/(2R1))
The accelerations of the two satellites can be calculated by substituting their respective velocities into the equation for acceleration:
a1 = v1^2/R1
a2 = v2^2/R2
Substituting the expressions for v1 and v2 into the equations for a1 and a2, we get:
a1 = (GM/R1)^2/R1
a2 = (GM/(2R1))^2/(2R1)
Simplifying these equations, we have:
a1 = (GM^2)/R1^3
a2 = (GM^2)/(8R1^3)
Dividing a2 by a1, we get:
a2/a1 = [(GM^2)/(8R1^3)] / [(GM^2)/R1^3]
= (GM^2)/(8R1^3) * R1^3/(GM^2)
= 1/8
Therefore, the accelerations A2 and A1 of the two satellites are related by A2 = (1/8)A1.
So, the correct answer is:
A2 = (1/8)A1