The big idea: Throw a ball straight up and it slows, stops, and drops back — near a planet you sit in a gravitational well, and climbing out costs energy.
Gravitational potential V
- The energy per kilogram at a point
Gravitational potential energy E_{p}
- The energy of a particular object of mass m there
Both share two features: Both are negative, and both are zero infinitely far away (where gravity has faded to nothing).
A planet of mass M. Its gravitational field points inwards (gravity always attracts), so an object near it sits in a 'potential well'. To escape, it must gain enough energy to climb all the way out to where the field has faded to nothing.
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Why negative?: We set the energy to zero at infinity (right out of the well).
Anywhere closer in, gravity has already pulled the object 'downhill', so it has less than zero energy — hence the minus sign. Closer in = more negative = deeper in the well.
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Gravitational potential is the gravitational potential energy per kilogram at a distance r from a mass M:
- gravitational potential (J kg⁻¹)
- gravitational constant, 6.67 × 10⁻¹¹ N m² kg⁻² (given)
- mass of the planet or star (kg)
- distance from the centre of that mass (m)
Multiply by the object's mass m to get the gravitational potential energy of that object:
- gravitational potential energy (J)
- gravitational constant, 6.67 × 10⁻¹¹ N m² kg⁻² (given)
- mass of the planet or star (kg)
- mass of the small object placed in the field (kg)
- distance from the centre of M (m)
What 'escape' means: To escape a planet, an object must be launched fast enough to climb out of the well — to reach the place where V = 0 (infinitely far away) and still be moving (or only just stop there).
Escape speed from energy: Using energy conservation, the kinetic energy at launch must equal the depth of the well. This gives the escape speed, and notice the object's own mass cancels out:
- escape speed — the launch speed needed to escape (m s⁻¹)
- gravitational constant, 6.67 × 10⁻¹¹ N m² kg⁻² (given)
- mass of the planet or star (kg)
- distance from the centre of M at launch — usually its radius (m)
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A planet has mass M = 6.0 × 10²⁴ kg and radius r = 6.4 × 10⁶ m. Find the escape speed from its surface. Take G = 6.67 × 10⁻¹¹ N m² kg⁻².
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How this is tested — gravitational energy and escape speed span both papers:
Paper 1A
- The sign and shape — Ep is negative and rises to 0 at infinity.
- What 'escape' means in energy terms.
Paper 2
- Plug numbers into v = √(2GM/r).
- Or use energy conservation (potential-energy change vs kinetic energy) to find a launch or impact speed.
The classic trap: Thinking escape speed depends on the launched object's mass. It does not — the mass cancels.
The energy method: Many questions are really energy conservation:
kinetic energy gained = depth of the well climbed.
To just escape from the surface: ½mv² = GMm/r. The m cancels, leaving vesc = √(2GM/r).
A 1200 kg probe rests on the surface of a moon of mass M = 7.3 × 10²² kg and radius r = 1.7 × 10⁶ m. (a) Find the gravitational potential energy of the probe. (b) Find the escape speed from the moon's surface. Take G = 6.67 × 10⁻¹¹ N m² kg⁻².
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