Question

An artificial satellite is moving around earth in a circular orbit with speed equal to one fourth the escape speed of a body from the surface of earth. The height of the satellite above earth's surface is (R is radius of earth).
A. 3R
B. 5R
C. 7R
D. 8R

Answer 1

Hint: To answer this question, we are required to know two formulas. The first one being escape velocity and the other one is the speed of the satellite. From equating these two formulas, we can obtain the height of the satellite.

Step by step answer:
Let us begin the question by finding the escape velocity.
Escape velocity is defined as a function of the mass of the body and the distance to the centre of mass of the body.

The escape velocity = $\sqrt{\text{2gR}}$
Now let us find the speed of the satellite.
The speed of the satellite is = $\dfrac{\sqrt{\text{2gR}}}{\text{4}}$

There are various quantities which are mentioned in the above equations. Let us know what they mean.

G represents the universal gravitational constant
R represents the radius of the Earth
Once we know the escape velocity and the speed of the satellite we need to move on to the process of finding the height of the satellite.

Now, applying the concept of force balance we get,

Where we have considered x is the height of the satellite,

Equating the following relations we get that,

$\dfrac{\text{GMm}}{{{\text{x}}^{\text{2}}}}\text{ = }\dfrac{\text{m}{{\left( \dfrac{\sqrt{\text{2gR}}}{\text{4}} \right)}^{\text{2}}}}{\text{x}}$

Now, from the relation mentioned above we can get the value of x.

$\text{x = }\dfrac{\text{GM}}{\text{2gR}}\text{ }\times \text{ 16}$
$\text{= 8}\dfrac{\text{GM}}{\text{gR}}$
$\text{=8R}$

So the value of x is 8R.

8R represents the total height of the satellite.

So, the height of the satellite above the earth’s surface is $\text{8R }-\text{ R =7R}$.

Finally, we know that the height of the satellite above the earth’s surface is 7R.
Therefore, among the mentioned options the correct one is Option C.

Note: We all should be having an idea about the concept of escape velocity that is mentioned in the answer. The escape velocity is defined as the minimum speed that is needed for a free, non-propellant object to escape from the gravitational pull of a very massive body. This means that the body wants to achieve an infinite distance from it.