Important Note : This Potential Energy Calculator provides educational estimates for gravitational potential energy and elastic spring potential energy. Gravitational mode uses PE = mgh and assumes a nearly constant gravitational acceleration over the selected height range, so it is best for near-surface height problems rather than large-scale orbital or space calculations. Elastic mode uses PE = ½kx² and assumes an ideal spring that follows Hooke’s law within its elastic range. The result does not include friction, air resistance, heat loss, damping, spring damage, material failure, non-linear spring behavior, or full system energy transfer. Height depends on the reference level you choose, and elastic displacement is measured from the spring’s unstretched equilibrium position. Use this calculator for homework, learning, and general comparisons, and use a full physics or engineering method for safety-critical springs, lifts, falls, machinery, structures, or large gravitational-field problems.

Use this Potential Energy Calculator to find gravitational or elastic potential energy using standard introductory physics formulas. It works for height-based energy using mass, gravity, and height, or spring energy using spring constant and displacement. The calculator then converts the result into joules, kilojoules, watt-hours, kilowatt-hours, and foot-pound force.

Reviewed by: AjaxCalculators Editorial Team
Last updated: May 1, 2026
Method source: Standard gravitational potential energy formula (PE = mgh) and elastic potential energy formula (PE = ½kx²), with common energy-unit conversions from SI and NIST conversion relationships
Editorial standards: AjaxCalculators Editorial Policy

What This Potential Energy Calculator Calculates

This calculator estimates potential energy, which is stored energy related to position, height, or deformation. It supports two common introductory-physics modes:

• Gravitational potential energy: energy stored because an object has height in a gravitational field
• Elastic potential energy: energy stored in a stretched or compressed spring

Depending on the mode selected, the calculator uses the appropriate formula and returns the result in common energy units:

• joules (J)
• kilojoules (kJ)
• watt-hours (Wh)
• kilowatt-hours (kWh)
• foot-pound force (ft·lbf)

This makes the calculator useful for physics homework, spring problems, height-energy comparisons, mechanics examples, and energy-unit conversion.

What Potential Energy Means

Potential energy is stored energy that depends on position or configuration. An object lifted above a reference level can store gravitational potential energy. A spring that is stretched or compressed can store elastic potential energy.

Potential energy is not always an energy you can “see” directly. It becomes easier to understand when it changes into another form, such as kinetic energy. For example, a raised object can fall and gain speed, while a compressed spring can push an object and make it move.

How the Potential Energy Calculator Works

The calculator uses one of two formulas depending on the selected mode.

1) Gravitational Potential Energy

The standard near-surface gravitational potential energy formula is:

PE = mgh

In this formula:

• PE = gravitational potential energy
• m = mass
• g = gravitational acceleration
• h = height relative to a chosen reference level

This formula is commonly used near the surface of Earth or another planet when gravitational acceleration can be treated as approximately constant over the height range.

2) Elastic Potential Energy

The standard ideal-spring potential energy formula is:

PE = ½kx²

In this formula:

• PE = elastic potential energy
• k = spring constant
• x = displacement from the spring’s unstretched equilibrium position

This formula assumes the spring behaves like an ideal Hooke’s-law spring within the range being used.

Formula Summary

Energy Type	Formula	Known Values Needed
Gravitational potential energy	PE = mgh	Mass, gravity, and height
Elastic potential energy	PE = ½kx²	Spring constant and displacement
Mass from gravitational PE	m = PE / gh	Potential energy, gravity, and height
Height from gravitational PE	h = PE / mg	Potential energy, mass, and gravity
Spring constant from elastic PE	k = 2PE / x²	Potential energy and displacement
Spring displacement from elastic PE	x = √(2PE / k)	Potential energy and spring constant

Gravitational Potential Energy: Key Ideas

Gravitational potential energy increases when mass, gravity, or height increases. A heavier object, a larger gravitational field, or a greater height all increase the stored gravitational energy.

Variable	Effect on Gravitational PE	Example
Mass increases	PE increases directly	A 20 kg object has twice the PE of a 10 kg object at the same height and gravity
Gravity increases	PE increases directly	The same object at the same height has different PE on different planets
Height increases	PE increases directly	Lifting an object twice as high gives twice the PE, if mass and gravity stay the same

Height is measured relative to a chosen zero-reference level. If you choose the floor as zero, a shelf above the floor has positive height. If you choose a higher reference point, the same object may have a smaller or even negative height value relative to that reference.

Elastic Potential Energy: Key Ideas

Elastic potential energy increases when the spring is stiffer or when it is stretched or compressed farther from its equilibrium position.

Variable	Effect on Elastic PE	Example
Spring constant increases	PE increases directly	A stiffer spring stores more energy for the same displacement
Displacement increases	PE increases with displacement squared	Doubling displacement makes elastic PE four times larger
Displacement is negative	PE stays non-negative because x² is used	Compression and stretch can store energy when measured from equilibrium

Because displacement is squared, elastic potential energy depends on how far the spring is displaced, not on whether the displacement is written as positive or negative.

