The maximum and minimum distance of a comet from the sun are ` 14 xx 10^9` m and `7 xx 10^7`m respectively. If the maximum velocity of the comet is ` 5 xx10^2` km/sec, its minimum velcity will be:-

Options

A. `2.5km//sec`
B. `4 km//sec`
C. `6 km//sec`
D. `8 km//sec`

Answer

To solve the problem, we will use the principle of conservation of angular momentum. The angular momentum of a comet around the sun is given by the product of its mass, velocity, and distance from the sun. Since there are no external torques acting on the system, the angular momentum will remain constant.

### Step-by-Step Solution:

1. **Identify Given Values:**
- Maximum distance from the sun, \( D_{max} = 14 \times 10^9 \) m
- Minimum distance from the sun, \( D_{min} = 7 \times 10^7 \) m
- Maximum velocity of the comet, \( V_{max} = 5 \times 10^2 \) km/s = \( 5 \times 10^5 \) m/s (convert km/s to m/s)

2. **Use the Conservation of Angular Momentum:**
The angular momentum at maximum distance and maximum velocity is equal to the angular momentum at minimum distance and minimum velocity:
\[ M \cdot V_{max} \cdot D_{min} = M \cdot V_{min} \cdot D_{max} \]
Since the mass \( M \) of the comet is constant, it can be canceled out:
\[ V_{max} \cdot D_{min} = V_{min} \cdot D_{max} \]

3. **Rearranging the Equation:**
To find the minimum velocity \( V_{min} \), rearrange the equation:
\[ V_{min} = \frac{V_{max} \cdot D_{min}}{D_{max}} \]

4. **Substituting the Values:**
Substitute the known values into the equation:
\[ V_{min} = \frac{(5 \times 10^5 \, \text{m/s}) \cdot (7 \times 10^7 \, \text{m})}{14 \times 10^9 \, \text{m}} \]

5. **Calculating the Result:**
- Calculate the numerator:
\[ 5 \times 10^5 \cdot 7 \times 10^7 = 35 \times 10^{12} \, \text{m}^2/\text{s} \]
- Calculate the denominator:
\[ 14 \times 10^9 \, \text{m} \]
- Now divide:
\[ V_{min} = \frac{35 \times 10^{12}}{14 \times 10^9} = \frac{35}{14} \times 10^{12 - 9} = 2.5 \times 10^3 \, \text{m/s} \]

6. **Convert the Result to km/s:**
\[ V_{min} = 2.5 \times 10^3 \, \text{m/s} = 2.5 \, \text{km/s} \]

### Final Answer:
The minimum velocity of the comet is \( 2.5 \, \text{km/s} \).

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