Gravitational Potential and Escape Speed
Gravitational potential is the signed external work per kilogram to move a small mass slowly from infinity to a point. With zero at infinity it is negative. Escape speed is the minimum outward launch speed that reaches infinity under gravity alone, without drag or collision.
What a learner can do afterwards
- Calculates potential as minus GM over r and potential energy as minus GMm over r at a stated position
- Explains the negative sign by naming infinity as the zero of the scale
- Gets escape speed by setting kinetic energy equal to the depth of the well
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Gravitational potential V is the work per kilogram done by an external force to move a small mass slowly from infinity to a point. With zero at infinity, V = -GM/r outside a spherical source of mass M. G is the gravitational constant and r is distance from its centre. V is in J/kg and is negative: gravity does positive work inward, so the external work is negative.
Lifting a 9000 kg craft from Earth's surface to 400 km up raises U by about 3.32 x 10^10 J. Use change in U = GMm(1/r1 - 1/r2), with r1 = 6.37 x 10^6 m, r2 = r1 + 4.00 x 10^5 m, M = 5.96 x 10^24 kg and G = 6.67 x 10^-11 N m^2/kg^2. This excludes the kinetic energy needed for orbit. This is about ten months' energy for a household using 909 kWh per month.
The height is added to Earth's radius. Moving outward makes U less negative.
Potential energy U = mV = -GMm/r, in joules. Kinetic energy K = m v^2 / 2. At starting distance R, GMm/R is the depth of the potential-energy well below zero. Escape speed is the minimum outward launch speed to reach infinity under this source's gravity alone, without drag, further thrust or collision. At this threshold K + U = 0, so m v^2 / 2 = GMm/R and v = sqrt(2GM/R). Measure R from the source's centre. From Earth's surface it is about 11.2 km/s.
At minimum escape speed, K equals the energy needed to raise U from -GMm/R to zero.
Total mechanical energy is E = K + U. With gravity alone, an inward fall lowers U and raises K by the same amount, so E stays constant. Negative E means gravitationally bound, not necessarily in orbit. E of zero or more means unbound, not necessarily moving outward. An unbound object can escape if its path avoids collision. Separating masses raises U and needs positive work against gravity.
For a spherical planet of mass M and radius R, surface escape speed is v = sqrt(2GM/R), under the gravity-only launch assumptions. Compare planets with the same mass by keeping 2GM fixed. A smaller radius gives a larger 2GM/R and therefore a greater escape speed. Its surface potential energy U = -GMm/R is more negative, so its potential-energy well is deeper.
With zero at infinity, V = -GM/r. Escape speed supplies enough kinetic energy to approach infinity with zero speed under gravity alone.
Includes a lifting example adapted from OpenStax, University Physics Volume 1, section 13.3, Gravitational Potential Energy and Total Energy. The inward-fall question was written for this dataset.
Reading passages adapted from OpenStax, University Physics Volume 1, sections 13.3 and 13.4, and College Physics, section 7.3. Video: Khan Academy, Energy of satellite systems.
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