Question

The radius in kilometers to which the present radius of the earth (R=6400 km) is to be compressed so that the escape velocity is increased to ten times is
A. 8
B. 64
C. 640
D. 4800

Answer 1

To solve this problem, use the formula for escape velocity of a planetary system. Substitute the radius of the Earth in the formula and find its escape velocity. Then, write the expression for escape velocity when Earth is compressed. Now, use the condition which is given that the escape velocity increases by ten times. So, the new escape velocity will be two times the earlier escape velocity. Substitute the values and find the radius of earth when it is compressed.
Formula used:
$V=\sqrt { \dfrac { 2GM }{ R } }$

Complete answer:
Given: R= 6400 km
Escape velocity of a planetary system is given by,
$V=\sqrt { \dfrac { 2GM }{ R } }$ …(1)
Where, M is the mass of the planet
R is the radius of the planet
Substituting values in above equation we get,
${ V }_{ 1 }=\sqrt { \dfrac { 2GM }{ 6400 } }$ …(2)
After compression the radius of the Earth will be,
${ V }_{ 2 }=\sqrt { \dfrac { 2GM }{ { R }_{ 2 } } }$ …(3)
Where, ${R}_{2}$ is the radius of earth after compression
If the escape velocity is increased to 10 times then,
${V}_{2}=10{V}_{1}$
Substituting equation. (2) and (3) in above equation we get,
$\sqrt { \dfrac { 2GM }{ { R }_{ 2 } } } =10\sqrt { \dfrac { 2GM }{ 6400 } }$
Squaring both the sides we get,
$\dfrac { 2GM }{ { R }_{ 2 } } =100\times \dfrac { 2GM }{ 6400 }$
$\Rightarrow \dfrac { 2GM }{ { R }_{ 2 } } =\dfrac { 2GM }{ 64 }$
Now, cancelling the common terms on both the sides we get,
$\dfrac {1}{{R}_{2}}= \dfrac {1}{64}$
$\Rightarrow {R}_{2}= 64 km$
Thus, the radius in kilometers to which the present radius of the earth (R=6400 km) is to be compressed so that the escape velocity is increased to ten times is 64 km.

So, the correct answer is option B i.e. 64.

Note:
To solve these types of questions, students must know the formula for escape velocity and also what the escape velocity means. Escape velocity is defined as the minimum amount of energy that is required for any free or non-propelled object to free itself from the gravitational influence of a huge body. Even if it is not mentioned, students must remember that the escape velocity of earth is $11 {km}/{s}$.