Section 2.5: Escape Velocity
Escape velocity is the minimum speed required for an object to break free from the gravitational pull of a planet or celestial body without further propulsion.
\( v_{esc} = \sqrt{\frac{2GM}{r}} \), where \( M \) is the mass of the planet and \( r \) is the distance from the center.
Example 1
Find the escape velocity from Earth’s surface.
Earth mass \( M = 5.97\times10^{24} \text{ kg} \), radius \( R = 6.37\times10^6 \text{ m} \)
\( v_{esc} = \sqrt{\frac{2GM}{R}} = \sqrt{\frac{2*6.674\times10^{-11}*5.97\times10^{24}}{6.37\times10^6}} \approx 11.2 \text{ km/s} \)
Example 2
Compute the escape velocity from the Moon (mass \( 7.35×10^{22} \) kg, radius 1.737×10^6 m).
\( v_{esc} = \sqrt{\frac{2*6.674\times10^{-11}*7.35\times10^{22}}{1.737\times10^6}} \approx 2.38 \text{ km/s} \)
Practice Problems
- Find escape velocity from a planet with mass \( 1×10^{25} \) kg and radius \( 8×10^6 \) m.
- Determine escape velocity at 500 km above Earth’s surface.
- A spacecraft needs 5 km/s to escape a planet. Calculate the planet's mass if radius is 6×10^6 m.
- Find the escape velocity for Mars (mass \( 6.42×10^{23} \) kg, radius 3.39×10^6 m).
- Calculate escape speed for a small asteroid of radius 500 m, mass \( 1×10^{12} \) kg.
- If a rocket launches at 10 km/s from Earth, determine if it escapes.
- Find escape velocity for a satellite orbiting at 1000 km above Earth.
- A planet has escape velocity 7 km/s and radius 5×10^6 m. Find its mass.
- Compare escape velocities of Earth and Jupiter.
- Calculate the minimum launch speed from a height of 2R above a planet.