Two satellite are monitored as they orbite the earth,satellites Y is foure times as far from the earth's centere as is satellite X. if the period of revolution of satellite X is T. What is the period of revolution of Y?
To determine the period of revolution of satellite Y, we need to use the concept of Kepler's third law of planetary motion, which states that the square of the period of revolution is proportional to the cube of the average distance from the object being orbited.
Let's denote the distance of satellite X from the Earth's center as rX and the distance of satellite Y as rY. Given that satellite Y is four times as far from the Earth's center as satellite X, we can express this relationship as:
rY = 4 * rX
Now, since the period of revolution of satellite X is denoted as T, we can use Kepler's third law to set up the following equation:
(TY)^2 = k * rY^3
and
(TX)^2 = k * rX^3
where TY represents the period of revolution of satellite Y, TX represents the period of revolution of satellite X, and k is a constant of proportionality.
Dividing the two equations, we get:
(TY/TX)^2 = (rY/rX)^3
Substituting the given relationship of rY = 4 * rX:
(TY/TX)^2 = (4 * rX / rX)^3
Simplifying, we have:
(TY/TX)^2 = (4^3)
(TY/TX)^2 = 64
Now, taking the square root of both sides:
TY/TX = ±√64
Since we are dealing with a period of revolution, which is a positive quantity, we can disregard the negative sign.
TY/TX = √64
TY/TX = 8
To find the period of revolution of satellite Y (TY), we multiply this ratio by the known period of revolution of satellite X (TX):
TY = TX * 8
Therefore, the period of revolution of satellite Y is 8 times the period of revolution of satellite X.