Gravitational potential energy and escape speed
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Question
Define gravitational potential V.
Answer
The gravitational potential energy **per kilogram** at a point: $V = -\dfrac{GM}{r}$. Unit: J kg⁻¹. Negative everywhere, zero at infinity.
Question
Formula for gravitational potential energy E_{p}?
Answer
$E_{p} = -\dfrac{GMm}{r}$ — for a mass m at distance r from a mass M. Unit: joules (J).
Question
Why is gravitational potential energy negative?
Answer
We set it to **zero at infinity**; anywhere closer in, gravity has already pulled the object 'downhill', so it has less than zero — it sits in a **well**.
Question
Where is gravitational potential energy zero?
Answer
**At infinity** — infinitely far from the mass, where the field has faded to nothing.
Question
Define escape speed.
Answer
The minimum launch speed needed for an object to escape a planet's gravity — to reach where V = 0 (infinitely far) and just stop there.
Question
Formula for escape speed?
Answer
$v_{esc} = \sqrt{\dfrac{2GM}{r}}$ — from energy conservation. r is usually the planet's radius.
Question
Does escape speed depend on the escaping object's mass?
Answer
**No** — the mass cancels in v_{esc} = √(2GM/r). It depends only on the planet's mass M and radius r.
Question
In energy terms, what does 'escape' mean?
Answer
Supplying enough **kinetic energy** to climb out of the gravitational well to where V = 0 (infinitely far away): ½mv² = GMm/r.
Question
How does escape speed change if a planet's mass quadruples (same radius)?
Answer
It **doubles** — v_{esc} ∝ √M, so √4 = 2.
Question
As an object moves further from a planet, what happens to E_{p}?
Answer
E_{p} becomes **less negative** (rises towards 0), because r increases in E_{p} = -GMm/r.
Question
Difference between gravitational potential V and potential energy E_{p}?
Answer
V is the energy **per kilogram** (J kg⁻¹); E_{p} = mV is the energy of a specific object of mass m (J).
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Gravitational fields
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