Calculate the escape velocity required for an object to break free from the gravitational field of Earth, considering Earth's mass and radius.
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To calculate the escape velocity required for an object to break free from the gravitational field of Earth, we need to use the formula for escape velocity:
\[ v_e = \sqrt{\frac{{2G M_e}}{{R_e}}} \]
where:
- \( v_e \) is the escape velocity
- \( G \) is the gravitational constant (\(6.67430 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2\))
- \( M_e \) is the mass of Earth (\(5.972 \times 10^{24} \, \text{kg}\))
- \( R_e \) is the radius of Earth (\(6.371 \times 10^6 \, \text{m}\))
Substituting the values into the formula:
\[ v_e = \sqrt{\frac{{2 \cdot 6.67430 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2 \cdot 5.972 \times 10^{24} \, \text{kg}}}{{6.371 \times 10^6 \, \text{m}}}} \]
Simplifying:
\[ v_e = \sqrt{\frac{{2 \cdot 6.67430 \times 5.972}}{{6.371}}} \, \text{m/s} \]
Calculating the value:
\[ v_e = \sqrt{2 \cdot 6.67430 \times 5.972} \times 10^4 \, \text{m/s} \]
\[ v_e \approx 1.12 \times 10^4 \, \text{m/s} \]
Therefore, the escape velocity required for an object to break free from the gravitational field of Earth is approximately \(1.12 \times 10^4 \, \text{m/s}\).
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