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Gravitational potential energy at large distances review

Review the equations and skills related to gravitational potential energy at large distances, including how to apply conservations laws to objects in orbit.

Equations

EquationSymbolsMeaning in words
U, start subscript, G, end subscript, equals, minus, start fraction, G, m, start subscript, 1, end subscript, m, start subscript, 2, end subscript, divided by, r, end fractionU, start subscript, G, end subscript is gravitational potential energy, G is the gravitational constant, m, start subscript, 1, end subscript and m, start subscript, 2, end subscript are masses, and r is the distance between centers of mass of the two objectsGravitational potential energy at large distances is directly proportional to the masses and inversely proportional to the distance between them. The gravitational potential energy increases as r increases.

How to apply conservation laws to orbits

Although the Earth orbits the Sun, it does not go around in a perfect circle, but rather takes an elliptical path (Figure 1).
Figure 1: An elliptical path of a planet around the Sun. When the planet is closest to the Sun, speed v and kinetic energy are the highest, and gravitational potential energy is the lowest. When the planet moves farther away, the speed and kinetic energy decrease, and the gravitational potential energy increases. At all points in the orbit, angular momentum and energy are conserved.
This means that the Earth’s distance from Sun r varies throughout the orbit. There is no net external force or torque acting on the Sun-planet system, and the only force is gravity between the Sun and planet. Therefore, angular momentum and energy remain constant. However, the gravitational potential energy does change, because it depends on distance. As a result, kinetic energy also changes throughout an orbit, resulting in a higher speed when a planet is closer to the Sun.
When dealing with gravitational potential energy over large distances, we typically make a choice for the location of our zero point of gravitational potential energy at a distance r of infinity. This makes all values of the gravitational potential energy negative.
If we make our zero of potential energy at infinity, then the gravitational potential energy as a function of r is:
U, start subscript, G, end subscript, equals, minus, start fraction, G, m, start subscript, 1, end subscript, m, start subscript, 2, end subscript, divided by, r, end fraction
For example, imagine we are landing on a planet. As we come closer to the planet, the radial distance between us and the planet decreases. As r decreases, we lose gravitational potential energy - in other words, U, start subscript, G, end subscript becomes more negative. Because energy is conserved, the velocity must increase, resulting in an increase in kinetic energy.

Common mistakes and misconceptions

  1. Students forget that there must be two separated objects considered as the system to have potential energy. A single object cannot have potential energy with itself, but only with respect to another object. For example, the Moon only has gravitational potential energy relative to the Earth (or another object).
  2. Sometimes people forget that gravitational potential energy at large distances is negative. We typically make a choice for the location of our zero point of gravitational potential energy at a distance r of infinity. This makes all values of the gravitational potential energy negative.

Learn more

For deeper explanations of these concepts, see our video about gravitational potential energy at large distances.
To check your understanding and work toward mastering these concepts, check out our exercises:

Want to join the conversation?

  • aqualine ultimate style avatar for user Yes yes
    Why does the gravitational potential energy increases as r increases if the equation states that they are inversely proportional?
    (2 votes)
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    • hopper cool style avatar for user obiwan kenobi
      Think about it, the formula gmh applies for small distances. But for ridiculously large distances, it makes sense that the further away an object gets, the smaller the gravitational potential energy it has. The further away the object gets, the less influence it will feel gravitationally. Hope this helps!
      (2 votes)
  • aqualine ultimate style avatar for user Muhammad Olim
    I saw practice questions about the change of maximum gravitational potential energy of the Sun-planet system as a planet is struck by a large asteroid. In this situation:

    1. How minimum gravitational potential energy changes?
    2. How elliptical orbit of planet changes?
    (2 votes)
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  • blobby green style avatar for user Vihaan Gargava
    I didn't understood how can gravitational potential energy increase if distance between the centre of masses increases....it(potential energy) should particularly decrease as the distance is increasing?
    (1 vote)
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    • blobby green style avatar for user samraddhisagar07
      the gravitational potential energy is inversely proportional to radius and yea it should decrease as the radius increases but what you are forgetting here is a -ve sign so like for example the radius increases and the potential energy becomes -5 and then radius is reduced by half so the potential energy becomes -10 and as we know -10<-5
      (1 vote)
  • blobby green style avatar for user Bikash Adhikari
    Can anyone explain the conservation of energy in the following case? I suppose throw a body in air, which means I did work on the body to increase its potential energy by mgh. And it comes down with increasing velocity and as it reaches down its energy is 1/2 mv^2 which is equal to mgh. But when it reaches down, it doesn't have any velocity nor does it have potential energy. So, where does that energy go?
    (1 vote)
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  • blobby green style avatar for user 이태경
    I also saw the practice questions about the change of maximum gravitational potential energy of the planet and the sun-planet system and i don't really get the idea why the gravitational energy decreases in the 1st question
    (1 vote)
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    • marcimus purple style avatar for user Fiona T
      The gravitational potential energy decreases as the distance between the two bodies decreases.
      Looking at the equation in this article, there is a negative sign in front of the expression (G*m1*m2) / r.
      Ug and r are inversely proportional, so Ug decreases as the absolute value of r increases (and -r decreases, getting farther away from 0).
      (1 vote)