A body is moving in a low circular orbit about a planet of mass M and radius R. The radius of the orbit can be taken to be R itself. Then the ratio of the speed of this body in the orbit to the escape velocity from the planet is :

Options

A. `(1)/(sqrt2)`
B. `2`
C. `sqrt2`
D. `1`

Answer

To solve the problem, we need to find the ratio of the speed of a body in a low circular orbit around a planet to the escape velocity from that planet. Let's break this down step by step.

### Step 1: Determine the Orbital Speed (V₀)
The formula for the orbital speed \( V_0 \) of a body in a circular orbit at a distance \( R \) from the center of the planet is given by:

\[ V_0 = \sqrt{\frac{GM}{R}} \]

where:
- \( G \) is the gravitational constant,
- \( M \) is the mass of the planet,
- \( R \) is the radius of the planet (which is also the radius of the orbit in this case).

### Step 2: Determine the Escape Velocity (Vₑ)
The escape velocity \( V_e \) from the surface of the planet is given by:

\[ V_e = \sqrt{2 \frac{GM}{R}} \]

### Step 3: Calculate the Ratio of Orbital Speed to Escape Velocity
Now, we need to find the ratio \( \frac{V_0}{V_e} \):

\[ \frac{V_0}{V_e} = \frac{\sqrt{\frac{GM}{R}}}{\sqrt{2 \frac{GM}{R}}} \]

### Step 4: Simplify the Ratio
We can simplify this ratio:

\[ \frac{V_0}{V_e} = \frac{\sqrt{\frac{GM}{R}}}{\sqrt{2} \sqrt{\frac{GM}{R}}} = \frac{1}{\sqrt{2}} \]

### Conclusion
Thus, the ratio of the speed of the body in the orbit to the escape velocity from the planet is:

\[ \frac{V_0}{V_e} = \frac{1}{\sqrt{2}} \]

### Final Answer
The correct answer is option 1: \( \frac{1}{\sqrt{2}} \).

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