Calculate the escape velocity from the moon. The mass of the moon `=7.4xx10^22kg` and radius of the moon `=1740km

Explanation

To calculate the escape velocity from the moon, we can use the formula for escape velocity, which is given by:

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

Where:
- \( v_e \) is the escape velocity,
- \( G \) is the universal gravitational constant, approximately \( 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \),
- \( M \) is the mass of the moon,
- \( R \) is the radius of the moon.

### Step 1: Identify the values
From the problem, we have:
- Mass of the moon, \( M = 7.4 \times 10^{22} \, \text{kg} \)
- Radius of the moon, \( R = 1740 \, \text{km} = 1740 \times 10^3 \, \text{m} = 1.74 \times 10^6 \, \text{m} \)

### Step 2: Substitute the values into the formula
Now, we can substitute the values into the escape velocity formula:

\[ v_e = \sqrt{\frac{2 \times (6.67 \times 10^{-11}) \times (7.4 \times 10^{22})}{1.74 \times 10^6}} \]

### Step 3: Calculate the numerator
First, calculate the numerator:

\[ 2 \times (6.67 \times 10^{-11}) \times (7.4 \times 10^{22}) = 9.848 \times 10^{12} \]

### Step 4: Calculate the escape velocity
Now, substitute this back into the formula:

\[ v_e = \sqrt{\frac{9.848 \times 10^{12}}{1.74 \times 10^6}} \]

Calculating the division:

\[ \frac{9.848 \times 10^{12}}{1.74 \times 10^6} \approx 5.66 \times 10^6 \]

Now take the square root:

\[ v_e \approx \sqrt{5.66 \times 10^6} \approx 2381.4 \, \text{m/s} \]

### Step 5: Convert to km/s
Finally, convert the velocity from m/s to km/s:

\[ v_e \approx 2.38 \, \text{km/s} \]

### Final Answer
The escape velocity from the moon is approximately \( 2.38 \, \text{km/s} \).

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