Two satellites A and B are in different circular orbits about the earth. The orbit speed of satellite A is three times that of satellite B. Find the ratio (TA/TB) of the periods of the satellite.
Adinan
To find the ratio of the periods of the satellites (TA/TB), we need to understand the relationship between the speed, period, and radius of a circular orbit.
The speed of a satellite in a circular orbit depends on the mass of the central body and the radius of the orbit. The period of a satellite is the time taken for it to complete one full revolution around the central body.
Let's denote the speed of satellite A as VA, the speed of satellite B as VB, the period of satellite A as TA, the period of satellite B as TB, and the radius of satellite A's orbit as RA, and the radius of satellite B's orbit as RB.
Given that VA = 3VB (orbit speed of satellite A is three times that of satellite B), we can set up the following relationship:
VA = 2πRA/TA (1) [Speed of satellite A in terms of its period]
VB = 2πRB/TB (2) [Speed of satellite B in terms of its period]
Since we want to find the ratio of the periods (TA/TB), we can rearrange equations (1) and (2) to solve for TA and TB:
TA = 2πRA/VA (3) [Rearranging equation (1) to solve for TA]
TB = 2πRB/VB (4) [Rearranging equation (2) to solve for TB]
Now, let's substitute the given information VA = 3VB into equations (3) and (4):
TA = 2πRA/(3VB) (5) [Substitute VA = 3VB in equation (3)]
TB = 2πRB/VB (6) [Substitute VA = 3VB in equation (4)]
To find the ratio TA/TB, we divide equation (5) by equation (6):
(TA/TB) = (2πRA/(3VB)) / (2πRB/VB)
Simplifying the equation further:
(TA/TB) = (2πRA/(3VB)) * (VB/(2πRB))
(TA/TB) = (2πRA/(3RB))
Therefore, the ratio of the periods of the satellites (TA/TB) is:
(TA/TB) = (2πRA/(3RB))