Question

If a body is projected with a velocity ‘\[V\]’ which is greater than escape velocity ‘\[{V_e}\]’ then with what velocity body will move in the interstellar space:
(A) \[\sqrt {{V^2} - {V_e}^2} \]
(B) \[\sqrt {{V^2} + {V_e}^2} \]
(C) \[\sqrt {V/{V_e}} \]
(D) \[\sqrt {{V_e}/V} \]

Answer 1

Hint Apply law of conservation of energy, by applying the law for the energy experienced by the body in earth and energy experienced by the body in interstellar space. On solving the equation you will arrive at the desired answer.

Complete Step by Step Solution
Let ‘\[v\]’ be the velocity of the body projected in the interstellar space. It is noteworthy to remember that in interstellar space, the potential energy i.e. the gravitational energy experienced by the body is equal to zero.
We need to find the velocity ‘\[{v_f}\]’ experienced by the body in the space.
Now, we know that the body experiences a velocity of \[{v_{earth}}\] when it is on the surface of the earth. So, when the object is on the earth’s surface, it experiences both kinetic energy and potential energy due to gravity.
Now, Total energy experienced by the body on earth’s surface is given as sum of kinetic energy and potential energy.
\[ \Rightarrow T = K.E + P.E\]
\[ \Rightarrow \dfrac{1}{2}m{v_{earth}}^2 + (\dfrac{{ - GMm}}{R})\]
 Potential energy experienced by the body is the gravitational force. Therefore, we use Newton's law of gravitation equation for gravitational force.
\[ \Rightarrow T = \dfrac{1}{2}m{v_{earth}}^2 - \dfrac{{GMm}}{R}\]
Now, Find the total energy of the body on interstellar space.
\[ \Rightarrow T = K.E + P.E\]
Here, the potential energy experienced will be zero due to absence of gravity
\[ \Rightarrow T = \dfrac{1}{2}m{v_f}^2\]
According to law of conservation of energy,
\[ \Rightarrow {T_{earth}} = {T_{Space}}\]
\[ \Rightarrow \dfrac{1}{2}m{v_{earth}}^2 - \dfrac{{GMm}}{R} = \dfrac{1}{2}m{v_f}^2\]
\[ \Rightarrow \dfrac{{mR{v_{earth}}^2 - 2GMm}}{R} = m{v_f}^2\]
Cancelling m on both sides
\[ \Rightarrow {v_{earth}}^2 - {v_f}^2 = \dfrac{{2GMm}}{R}\]

We know that escape velocity \[{v_e} = \sqrt {\dfrac{{2GMm}}{R}} \]
Substituting this in above equation,
\[ \Rightarrow {v_{earth}}^2 - {v_f}^2 = {v_e}^2\]
\[ \Rightarrow {v_{earth}} = v\]
\[ \Rightarrow {v^2} - {v_f}^2 = {v_e}^2\]
\[ \Rightarrow \sqrt {{v^2} - {v_e}^2} = {v_f}\]

Hence , Option (a) is the right answer for the given question.

Note
Escape velocity is defined as the minimum velocity required by a free body to escape the gravitational force getting influenced on the body. Rockets and other space vehicles use this velocity to escape out of earths' atmosphere .