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?

Answer 1

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.