Question

If a satellite orbits as close to the earth’s surface as possible,
A. its speed is maximum
B. time period of its rotation is minimum
C. the total energy of the earth plus satellite system is minimum
D. the total energy of the earth plus satellite system is maximum

Answer 1

Hint: The above problem can be resolved using the mathematical relation for the orbiting velocity, the time period of the satellite and the energy of the satellite. The orbit around the earth is that circular or the elliptical path, around which the satellites tend to move. When this orbital radius is reduced, then one can notice the variation in velocity, time period and the energy.

Complete step by step answer:
We know the expression for the time period of the satellite is,
\[T = \dfrac{{2\pi r}}{v}\] …….(1)
Here, r is the radius of orbit, along which the satellite is moving and v is the velocity of the satellite.
We know that the total energy of the earth and satellite shows the relationship as,
\[E \propto r\]
The above relation clears that when the orbiting radius is less, then the energy of the satellite and the earth is minimum. And, from equation 1, it is clear that, when the orbiting radius decreases, the time period of the satellite will also reduce.
And the expression for the orbiting velocity is given as,
\[v = \sqrt {\dfrac{{GM}}{r}} \]
Here, G is a universal gravitational constant and the M is the mass of earth.
Hence this relation shows that, decreasing the radius of orbit, the speed of orbiting also reduces.
Therefore, option (A), option (B) and option (C) are correct for the given statement.

Note: To resolve the given problem, one must remember the formula for the time period of the satellite along with some relations of the orbital radius with the time period, and orbital radius with velocity and many more. Moreover, these relations will help us to predict the exact variation for these variables.