Potential Energy Calculator
Calculate gravitational or elastic potential energy, or solve the standard near-surface equation for mass, height, or gravity.
Calculator is for informational purposes only. Terms and Conditions
Use this near a body’s surface when gravitational acceleration can be treated as constant over the height change.
Choose the calculation setup
Start with standard gravitational potential energy, or choose another physically distinct potential-energy model.
Enter the known values
Required inputs change with the selected equation and unknown.
Fields marked required must be completed. Heights are measured relative to the reference level used for the problem.
Result
Primary answer first, followed by conversions, warnings, and transparent calculation steps.
Result details
- Check—
Show calculation steps Review conversions, equations, substitutions, assumptions, and checks
- Enter valid values to see the complete calculation.
Potential energy visualization
The diagram changes with the selected calculation type and current valid inputs.
Method, Sources, and Assumptions
Calculation basis, verified constants, model limits, and reference information.
The calculator uses standard classical-mechanics relationships for gravitational and elastic potential energy.
- Near-surface gravitational calculations assume gravitational acceleration is effectively constant over the height interval; Earth standard gravity is 9.80665 m/s².
- Planet gravity presets use NASA/JPL equatorial surface-gravity data where available; the Moon preset is 1.62 m/s².
- General gravitational potential energy uses G = 6.67430 × 10⁻¹¹ m³/(kg·s²), with zero potential defined at infinite separation.
- Potential energy depends on the selected reference level; only potential-energy differences are reference-independent for the near-surface model.
- Elastic calculations assume an ideal linear spring that follows Hooke’s law over the entered displacement.
Calculator guide
Understanding Your Potential Energy Result
The calculator above finds gravitational potential energy from mass, height, and gravity, or solves the near-surface equation for mass, height, or gravity. For the most common case, potential energy is calculated as \(PE=U=mgh\). The calculator also supports change in gravitational potential energy, ideal spring energy, and the general two-body gravitational model.
For standard gravitational potential energy, the result is measured relative to a chosen zero-height reference. A negative value can therefore be valid when the object is below that reference, while a positive value means it is above it.
- Most common use
- Mass + gravity + vertical height → gravitational potential energy
- Outputs
- Energy, mass, height, gravity, spring constant, or displacement depending on the solve mode
- Key check
- Use one consistent reference height and the correct energy model for the physical problem
How to Use the Potential Energy Calculator
Choose the physical model first, then choose the unknown when that model supports inverse solving. The calculator changes the required fields and answer units to match the active setup.
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Choose the potential energy type
Use Gravitational: U = mgh for ordinary near-surface elevation problems, Change in gravitational PE for an initial-to-final height change, Elastic / spring PE for an ideal linear spring, or General gravitational PE when center-to-center distance matters explicitly.
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Select what to solve for
The standard gravitational mode can solve for potential energy, mass, height, or gravity. Elastic mode can solve for energy, spring constant, or displacement. The change-in-height and general-gravity modes return energy.
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Enter the known quantities and units
Use the unit selectors beside each active field. The SI / Metric and U.S. Customary presets convert existing values while preserving the same physical quantities.
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Check gravity and the reference level
For near-surface gravity, Earth standard gravity is available as \(9.80665\ \mathrm{m/s^2}\), along with planet and Moon presets or a custom value. Height is vertical position relative to your chosen zero-potential reference, not the path length traveled.
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Review the result, warnings, and steps
Use the main result first, then open the calculation steps to verify conversions and substitution. If the sign or magnitude looks surprising, recheck the model, reference height, and units before accepting the answer.
Potential Energy Methods and Formulas
For an object near Earth’s surface, gravitational potential energy is \(PE=U=mgh\), where \(m\) is mass, \(g\) is gravitational acceleration, and \(h\) is vertical height from the chosen reference level. Other potential-energy problems require different relationships.
Near-surface gravitational potential energy
Potential energy equals mass × gravitational acceleration × vertical height relative to the selected reference level.
This constant-\(g\) form is appropriate when gravitational acceleration changes negligibly over the height interval. The same equation can be rearranged to solve for \(m\), \(h\), or \(g\).
Change in gravitational potential energy
The energy change depends on the final height minus the initial height. Moving downward makes \(h_2-h_1\) negative, so \(\Delta U\) is negative.
Elastic potential energy
Ideal spring energy equals one-half the spring constant multiplied by displacement squared.
Because \(x\) is squared, equal-magnitude compression and extension store the same energy in an ideal linear spring. Doubling displacement quadruples the stored energy when \(k\) is unchanged.
General gravitational potential energy
For two masses separated by center-to-center distance \(r\), Newtonian gravitational potential energy is negative when zero potential is defined at infinite separation.
The calculator uses \(G=6.67430\times10^{-11}\ \mathrm m^3\,\mathrm{kg}^{-1}\,\mathrm s^{-2}\).
- \(PE,U,\Delta U\)
- Potential energy or change in potential energy, with SI unit joule (J).
