In the question number 15, the ratio of the velocity of the satellite at apogee and perigee is

Options

A. `(1)/(2)`
B. `(1)/(3)`
C. `(1)/(4)`
D. `(1)/(6)`

Explanation

To find the ratio of the velocity of the satellite at apogee and perigee, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Definitions**:
- **Apogee**: The point in the orbit of a satellite where it is farthest from the Earth.
- **Perigee**: The point in the orbit of a satellite where it is closest to the Earth.

2. **Use the Conservation of Angular Momentum**:
- According to the law of conservation of angular momentum, the angular momentum at perigee (closest point) is equal to the angular momentum at apogee (farthest point).
- Mathematically, this can be expressed as:
\[ M \cdot V_p \cdot R_p = M \cdot V_a \cdot R_a \]
where:
- \( M \) is the mass of the satellite (which cancels out),
- \( V_p \) is the velocity at perigee,
- \( R_p \) is the distance from the center of the Earth to the satellite at perigee,
- \( V_a \) is the velocity at apogee,
- \( R_a \) is the distance from the center of the Earth to the satellite at apogee.

3. **Rearranging the Equation**:
- Canceling the mass \( M \) from both sides gives:
\[ V_p \cdot R_p = V_a \cdot R_a \]
- Rearranging this gives:
\[ \frac{V_a}{V_p} = \frac{R_p}{R_a} \]

4. **Substituting the Values**:
- From the problem, we know:
- \( R_p = 2r \) (the distance at perigee),
- \( R_a = 6r \) (the distance at apogee).
- Substituting these values into the equation gives:
\[ \frac{V_a}{V_p} = \frac{2r}{6r} \]

5. **Simplifying the Ratio**:
- The \( r \) cancels out:
\[ \frac{V_a}{V_p} = \frac{2}{6} = \frac{1}{3} \]

6. **Conclusion**:
- Thus, the ratio of the velocity of the satellite at apogee to the velocity at perigee is:
\[ \frac{V_a}{V_p} = \frac{1}{3} \]

### Final Answer:
The ratio of the velocity of the satellite at apogee to perigee is \( \frac{1}{3} \).

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