Two satellites are monitored as they orbit the earth,satellites Y is foure times as far from earth's centre as is satellite X. IF the period of revolution of satellite X is T. WHAT IS THE PERIOD OF REVOLUTION OF OF Y?
To find the period of revolution of satellite Y, we can use Kepler's Third Law of Planetary Motion, which states that the square of the period of revolution of a satellite is directly proportional to the cube of its average distance from the center of the earth.
Let's assume that the average distance of satellite X from the center of the earth is "r." Given that satellite Y is four times as far from the earth's center as satellite X, the average distance of satellite Y would be 4r.
Now, according to Kepler's Third Law, we can write the following equation:
(Tx)^2 = k * (Rx)^3
(Ty)^2 = k * (Ry)^3
Where:
Tx = period of revolution of satellite X
Ty = period of revolution of satellite Y
Rx = average distance of satellite X from the center of the earth
Ry = average distance of satellite Y from the center of the earth
k = constant of proportionality
Since we are comparing the two satellites, we can establish the following relationship:
(Tx)^2 = k * (Rx)^3
(Ty)^2 = k * (Ry)^3
By substituting the given information that Ry = 4Rx, we can rewrite the equation for satellite Y:
(Ty)^2 = k * (4Rx)^3
Now, we can divide both sides of the equation by (Tx)^2 to eliminate the constant of proportionality:
(Ty/Tx)^2 = (4Rx)^3
To solve for (Ty/Tx), we need to take the square root of both sides:
Ty/Tx = sqrt((4Rx)^3)
The square root of a number raised to an exponent can be calculated by taking the square root of the number and then raising it to half of the exponent:
Ty/Tx = (4Rx)^(3/2)
Since we are given that the period of revolution of satellite X is T, we can substitute Tx with T:
Ty/T = (4Rx)^(3/2)
Finally, substituting Rx with r and simplifying, the equation becomes:
Ty/T = (4r)^(3/2)
Therefore, the period of revolution of satellite Y (Ty) is equal to the product of the period of revolution of satellite X (T) and (4r) raised to the power of 3/2.