A comet of mass `10^(8)` kg travels around the sun in an elliptical orbit . When it is closed to the sun it is `2.5 xx 10 ^(11)` m away and its speed is `2 xx 10 ^(4) m s^(-1)` Find the change in kinetic energy when it is farthest from the sun and is `5 xx 10 ^(10)` m away from the sun

Options

A. `38 xx10^(8) J`
B. `48 xx10^8J`
C. `58 xx10^8 J`
D. `56xx10^8J`

Answer

To solve the problem step by step, we need to find the change in kinetic energy of the comet when it moves from its closest point to the Sun to its farthest point.

### Step 1: Identify the given data
- Mass of the comet, \( m = 10^8 \, \text{kg} \)
- Distance from the Sun when closest, \( r_1 = 2.5 \times 10^{11} \, \text{m} \)
- Speed when closest, \( v_1 = 2 \times 10^4 \, \text{m/s} \)
- Distance from the Sun when farthest, \( r_2 = 5 \times 10^{10} \, \text{m} \)

### Step 2: Use conservation of angular momentum to find the speed at the farthest point
The angular momentum \( L \) of the comet is conserved. Therefore, we can write:
\[ L = m \cdot v_1 \cdot r_1 = m \cdot v_2 \cdot r_2 \]
Where \( v_2 \) is the speed at the farthest point. Rearranging gives:
\[ v_2 = \frac{v_1 \cdot r_1}{r_2} \]

### Step 3: Substitute the values to find \( v_2 \)
Substituting the known values:
\[ v_2 = \frac{(2 \times 10^4) \cdot (2.5 \times 10^{11})}{5 \times 10^{10}} \]

### Step 4: Simplify the equation
Calculating \( v_2 \):
\[ v_2 = \frac{(2 \times 2.5) \times 10^{15}}{5 \times 10^{10}} = \frac{5 \times 10^{15}}{5 \times 10^{10}} = 10^5 \, \text{m/s} \]

### Step 5: Calculate the kinetic energy at both positions
The kinetic energy \( KE \) is given by the formula:
\[ KE = \frac{1}{2} m v^2 \]

Calculating \( KE_1 \) (at closest point):
\[ KE_1 = \frac{1}{2} \cdot (10^8) \cdot (2 \times 10^4)^2 = \frac{1}{2} \cdot (10^8) \cdot (4 \times 10^8) = 2 \times 10^{16} \, \text{J} \]

Calculating \( KE_2 \) (at farthest point):
\[ KE_2 = \frac{1}{2} \cdot (10^8) \cdot (10^5)^2 = \frac{1}{2} \cdot (10^8) \cdot (10^{10}) = 5 \times 10^{17} \, \text{J} \]

### Step 6: Find the change in kinetic energy
The change in kinetic energy \( \Delta KE \) is given by:
\[ \Delta KE = KE_2 - KE_1 \]
Substituting the values:
\[ \Delta KE = (5 \times 10^{17}) - (2 \times 10^{16}) = 5 \times 10^{17} - 0.2 \times 10^{17} = 4.8 \times 10^{17} \, \text{J} \]

### Final Answer
The change in kinetic energy when the comet is farthest from the Sun is \( 4.8 \times 10^{17} \, \text{J} \).