- \(m\)
- Object mass. In SI calculations, mass is expressed in kilograms.
- \(g\)
- Gravitational acceleration, expressed in \(\mathrm{m/s^2}\) in SI form.
- \(h,h_1,h_2\)
- Vertical position relative to a chosen datum; initial and final heights must use the same datum.
- \(k\)
- Linear spring constant, with SI unit \(\mathrm{N/m}\).
- \(x\)
- Spring extension or compression magnitude measured from the undeformed position.
- \(G\)
- Newtonian constant of gravitation.
- \(M,r\)
- Central-body mass and center-to-center separation distance used in the general gravitational model.
Gravitational potential energy vs gravitational potential
Gravitational potential energy depends on the mass of the object. Gravitational potential is potential energy per unit mass:
Near a surface with constant \(g\), \(\phi=gh\). In the general gravitational field, \(\phi=-GM/r\). This distinction matters when comparing a gravitational field independently of the test mass placed in it.
| Model | Equation | Best use | Key behavior |
|---|---|---|---|
| Near-surface gravitational | \(U=mgh\) | Ordinary elevation changes with nearly constant \(g\) | Linear with \(m\), \(g\), and \(h\) |
| Change in gravitational PE | \(\Delta U=mg\Delta h\) | Comparing two elevations | Sign follows the direction of \(\Delta h\) |
| Elastic | \(U_s=\frac12kx^2\) | Ideal linear springs | Quadratic with displacement |
| General gravitational | \(U=-GMm/r\) | Large gravitational distances or orbital-scale problems | Magnitude decreases as \(r\) increases |
Worked Potential Energy Example
A 10 kg object is raised to a height of 5 m relative to the selected zero-height reference. Using Earth standard gravity, find its gravitational potential energy.
Substitute the values
Calculate the energy
Result
About 490 J
The unrounded calculation is \(490.3325\ \mathrm{J}\). Reporting about \(490\ \mathrm{J}\) avoids implying more measurement precision than the simple 10 kg and 5 m example provides. The same unrounded energy is approximately \(0.49033\ \mathrm{kJ}\) or \(0.13620\ \mathrm{Wh}\).
How to Interpret the Result
A potential-energy result is an energy quantity, not a force or power rating. Its meaning depends on the selected reference state and on the model used to calculate it.
Reference height controls the sign
For \(U=mgh\), choosing \(h=0\) defines the zero of potential energy. A negative height therefore gives negative \(U\) relative to that datum; it does not mean the mathematics failed.
Mass, height, and gravity are linear
Holding the other two quantities constant, increasing mass, height, or \(g\) by 10% changes \(U=mgh\) by 10%. Doubling height doubles gravitational potential energy relative to the same zero level.
Spring displacement is quadratic
Holding \(k\) constant, doubling spring displacement makes \(U_s=\tfrac12kx^2\) four times larger. Displacement-unit errors therefore have an amplified effect on spring-energy results.
| Input change | Held constant | Effect on result |
|---|---|---|
| Mass ×2 | \(g,h\) | \(U\) ×2 |
| Height ×2 | \(m,g\) | \(U\) ×2 |
| Gravity ×2 | \(m,h\) | \(U\) ×2 |
| Spring displacement ×2 | \(k\) | \(U_s\) ×4 |
| General-gravity distance ×2 | \(M,m\) | \(|U|\) ÷2 |
Why change in potential energy is often the better comparison
In many mechanics problems, the useful quantity is the change between two states rather than the arbitrary absolute zero level. If an object rises, \(\Delta U\) is positive; if it descends, \(\Delta U\) is negative.
Connection to kinetic and mechanical energy
When only conservative forces act, mechanical energy can transfer between kinetic and potential forms while the total remains constant:
For an ideal fall from rest with negligible losses, gravitational potential energy can convert to kinetic energy:
Real systems may also transfer energy to heat, sound, drag, damping, or deformation.
Energy is not power
A result of 50 kJ tells you how much energy is associated with the state change, not how quickly that energy must be delivered. Power also requires a time interval: \(P=\Delta E/\Delta t\). For lifting equipment, an idealized lifting-power check is \(P_{\mathrm{ideal}}=mgh/t\), before efficiency and other losses are considered.
Units, Input Checks, and Common Mistakes
The calculator converts supported units internally, but the physical meaning of each input still matters. The most common mistakes are using the wrong datum, mixing up mass and force, entering path length instead of vertical height, or using the wrong distance for the selected model.
Mass and weight are not interchangeable
Mass is the quantity \(m\) in \(PE=mgh\). Weight is a force:
If mass is known, use \(PE=mgh\). If weight \(W\) is already known as a force, then near a constant gravitational field the potential-energy change can be written as \(PE=Wh\). Do not multiply a force value by \(g\) a second time.
- Use vertical height, not path length. A 20 m ramp that rises only 5 m contributes 5 m of elevation change to \(mgh\).
