The International Space Stations (ISS), a habitable artificial satellite, is orbiting around earth at an altitude of 400 km. Calculate the additional velocity required to be given to escape the ISS from gravitational pull of earth.

Answer

To solve the problem of calculating the additional velocity required for the International Space Station (ISS) to escape Earth's gravitational pull, we can follow these steps:

### Step 1: Understand the Problem
The ISS is orbiting Earth at an altitude of 400 km. We need to find the additional velocity required for it to escape Earth's gravitational pull.

### Step 2: Determine the Radius of Orbit
The radius of the Earth (R) is approximately 6400 km. The altitude (h) of the ISS is 400 km. Therefore, the total radius (r) from the center of the Earth to the ISS is:
\[ r = R + h = 6400 \, \text{km} + 400 \, \text{km} = 6800 \, \text{km} = 6.8 \times 10^6 \, \text{m} \]

### Step 3: Calculate the Orbital Velocity
The orbital velocity (v_o) of the ISS can be calculated using the formula:
\[ v_o = \sqrt{\frac{GM}{r}} \]
where:
- \( G \) is the gravitational constant, \( G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \)
- \( M \) is the mass of the Earth, \( M = 6 \times 10^{24} \, \text{kg} \)

Substituting the values:
\[ v_o = \sqrt{\frac{(6.67 \times 10^{-11}) \times (6 \times 10^{24})}{6.8 \times 10^6}} \]

### Step 4: Calculate the Escape Velocity
The escape velocity (v_e) from the surface of the Earth is given by:
\[ v_e = \sqrt{\frac{2GM}{R}} \]
Substituting \( R = 6400 \, \text{km} = 6.4 \times 10^6 \, \text{m} \):
\[ v_e = \sqrt{\frac{2 \times (6.67 \times 10^{-11}) \times (6 \times 10^{24})}{6.4 \times 10^6}} \]

### Step 5: Find the Additional Velocity Required
The additional velocity required (Δv) to escape from the ISS can be calculated as:
\[ \Delta v = v_e - v_o \]

### Step 6: Calculate Numerical Values
1. Calculate \( v_o \):
\[ v_o = \sqrt{\frac{(6.67 \times 10^{-11}) \times (6 \times 10^{24})}{6.8 \times 10^6}} \approx 7.67 \, \text{km/s} \]

2. Calculate \( v_e \):
\[ v_e = \sqrt{\frac{2 \times (6.67 \times 10^{-11}) \times (6 \times 10^{24})}{6.4 \times 10^6}} \approx 11.2 \, \text{km/s} \]

3. Calculate \( \Delta v \):
\[ \Delta v = v_e - v_o \approx 11.2 \, \text{km/s} - 7.67 \, \text{km/s} \approx 3.53 \, \text{km/s} \]

### Final Answer
The additional velocity required for the ISS to escape Earth's gravitational pull is approximately **3.53 km/s**.

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