Question

A space station is set up in a space at a distance equal to the earth’s radius from the surface of the earth. Suppose a satellite can be launched from the space station. Let ${\nu _1}$and ${\nu _2}$ be the escape velocities of the satellite on the earth’s surface and space station respectively. Then, find which of these are true:
A. ${\nu _1} = {\nu _2}$
B. ${\nu _2} < {\nu _1}$
C. ${\nu _2} > {\nu _1}$
D. (A), (B) and (C) are valid depending upon the mass of the satellite.

Answer 1

Hint:
1. Total energy of the planet + satellite system is conserved.
2. Escape velocity of any satellite from the planet is given by velocity at which kinetic energy of satellite equals potential energy with respect to planet as reference. i.e.
$K.E + P.E = T.E = 0$ (this case)
$ \Rightarrow K.E = - P.E$
3. Space station still lies under earth’s gravitational pull.

Formula Used:
1. Work-Energy theorem for conservative system: $K.E + P.E = T.E = 0$
2. P.E of a body under gravitational influence: $P.E = - \dfrac{{GMm}}{r}$ ……. (a)
where r is distance between body, M and m are mass of Planet and body respectively.

Complete step by step answer:

Given:
1. ${\nu _1}$and ${\nu _2}$are escape velocities from earth and space station respectively.
2. Distance between earth and space station is 2R
To find: relation between ${\nu _1}$and ${\nu _2}$if masses are known

Step 1 of 4:

Since no external force is applied, the planet and satellite form a conservative system.
Hence, $K.E + P.E = 0$ …… (1)

Step 2 of 4:

Putting value of K.E and P.E in the equation (1) by using equation (a) we get value of ${\nu _1}$ where r= R,
$\dfrac{1}{2}m{\nu _1}^2 - \dfrac{{GMm}}{R} = 0$
$ \Rightarrow {\nu _1} = \sqrt {\dfrac{{2GM}}{R}} $ …… (2)

Step 3 of 4:

In case of launch from the space station, total distance between planet and satellite: r= 2R
Putting value of r in equation (1) by using equation (a) we get value of ${\nu _2}$:
$\dfrac{1}{2}m{\nu _2}^2 - \dfrac{{GMm}}{{2R}} = 0$
$ \Rightarrow {\nu _2} = \sqrt {\dfrac{{2GM}}{{2R}}} = \sqrt {\dfrac{{GM}}{R}} $ …… (3)

Step 4 of 4:

Comparing equation (3) and (2) we get,
$ \Rightarrow {\nu _2} < {\nu _1}$

Correct Answer: B. ${\nu _2} < {\nu _1}$

Additional Information: Here we assumed the space station has no mass otherwise, we had to count potential energy due to the space station as well in the total energy equation.

Note: Escape velocity of any body from the surface of a planet is independent of body mass. It just depends upon the mass and radius of the planet.