The two artificial satellites, 1 and 2 orbit the Eart in circular orbits having radii R1 and R2, respectively. If R2 = 2R1, how are the accelerations a2 and a1 of the two satellites related?
Since the satellites are in circular orbits, we can use the formula for centripetal acceleration:
a = v²/r
where a is the acceleration, v is the velocity, and r is the radius of the orbit.
For satellite 1, we can write:
a1 = v1²/R1
For satellite 2, we can write:
a2 = v2²/R2
The velocities of the satellites can be related using the fact that the period of the orbit is the same for both satellites:
T1 = T2
The period T is related to the velocity and radius by the equation:
T = 2πr/v
Solving for v, we get:
v1 = 2πR1/T1
and
v2 = 2πR2/T2
Substituting these expressions back into the equations for acceleration:
a1 = (2πR1/T1)²/R1
a2 = (2πR2/T2)²/R2
Simplifying:
a1 = (4π²R1²)/T1²R1
a2 = (4π²R2²)/T2²R2
Since T1 = T2, we can cancel out the T² terms:
a1 = (4π²R1²)/R1
a2 = (4π²R2²)/R2
Since R2 = 2R1:
a2 = (4π²(2R1)²)/(2R1)
Simplifying:
a2 = (4π²(4R1²))/(2R1)
a2 = (16π²R1²)/(2R1)
a2 = (8π²R1)
Therefore, the acceleration a2 of satellite 2 is 8 times greater than the acceleration a1 of satellite 1.