Question

A satellite has a kinetic energy X, potential energy Y and total energy Z in an orbit. How are they related?
$$\begin{gathered} \text{A}\text{. }Z=Y=-2X \\ \text{B}\text{. }Z={\displaystyle \frac{Y}{2}}=-X \\ \text{C}\text{. }Z=2Y=-2X \\ \text{D}\text{. }2Z=Y=-2X \end{gathered}$$

Answer 1

Hint: In order to solve this question, we need to find the kinetic energy, potential energy and the total energy of the satellite revolving in earth’s orbit. We know that total energy is equal to the sum of kinetic energy and the potential energy.

Complete step by step answer:
A satellite is able to revolve around in a closed orbit because of the gravitational force acting on the satellite due to earth. The gravitational force provides the centripetal force required to make the satellite move in a circular orbit.
Therefore, we can write the following expression for a satellite where the centripetal force on the satellite is equal to the gravitational force acting on the satellite.

$${\displaystyle \frac{m{v}^2}{r}}={\displaystyle \frac{GMm}{r^2}}$$

Here m is the mass of the satellite, M is the mass of the earth, v is the velocity with which the satellite moves in its orbit and r is the distance of the satellite from the centre of the earth. From this expression we can calculate the kinetic energy X of the satellite which is given as

$X={\displaystyle \frac{1}{2}}m{v}^2={\displaystyle \frac{GMm}{2r}}...\text{(i)}$

Now, the expression for the gravitational potential energy is given as

$Y=-{\displaystyle \frac{GMm}{r}}...\text{(ii)}$

The total energy of the satellite revolving in a closed orbit around earth is given as the sum of the kinetic energy and the potential energy of the satellite. So, by using the equation (i) and (ii), we get

$Z=X+Y={\displaystyle \frac{GMm}{2r}}-{\displaystyle \frac{GMm}{r}}=-{\displaystyle \frac{GMm}{2r}}...\text{(iii)}$

Now we have all the three values, by comparing the equations (i), (ii) and (iii), we get
$Z={\displaystyle \frac{Y}{2}}=-X$
Hence, the correct answer is option B.

Note:
A body cannot move in a circular orbit without a centripetal force acting on that body. In a gravitational system of earth and a satellite, the centripetal force is provided by the gravitational force acting on the satellite.