Learn the concept and calculation of cosmic velocities, which determine whether an object orbits Earth or completely escapes from it.
Basic Knowledge
First cosmic velocity v1: The minimum speed required to maintain a circular orbit just above Earth's surface.v1 = √(gR) = √(GM/R) (from g = 9.8 m/s² and R = 6.4×10⁶ m)
v1 ≈ 7.9 km/s
Second cosmic velocity (escape velocity) v2: The minimum speed required to completely overcome Earth's gravity and reach infinity.
Conservation of energy (the kinetic energy at Earth's surface equals the difference in potential energy at infinity):
½mv2² − GMm/R = 0 (the kinetic energy at infinity is taken as 0)
→ v2 = √(2GM/R) = √(2gR) = √2 × v1
v2 ≈ 11.2 km/s
(Reference) Third cosmic velocity v₃ ≈ 16.7 km/s: The speed required to escape the solar system (taking Earth's orbital velocity into account).
First cosmic velocity: v1 = √(gR) ≈ 7.9 km/s
Second cosmic velocity: v2 = √(2gR) = √2 × v1 ≈ 11.2 km/s
Derivation of escape velocity: Conservation of mechanical energy (Earth's surface → infinity)
Gravitational potential: U = −GMm/r (with infinity as the reference U = 0)
v2 = √2 × v1 (remember the numerical relationship)
Further Exploration (Background and Significance)
Deriving the second cosmic velocity requires the concept of gravitational potential energy U = −GMm/r. Mechanical energy at Earth's surface = ½mv2² + (−GMm/R). At infinity, both KE and PE are zero (the minimum condition). By conservation of energy, ½mv2² = GMm/R → v2 = √(2GM/R).
A black hole is a celestial body whose escape velocity exceeds the speed of light c: √(2GM/r_s) = c → Schwarzschild radius r_s = 2GM/c². A black hole with the same mass as the Sun has r_s ≈ 3 km, extremely small compared with the Sun's current radius of 700,000 km.
The ratio of the second cosmic velocity to the first cosmic velocity, √2, is frequently tested in entrance examinations. The escape velocity from another planet can also be calculated by substituting that planet's values of g and R.
Actual rocket launches require techniques such as overcoming air resistance, using Earth's rotation (launching toward the equator), and using multistage rockets, so more fuel is needed than the theoretical value suggests.
- First cosmic velocity: surface circular orbit v1 = √(gR)
- Second cosmic velocity: escape velocity v2 = √(2gR)
- v2 = √2 × v1 (numerical relationship)
- Derivation of escape velocity: conservation of mechanical energy
- U = −GMm/r (infinity is the reference = 0)
- v ≥ v2 → escape from Earth
- Black hole: escape velocity = speed of light
Points of Caution (Common Confusions)
① Do not confuse the first and second cosmic velocities. The first is for orbiting (a satellite), while the second is for escaping (v2 is greater). ② Potential energy U = −GMm/r is a negative value. It is most negative at Earth's surface (a bound state). ③ The derivation of escape velocity uses “KE = 0 at infinity” (strictly, KE ≥ 0, with equality representing the minimum escape condition). ④ Remember the representative values 7.9 km/s and 11.2 km/s.
Practice
- Given Earth's g = 9.8 m/s² and R = 6.4×10⁶ m, calculate the first and second cosmic velocities.
- Express the escape velocity from the Moon (g_月 = g/6, R_月 = R/4, where R is Earth's value) using Earth's v2.
- Using conservation of mechanical energy, derive Earth's escape velocity v2 = √(2GM/R).