Question

Escape velocity of a satellite of the earth at an altitude equal to radius of the earth is v. What will be the escape velocity at an altitude equal to 7R, where R = radius of the earth?
A. $\dfrac{v}{4}$
B. $\dfrac{v}{2}$
C. $8v$
D. $4v$

Answer 1

Hint: In this question, we will use the concept of the escape velocity of a body from the surface of the earth. The expression for the escape velocity is given by, ${v_e} = \sqrt {\dfrac{{2GM}}{R}} $.
Formula used: ${v_e} = \sqrt {\dfrac{{2GM}}{R}} $

Complete Step-by-Step solution:
The velocity of escape is the minimum velocity at which a body will be projected vertically upwards in order to escape the earth's gravitational field.
We know that, the total work done in moving the body from the surface of the earth (x = R) to a region beyond the gravitational field of the earth is given by,
$ \Rightarrow W = \dfrac{{GMm}}{R}$ ………(i)
If ${v_e}$ is the escape velocity of the body, then the kinetic energy imparted to the body at the surface of the earth will be just sufficient to perform work W.
$
   \Rightarrow \dfrac{1}{2}m{v_e}^2 = \dfrac{{GMm}}{R} \\
   \Rightarrow {v_e}^2 = \dfrac{{2GM}}{R} \\
$
Then escape velocity, ${v_e} = \sqrt {\dfrac{{2GM}}{R}} $. ……….(ii)
According to the question, the escape velocity at an altitude equal to the radius of the earth is v.
So the altitude equal to the radius of earth, then we have
$R = {R_e} + {R_e} = 2{R_e}$, where ${R_e}$is the radius of earth.
Its expression can be given by,
$v = \sqrt {\dfrac{{2G{M_e}}}{{2{R_e}}}} $ ………(iii)
If the escape velocity at an altitude equal to 7R then,
R = ${R_e}$+7${R_e}$= 8${R_e}$ substituting this value in equation (ii), we get
 $
   \Rightarrow {v_e} = \sqrt {\dfrac{{2G{M_e}}}{{8{R_e}}}} \\
   \Rightarrow {v_e} = \dfrac{1}{2}\sqrt {\dfrac{{2G{M_e}}}{{2{R_e}}}} \\
$
We have $\sqrt {\dfrac{{2G{M_e}}}{{2{R_e}}}} = v$, then
$ \Rightarrow {v_e} = \dfrac{v}{2}$.
Hence, the correct answer is option (B).

Note: In this type of questions, we have to remember the basic concept of escape velocity. First we have to identify all the given details then we will use the formula of escape velocity. We will add the radius of earth with the altitude because the altitude given is from the surface only but the distance, R in the formula is from the centre of the earth, so we will add the radius of earth also. After that we will substitute all the values in the formula and then we will get the answer.