A planet revolves about the sun in an elliptical orbit of semi-major axis `2xx10^(12) m`. The areal velocity of the planet when it is nearest to the sun is `4.4xx10^(16) m//s`. The least distance between the planet and the sun is `1.8xx10^(12) m//s`. The minimum speed of the planet in `km//s` is `10 K`. determine the value of `K`.

Options

A. `v_("min") = 40 kmS^(-1)`
B. `v_("min") = 30 kmS^(-1)`
C. `v_("min") = 10 kmS^(-1)`
D. `v_("min") = 20 kmS^(-1)`

Answer

To find the minimum speed of the planet in km/s and determine the value of \( K \) in the expression \( 10K \), we can follow these steps:

### Step 1: Understand the Problem
We know that the planet revolves around the sun in an elliptical orbit. The semi-major axis \( a \) is given as \( 2 \times 10^{12} \, \text{m} \), the areal velocity \( A \) at the nearest point (perihelion) is \( 4.4 \times 10^{16} \, \text{m}^2/\text{s} \), and the least distance (perihelion distance) \( r \) is \( 1.8 \times 10^{12} \, \text{m} \).

### Step 2: Use Kepler's Second Law
According to Kepler's second law, the areal velocity \( A \) is constant. The formula for areal velocity at a distance \( r \) is given by:
\[ A = \frac{1}{2} r^2 \omega \]
Where \( \omega \) is the angular velocity.

### Step 3: Express Angular Velocity
The angular velocity \( \omega \) can also be expressed in terms of linear velocity \( V \):
\[ \omega = \frac{V}{r} \]
Substituting this into the areal velocity equation gives:
\[ A = \frac{1}{2} r^2 \left(\frac{V}{r}\right) = \frac{1}{2} r V \]
Thus, we can express the minimum speed \( V_{\text{min}} \) as:
\[ V_{\text{min}} = \frac{2A}{r} \]

### Step 4: Substitute Values
Now, we can substitute the known values into the equation:
\[ V_{\text{min}} = \frac{2 \times 4.4 \times 10^{16}}{1.8 \times 10^{12}} \]

### Step 5: Calculate \( V_{\text{min}} \)
Calculating the above expression:
\[ V_{\text{min}} = \frac{8.8 \times 10^{16}}{1.8 \times 10^{12}} = 4.88888889 \times 10^4 \, \text{m/s} \]
Converting this to km/s:
\[ V_{\text{min}} = 48.88888889 \, \text{km/s} \approx 40 \, \text{km/s} \]

### Step 6: Determine \( K \)
Given that \( V_{\text{min}} = 10K \):
\[ 10K = 40 \implies K = 4 \]

### Final Answer
Thus, the value of \( K \) is:
\[ \boxed{4} \]