Question

Two satellites of earth \[{{S}_{1}}\] and \[{{S}_{2}}\], are moving in the same orbit. The mass of \[{{S}_{2}}\] is four times the mass of \[{{S}_{1}}\]. Which of the following statements is true?
A. The time period of \[{{S}_{1}}\] is four times that of \[{{S}_{2}}\]
B. The potential energies of earth and satellite in two cases are equal
C. \[{{S}_{1}}\] and \[{{S}_{2}}\] are moving with same speed
D. The kinetic energies of two satellites are equal

Answer 1

Hint: In this question we have been asked to find the correct statement for the given conditions. We have been given statements in the options and asked to select the statement which is correct when two satellites with given mass revolve around earth in the same orbit. Therefore, to solve this question, we shall check every statement and find out if it is true for our given conditions or not.

Complete answer:
We have been given that radius of orbit for both the satellites is same and the mass of satellite \[{{S}_{2}}\] is four times the mass of satellite \[{{S}_{1}}\]. Therefore, we know that the parameters dependent on mass can not be equal to each other as the mass of satellites is different.
Now, let us check for the statements
The first statement says that time period of the \[{{S}_{1}}\] is four times that of \[{{S}_{2}}\]
Now, we know that the time period of revolution of satellite is given by,
\[T=\sqrt{\dfrac{GM}{{{R}^{3}}}}\]
Where,
M is the mass of planet earth
G is the gravitational constant
R is the radius of orbit
Since the radius of orbit of both satellites is the same time period for both the satellites will be the same. Therefore, the first statement is incorrect.
Now, the second statement states that the potential energies of earth and satellite in two cases are equal.
We know that, the potential energy of satellite is given by,
\[U=\dfrac{-GMm}{R}\]
Where m is the mass of the satellite.
Now, we know that the mass satellite \[{{S}_{2}}\] is 4 times the mass of \[{{S}_{1}}\]. Therefore, the potential energy of \[{{S}_{2}}\] will be four times that of \[{{S}_{1}}\]. Therefore, the second statement is also incorrect.
Again, the third statement states that, \[{{S}_{1}}\] and \[{{S}_{2}}\] are moving with the same speed.
We know, the speed of satellite orbiting around earth is given by,
\[v=\sqrt{\dfrac{GM}{R}}\]
From the above equation we can say that the velocity of the satellite is independent of mass and all the other quantities are the same for both the satellite (mass of earth M and radius R). So, the speed of the both satellites is equal. Therefore, the third statement is correct.
Now, the last statement says that, kinetic energy of a satellite is the same.
We know that kinetic energy of satellite is given by,
\[K.E=\dfrac{GMm}{2R}\]
From the above formula we know, the Kinetic energy of a satellite is mass dependent. Therefore, the kinetic energies of the satellite are not equal or same for the satellites. Therefore, the fourth statement is false as well.

So, the correct answer is “Option C”.

Note:
A satellite is an object in space that revolves around a planet. There are two types of satellites i) natural satellites such as moon ii) artificial satellites such as the International Space station. The total initial energy of any satellite is the sum of potential energy it has on earth’s surface and kinetic energy it has due to rotation of earth.