The ratio of the orbital speeds of two satellites of the earth if the satellite are at heights `6400 km` and `19200km`(Raduis of the earth=`6400 km`)

Options

A. `sqrt(2):1`
B. `sqrt(3):1`
C. `2:1`
D. `3:1`

Explanation

To find the ratio of the orbital speeds of two satellites at heights of 6400 km and 19200 km above the Earth's surface, we can follow these steps:

### Step 1: Understand the formula for orbital speed
The orbital speed \( V \) of a satellite at a height \( h \) above the Earth's surface can be expressed as:
\[ V = \sqrt{\frac{GM}{R + h}} \]
where:
- \( G \) is the universal gravitational constant,
- \( M \) is the mass of the Earth,
- \( R \) is the radius of the Earth,
- \( h \) is the height of the satellite above the Earth's surface.

### Step 2: Define the heights of the satellites
Given:
- The radius of the Earth \( R = 6400 \, \text{km} \).
- Height of the first satellite \( h_1 = 6400 \, \text{km} \).
- Height of the second satellite \( h_2 = 19200 \, \text{km} \).

### Step 3: Calculate the total distance from the center of the Earth for both satellites
For the first satellite:
\[ R + h_1 = 6400 \, \text{km} + 6400 \, \text{km} = 12800 \, \text{km} \]
For the second satellite:
\[ R + h_2 = 6400 \, \text{km} + 19200 \, \text{km} = 25600 \, \text{km} \]

### Step 4: Write the expressions for the orbital speeds
For the first satellite:
\[ V_1 = \sqrt{\frac{GM}{12800}} \]
For the second satellite:
\[ V_2 = \sqrt{\frac{GM}{25600}} \]

### Step 5: Calculate the ratio of the orbital speeds
Now, we need to find the ratio \( \frac{V_1}{V_2} \):
\[ \frac{V_1}{V_2} = \frac{\sqrt{\frac{GM}{12800}}}{\sqrt{\frac{GM}{25600}}} \]
This simplifies to:
\[ \frac{V_1}{V_2} = \sqrt{\frac{25600}{12800}} = \sqrt{2} \]

### Final Answer
The ratio of the orbital speeds of the two satellites is:
\[ \frac{V_1}{V_2} = \sqrt{2} \]

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