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What is escape velocity?

Class 11 PhysicsNCERT chapter: Gravitationइसे हिंदी में पढ़ें

Exam-ready answer

Escape velocity is the minimum speed with which a body must be projected from the surface of the Earth (or any planet) so that it escapes the gravitational pull and never returns. Its value is v_e = √(2GM/R) = √(2gR). For the Earth it is about 11.2 km/s, and it does not depend on the mass of the body.

Explanation

When a stone is thrown upward, gravity slows it down and it falls back. A faster throw takes it higher. If it is thrown fast enough, its kinetic energy is just enough to overcome the gravitational potential energy and it reaches infinity, far from the Earth, with zero speed. That minimum launch speed is the escape velocity. Air resistance is ignored in this calculation.

At the surface, the total energy of a body of mass m is ½mv² − GMm/R, where M and R are the mass and radius of the Earth. At infinity the potential energy is zero and the least possible kinetic energy is zero, so the total energy there is zero. Since total energy is conserved, the minimum condition is ½mv_e² − GMm/R = 0, giving v_e = √(2GM/R). Using g = GM/R², this becomes v_e = √(2gR).

Putting g ≈ 9.8 m s⁻² and R ≈ 6.4 × 10⁶ m gives v_e ≈ 11.2 km/s for the Earth. The mass m cancels out, so a pebble and a large spacecraft need the same escape speed. The Moon has much weaker gravity, so its escape velocity is only about 2.3 km/s, about five times smaller than Earth's.

Points to remember

  1. Escape velocity: the minimum initial speed needed for a body to escape the gravitational field of a planet.
  2. Formula: v_e = √(2GM/R) = √(2gR).
  3. Value for Earth ≈ 11.2 km/s; for the Moon ≈ 2.3 km/s.
  4. It is independent of the mass of the body being projected.
  5. Derived from energy conservation: ½mv_e² = GMm/R.

Exam tip

Write energy at the surface and at infinity, equate them to get ½mv_e² = GMm/R, then substitute GM = gR². End with the numerical value 11.2 km/s for the Earth.

Questions students also ask

Why does the Moon have no atmosphere?
The escape velocity on the Moon is only about 2.3 km/s. Gas molecules at the Moon's surface can reach such speeds, so over time they escaped into space. That is why the Moon has no atmosphere.
Does escape velocity depend on the direction of projection?
No. The derivation uses only energy, which is a scalar, so the direction does not appear. Ignoring air resistance and the Earth's rotation, a body needs the same speed of about 11.2 km/s whichever direction it is launched in, as long as it does not hit the Earth.
How is escape velocity related to orbital velocity near the Earth?
For a satellite orbiting just above the Earth's surface, the orbital speed is √(gR), about 7.9 km/s. The escape velocity is √(2gR), so it is √2 times the orbital speed near the surface.

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