Gravitational Potential
Work done per unit mass from infinity.
This public page keeps the free explanation visible and leaves premium worked solving, advanced walkthroughs, and saved study tools inside the app.
Core idea
Overview
Gravitational potential is a scalar quantity representing the potential energy per unit mass at a specific location within a gravitational field. It defines the work done per unit mass by gravity to move an object from an infinite distance to a specific point, resulting in a negative value as energy is released during the approach.
When to use: Apply this formula when analyzing the energy environment around spherical celestial bodies like planets or stars. It is essential when the distance from the source of gravity changes significantly, rendering the local g ≈ 9.81 approximation invalid.
Why it matters: This equation is the foundation for calculating escape velocities and the energy required for orbital maneuvers. It helps engineers determine the fuel needed for spacecraft to climb out of a planet's 'gravity well' and traverse the solar system.
Symbols
Variables
V = Potential, G = Grav Constant, M = Mass, r = Distance
Walkthrough
Derivation
Understanding Gravitational Potential
Work done per unit mass to bring a test mass from infinity to a point in a gravitational field.
- Potential is defined as zero at infinity.
State the Result for a Point/Spherical Mass:
Potential is negative because gravity is attractive; work must be done to move mass away to infinity.
Result
Source: AQA A-Level Physics — Gravitational Fields
Free formulas
Rearrangements
Solve for
Make V the subject
V is already the subject of the formula.
Difficulty: 1/5
Solve for
Make r the subject
Start from Gravitational Potential. To make r the subject, clear r, then divide by V.
Difficulty: 3/5
Solve for
Make M the subject
Start from Gravitational Potential. To make M the subject, clear r, then divide by G.
Difficulty: 3/5
Solve for
Make G the subject
Start from Gravitational Potential. To make G the subject, clear r, then divide by M.
Difficulty: 3/5
The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.
Visual intuition
Graph
The graph forms a hyperbola because the distance r appears in the denominator of the potential formula. For a physics student, this shape shows that potential magnitude is greatest when r is small near a mass, while it approaches zero as r becomes very large. The most important feature is that the curve never reaches zero, meaning the gravitational influence of a mass technically extends across all space regardless of distance.
Graph type: hyperbolic
Why it behaves this way
Intuition
Imagine a funnel-shaped 'gravity well' in space, where the potential is deepest (most negative) near the central mass and gradually flattens out to zero at infinite distance.
Signs and relationships
- -: The negative sign indicates that gravity is an attractive force. Gravitational potential is defined as zero at infinite separation. As a mass approaches the central body, work is done by the gravitational field, and the
Free study cues
Insight
Canonical usage
Units for all variables must be consistent within the International System of Units (SI) to yield gravitational potential in joules per kilogram.
Common confusion
A common mistake is using inconsistent units, such as kilometers for distance while using the SI value for G, or confusing gravitational potential (J/kg) with gravitational potential energy (J).
Unit systems
One free problem
Practice Problem
Calculate the gravitational potential on the surface of Earth. Assume Earth has a mass of 5.97 × 10²⁴ kg and a radius of 6,371,000 meters.
Solve for:
Hint: Multiply the gravitational constant by the mass of the Earth, then divide by the radius and add a negative sign.
The full worked solution stays in the interactive walkthrough.
Where it shows up
Real-World Context
Comparing gravitational potential at Earth’s surface vs higher altitude.
Study smarter
Tips
- The value is always negative because the potential at infinity is defined as zero.
- Ensure r is the distance from the center of the mass, not its surface.
- The units are Joules per kilogram (J/kg) or m² s⁻².
Avoid these traps
Common Mistakes
- Dropping the negative sign.
- Using r² instead of r.
Common questions
Frequently Asked Questions
Work done per unit mass to bring a test mass from infinity to a point in a gravitational field.
Apply this formula when analyzing the energy environment around spherical celestial bodies like planets or stars. It is essential when the distance from the source of gravity changes significantly, rendering the local g ≈ 9.81 approximation invalid.
This equation is the foundation for calculating escape velocities and the energy required for orbital maneuvers. It helps engineers determine the fuel needed for spacecraft to climb out of a planet's 'gravity well' and traverse the solar system.
Dropping the negative sign. Using r² instead of r.
Comparing gravitational potential at Earth’s surface vs higher altitude.
The value is always negative because the potential at infinity is defined as zero. Ensure r is the distance from the center of the mass, not its surface. The units are Joules per kilogram (J/kg) or m² s⁻².
References
Sources
- Halliday, Resnick, and Walker, Fundamentals of Physics
- Britannica: Gravitational potential
- Wikipedia: Gravitational potential
- NIST CODATA (2018) for the value of G
- Halliday, Resnick, and Walker, Fundamentals of Physics, 11th Edition (2018) for unit definitions and dimensional analysis
- Halliday, Resnick, Walker - Fundamentals of Physics, 10th Edition
- AQA A-Level Physics — Gravitational Fields