The value of escape speed from the surface of earth is

Options

A. 11.2 m/s
B. 11.2 km/s
C. 22.2 km/h
D. 110000 m/s

Answer

To find the escape speed from the surface of the Earth, we can use the formula for escape velocity:

### Step 1: Write down the formula for escape velocity
The escape velocity (v_e) is given by the formula:
\[ v_e = \sqrt{\frac{2GM}{R}} \]
where:
- \( G \) is the universal gravitational constant (\(6.674 \times 10^{-11} \, \text{N m}^2/\text{kg}^2\)),
- \( M \) is the mass of the Earth (\(5.972 \times 10^{24} \, \text{kg}\)),
- \( R \) is the radius of the Earth (approximately \(6.371 \times 10^6 \, \text{m}\)).

### Step 2: Substitute the values into the formula
We can also express the escape velocity in terms of the acceleration due to gravity (g) at the surface of the Earth:
\[ v_e = \sqrt{2gR} \]
where \( g \) is approximately \(9.8 \, \text{m/s}^2\).

### Step 3: Calculate the escape velocity
Now, substituting the values:
- \( g = 9.8 \, \text{m/s}^2 \)
- \( R = 6.371 \times 10^6 \, \text{m} \)

\[ v_e = \sqrt{2 \times 9.8 \, \text{m/s}^2 \times 6.371 \times 10^6 \, \text{m}} \]

### Step 4: Perform the calculation
Calculating the product:
\[ 2 \times 9.8 \times 6.371 \times 10^6 \approx 124,000,000 \, \text{m}^2/\text{s}^2 \]
Taking the square root:
\[ v_e \approx \sqrt{124,000,000} \approx 11,136 \, \text{m/s} \]

### Step 5: Convert to kilometers per second
To convert meters per second to kilometers per second:
\[ v_e \approx 11.2 \, \text{km/s} \]

### Conclusion
Thus, the value of escape speed from the surface of the Earth is approximately:
\[ \boxed{11.2 \, \text{km/s}} \]

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