Calculate the escape velocity of an atmospheric particle 1000 km above the surface of the earth. Given `R = 6.4 xx 10^(6)m` and `g = 9.8 ms^(-2)`.

Explanation

To calculate the escape velocity of an atmospheric particle located 1000 km above the surface of the Earth, we can use the formula for escape velocity:

\[ v_e = \sqrt{\frac{2GM}{R}} \]

Where:
- \( v_e \) is the escape velocity,
- \( G \) is the universal gravitational constant (\(6.67 \times 10^{-11} \, \text{m}^3/\text{kg s}^2\)),
- \( M \) is the mass of the Earth (\(6 \times 10^{24} \, \text{kg}\)),
- \( R \) is the distance from the center of the Earth to the particle.

### Step 1: Calculate the total distance \( R \)
The distance \( R \) is the sum of the Earth's radius and the height above the surface:

\[ R = R_{\text{Earth}} + h \]

Given:
- \( R_{\text{Earth}} = 6.4 \times 10^6 \, \text{m} \)
- \( h = 1000 \, \text{km} = 1000 \times 10^3 \, \text{m} = 1 \times 10^6 \, \text{m} \)

Now, calculate \( R \):

\[ R = 6.4 \times 10^6 \, \text{m} + 1 \times 10^6 \, \text{m} = 7.4 \times 10^6 \, \text{m} \]

### Step 2: Substitute values into the escape velocity formula
Now substitute \( G \), \( M \), and \( R \) into the escape velocity formula:

\[ v_e = \sqrt{\frac{2 \times (6.67 \times 10^{-11}) \times (6 \times 10^{24})}{7.4 \times 10^6}} \]

### Step 3: Calculate the numerator
Calculate the numerator:

\[ 2 \times (6.67 \times 10^{-11}) \times (6 \times 10^{24}) = 8.004 \times 10^{14} \]

### Step 4: Calculate the escape velocity
Now calculate \( v_e \):

\[ v_e = \sqrt{\frac{8.004 \times 10^{14}}{7.4 \times 10^6}} \approx \sqrt{1.080 \times 10^8} \approx 10.4 \, \text{m/s} \]

### Final Answer
The escape velocity of the atmospheric particle 1000 km above the surface of the Earth is approximately:

\[ \boxed{10.4 \, \text{m/s}} \]