- Keep the same reference level for \(h_1\) and \(h_2\). Mixing two different datums makes \(\Delta U=mg(h_2-h_1)\) invalid.
- Use center-to-center distance in the general gravity mode. The \(r\) in \(-GMm/r\) is separation between mass centers, not simply altitude above a surface.
- Use spring displacement from the undeformed position. The ideal spring equation uses extension or compression \(x\), not total spring length.
- Remember the square in spring energy. Using \(x\) instead of \(x^2\) can dramatically understate or overstate the result.
- Use a spring constant valid for the working range. A single constant \(k\) assumes force remains approximately proportional to displacement.
- Do not treat a planet gravity preset as an exact local measurement. Surface gravity varies with location, elevation, rotation, and local conditions. Presets are convenient reference values.
| Quantity | Supported units | Fast check |
|---|---|---|
| Mass | kg, g, lb, oz; general-gravity central mass also supports Earth mass and solar mass | Mass must be positive |
| Length / height | m, cm, mm, km, ft, in, yd, mi; available choices vary by field | Height may be negative relative to the chosen datum |
| Gravity | m/s², ft/s² | Must be positive in the near-surface model |
| Energy | J, kJ, MJ, Wh, kWh, ft·lbf, BTU | Gravitational energy may be negative by reference convention |
| Spring constant | N/m, N/mm, lbf/in | Use a value valid for the spring’s linear range |
When Each Potential Energy Model Applies
Choosing the right equation matters more than adding decimal places. The calculator includes four modes because the simple \(mgh\) relationship is not the right model for every potential-energy problem.
Use \(U=mgh\) near a surface
Use the constant-\(g\) model for ordinary lifting, lowering, classroom mechanics, buildings, hills, and similar problems where gravitational acceleration is effectively constant across the height change.
Use \(\Delta U\) for two elevations
When the question is about moving from one height to another, \(\Delta U=mg(h_2-h_1)\) avoids unnecessary concern about the arbitrary absolute zero level, provided both heights use the same datum.
Use the spring model only for linear behavior
\(U_s=\tfrac12kx^2\) assumes an ideal linear spring with a constant \(k\). It does not capture yielding, large geometric nonlinearity, hysteresis, friction, or damping losses.
Use \(-GMm/r\) when gravity varies materially
For satellites, orbital-scale distances, and other problems where gravitational acceleration changes materially with distance, use the general gravitational model with center-to-center separation.
When should you not use \(PE=mgh\)?
Do not treat \(mgh\) as a universal gravitational-energy equation. Its constant-\(g\) approximation becomes progressively less representative as the height interval becomes large enough for gravity to change materially. There is no single universal cutoff; the acceptable approximation depends on the required accuracy and the scale of the problem.
Potential Energy Calculator FAQ
These answers cover the questions that most often change how a potential-energy result should be entered or interpreted.
What is the formula for gravitational potential energy?
Near a surface where gravity can be treated as constant, use \(PE=U=mgh\). Here \(m\) is mass, \(g\) is gravitational acceleration, and \(h\) is vertical height relative to the selected zero-potential reference level.
Is height in \(mgh\) vertical height or distance traveled?
Use vertical height relative to the reference level, not the length of the path traveled. For example, an object moving 20 m along a ramp but rising only 5 m has a gravitational height change of 5 m.
Can gravitational potential energy be negative?
Yes. In the near-surface \(mgh\) model, the zero level is chosen by the analyst. If the object is below that datum, \(h\) and therefore \(U\) can be negative. In the general \(-GMm/r\) convention, gravitational potential energy is negative when zero is defined at infinite separation.
What units are used for potential energy?
The SI unit is the joule (J), equivalent to \(\mathrm{kg\,m^2/s^2}\) or \(\mathrm{N\,m}\). The calculator can also report kJ, MJ, Wh, kWh, ft·lbf, and BTU.
What happens to potential energy if height doubles?
For \(U=mgh\), if mass and gravity stay constant and height is measured from the same reference level, doubling \(h\) doubles \(U\). This linear relationship does not apply to spring displacement, because elastic energy varies with \(x^2\).
What is the difference between potential and kinetic energy?
Potential energy is associated with position or configuration; kinetic energy is associated with motion. In an ideal conservative system, energy can transfer between the two while total mechanical energy remains constant.
When should I use \(-GMm/r\) instead of \(mgh\)?
Use the general gravitational expression when center-to-center distance changes enough that treating \(g\) as constant is no longer appropriate, such as many satellite or orbital-scale problems. Use \(mgh\) for ordinary near-surface height changes where \(g\) is effectively constant.
How do I calculate elastic potential energy?
For an ideal linear spring, use \(U_s=\tfrac12kx^2\), where \(k\) is the spring constant and \(x\) is extension or compression from the undeformed position. The calculator can also rearrange this relationship to solve for \(k\) or displacement magnitude.