Question

Two identical satellites orbit the earth in stable orbits. One satellite orbits with a speed v at a distance r from the center of the earth. The second satellite travels at a speed that is less than v.At what distance from the center of the earth does the second satellite orbit?

	At a distance that is less than r.
	At a distance equal to r.
	At a distance greater than r.

Now assume that a satellite of mass m is orbiting the earth at a distance r from the center of the earth with speed v_e. An identical satellite is orbiting the moon at thesame distance with a speed v_m. How does the time T_m it takes the satellite circling the moon to make onerevolution compare to the time T_e it takes the satellite orbiting the earth to make onerevolution?

Tm is less than Te.   Tm is equal to Te.  Tm is greater than Te.

Answer 1

Concepts and reason

The concepts used to solve this problem are orbital speed, and satellite’s time period of revolution.

Initially, use the orbital speed expression to determine the condition of second satellite’s orbit.

Finally, use the time period expression to determine the time period of the satellite.

Fundamentals

Orbital speed of a satellite is the speed with which it goes around the earth in a periodic motion. The path of this motion is called as orbit.

Expression for the orbital speed is,

$v = \sqrt {\frac{{GM}}{r}}$

Here, orbital speed is $v$, universal gravitational constant is $G$, mass of earth is $M$, and the radius of orbit of the satellite is $r$.

The time period of satellite’s orbit is the time taken by the satellite to complete one orbit around the planet.

Expression for time period is,

$T = \frac{{2\pi r}}{v}$

Here, time period of orbit is $T$, radius of orbit is $r$, and orbital speed is $v$.

(1)

The two satellites have equal masses. One satellite is orbiting with an orbital speed $v$, and radius $r$. The second satellite has an orbital speed $v'$ that is less than $v$.

The expression for orbital speed of the first satellite is,

$v = \sqrt {\frac{{GM}}{r}}$

The condition for the orbital speed of the second satellite is,

$v' < v$ …… (1)

From the expression for the orbital speed,

$v\alpha \frac{1}{{\sqrt r }}$ …… (2)

The orbital speed is inversely proportional to the square root of the radius of the orbit.

From the expressions (1) and (2), if the orbital speed of second satellite is less than the orbital speed of the first satellite, then the radius of the second satellite should be greater than that of the first satellite.

The incorrect options are,

• At a distance that is less than $r$

• At a distance equal to $r$

Use the condition$v' < v$,

• At a distance that is less than $r$

This option is incorrect because the velocity of second satellite is less than the first satellite velocity and hence its radius of orbit cannot be less than that of first satellite’s radius of orbit.

• At a distance equal to $r$

This option is incorrect as the orbital speed of both satellites are different and hence they cannot have same radius of orbit.

Hence, the correct option is

• At a distance greater than $r$

The above option is correct because, the radius of orbit is inversely proportional to the orbital speed and hence its radius will be greater than that of first satellite.

(2)

Satellite of mass $m$ goes around the earth in an orbit having radius $r$ and orbital speed ${v_e}$.

The expression for the orbital speed for this orbit is,

${v_e} = \sqrt {\frac{{G{M_e}}}{r}}$ …… (3)

If the satellite goes around moon in an orbit having same radius $r$ and with orbital speed ${v_m}$, the orbital speed of the satellite around moon is,

${v_m} = \sqrt {\frac{{G{M_m}}}{r}}$ …… (4)

From the equations (3) and (4), as the mass of earth is greater than the mass of moon, the condition for the orbital velocities of the satellite around earth and moon is,

${v_e} > {v_m}$ …… (5)

The time period of the orbit of the satellite around earth is,

${T_e} = \frac{{2\pi r}}{{{v_e}}}$ …… (6)

The time period of orbit of the satellite around moon is,

${T_m} = \frac{{2\pi r}}{{{v_m}}}$ …… (7)

Use the condition (5) in equations (6) and (7)

As the orbital speed around earth is greater than the orbital speed around moon, the satellite will have the time period of revolution around moon ${T_m}$ to be greater than its time period around earth ${T_e}$.

The incorrect options are,

• ${T_m}$ is less than ${T_e}$

• ${T_m}$ is equal to ${T_e}$

Applying the condition ${v_e} > {v_m}$ for the orbital velocities,

• ${T_m}$ is less than ${T_e}$

The above option is incorrect because the orbital speed is inversely proportional to the time period of motion.

• ${T_m}$ is equal to ${T_e}$

The above option is incorrect because the condition ${v_e} > {v_m}$ says that the satellite cannot have same time period of revolution moon and earth.

Hence, the correct option is

• ${T_m}$ is greater than ${T_e}$

Ans: Part 1

The correct option is, at a distance that is greater than ${\bf{r}}$.

Part 2

The correct option is (C) ${{\bf{T}}_{\bf{m}}}$ is greater than ${{\bf{T}}_{\bf{e}}}$