Escape Velocity Earth

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Escape velocity from Earth surface (R, g)?

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1. **Energy Conservation**: For a mass $m$ to just escape Earth's gravity, its total energy at infinity must be at least zero.
2. **Initial Energy**: $K + U = \frac{1}{2} m v_e^2 - \frac{GMm}{R} = 0$.
3. **Isolate velocity**: $v_e = \sqrt{\frac{2GM}{R}}$.
4. **Linking to 'g'**: Near the surface, acceleration due to gravity is $g = \frac{GM}{R^2} \implies GM = g R^2$.
5. **Final Substitution**: $v_e = \sqrt{\frac{2 (gR^2)}{R}} = \sqrt{2gR}$.
6. **Numerical Value**: For Earth, $v_e \approx 11.2 	ext{ km/s}$.

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Escape Velocity Earth

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