Equipotentials

An equipotential curve is a level curve of the potential function.

Question#5

Question #6

• Let g(t) be a parametrization of the path which follows the equipotential between two such points. Without appeal to the Gradient Theorem, use your answers to Question #1 to give a geometric reason why the work integral is zero for this particular path.
• Use the Gradient Theorem to show that it takes no work to move along any path (that is, not necessarily a path which follows an equipotential) between two points which sit on the same equipotential curve.

Question #7

You now know that it takes zero work against the gravitational field to get to the asteroid, no matter what path you use. But you would surely need more fuel (more anti-matter, whatever) to fly along a path from the Earth to Pluto at (0,40) and then to the asteroid, than if you just cruised along the equipotential to get there.

• How much energy must we expend against the gravitational force field just to get from the Earth to Pluto, for example?
• Explain the apparent paradox here. Why do we need more fuel for the trip via Pluto, if the work is still zero?
• Now you want to to get from the Earth at (1,0) to Jupiter at (0,-5). Desribe a route that you think would require the smallest gas tank, keeping in mind the geometry of the gradient field of the potential from Question #5. Give reasons for choosing this trajectory. Do you think such a route is unique?