Energy Unit Conversion Summary

The calculator converts joules into other common energy units. The NIST Guide to the SI conversion factors lists relationships such as watt-hour, kilowatt-hour, and foot-pound force conversions.

Unit	Relationship	Common Use
joule (J)	Base SI energy unit	Physics, mechanics, science calculations
kilojoule (kJ)	1 kJ = 1,000 J	Larger energy values
watt-hour (Wh)	1 Wh = 3,600 J	Electrical-energy comparison
kilowatt-hour (kWh)	1 kWh = 3,600,000 J	Large energy comparisons and electricity contexts
foot-pound force (ft·lbf)	1 ft·lbf ≈ 1.355818 J	Imperial mechanical work and energy

Gravity Values and Reference Level

In gravitational mode, the value of g matters. Near Earth’s surface, a commonly used standard gravity value is:

g = 9.80665 m/s²

Some calculators may let you use a gravity preset or enter a custom gravity value. This is useful because gravitational acceleration is different on the Moon, Mars, and other bodies.

Gravity Setting	Approximate Use	Important Note
Earth standard gravity	9.80665 m/s²	Common value for near-Earth calculations
Custom gravity	User-entered value	Useful for other planets or textbook-specific values
Local gravity	May vary slightly by location	Useful only when higher precision is needed

The height value also depends on the zero-reference level you choose. Potential energy is most useful when comparing differences between heights, not just one absolute number.

Worked Example: Gravitational Potential Energy

Suppose a 10 kg object is lifted to a height of 5 m on Earth.

Step 1: Identify the known values
Mass, m = 10 kg
Gravity, g = 9.80665 m/s²
Height, h = 5 m

Step 2: Use the gravitational PE formula
PE = mgh

Step 3: Substitute the values
PE = 10 × 9.80665 × 5

Step 4: Calculate
PE = 490.33 J

Result: The object has about 490.33 joules of gravitational potential energy relative to the chosen height reference.

Worked Example: Convert 490.33 J to Other Units

Now convert 490.33 J into kJ, Wh, kWh, and ft·lbf.

Output Unit	Calculation	Result
kJ	490.33 ÷ 1,000	0.49033 kJ
Wh	490.33 ÷ 3,600	0.1362 Wh
kWh	490.33 ÷ 3,600,000	0.000136 kWh
ft·lbf	490.33 ÷ 1.355818	361.65 ft·lbf

Worked Example: Elastic Potential Energy

Suppose a spring has a spring constant of 200 N/m and is stretched by 0.10 m.

Step 1: Identify the known values
Spring constant, k = 200 N/m
Displacement, x = 0.10 m

Step 2: Use the elastic PE formula
PE = ½kx²

Step 3: Square the displacement
x² = 0.10² = 0.01

Step 4: Substitute the values
PE = 0.5 × 200 × 0.01

Step 5: Calculate
PE = 1 J

Result: The spring stores 1 joule of elastic potential energy.

Worked Example: Why Spring Displacement Has a Strong Effect

Elastic potential energy depends on displacement squared. This means doubling the stretch or compression makes the stored energy four times larger, assuming the spring constant stays the same.

Suppose a spring has k = 200 N/m.

Displacement	Formula	Elastic PE
0.10 m	0.5 × 200 × 0.10²	1 J
0.20 m	0.5 × 200 × 0.20²	4 J
0.30 m	0.5 × 200 × 0.30²	9 J

The displacement tripled from 0.10 m to 0.30 m, but the elastic potential energy increased nine times because displacement is squared.

Worked Example: Compare Gravitational PE on Different Gravity Values

Suppose a 5 kg object is lifted 2 m. Compare its gravitational potential energy using two different gravity values.

Gravity Setting	Formula	Potential Energy
Earth: 9.80665 m/s²	5 × 9.80665 × 2	98.07 J
Moon example: 1.62 m/s²	5 × 1.62 × 2	16.2 J

The same mass and height store less gravitational potential energy on the Moon because the gravitational acceleration is lower.

How to Use This Potential Energy Calculator

1. Select Gravitational PE or Elastic PE.
2. For gravitational mode, enter mass, gravity, and height.
3. For elastic mode, enter spring constant and displacement.
4. Select the correct input units.
5. If available, choose a gravity preset or enter a custom gravity value.
6. Click Calculate if the tool requires it.
7. Review the energy result in J, kJ, Wh, kWh, and ft·lbf.
8. Check whether your result should be interpreted relative to a height reference or spring equilibrium position.

How to Interpret the Result

The result tells you how much energy is stored because of height or spring deformation.

Result Type	Meaning	Example Interpretation
Positive gravitational PE	Object is above the chosen zero-reference level	A box on a shelf relative to the floor
Negative gravitational PE	Object is below the chosen zero-reference level	A basement level if ground floor is chosen as zero
Zero gravitational PE	Object is at the chosen reference level	An object on the floor if floor is zero
Positive elastic PE	Spring is stretched or compressed from equilibrium	A compressed spring ready to release
Zero elastic PE	Spring is at its unstretched equilibrium position	No deformation from rest length

Potential Energy and Kinetic Energy

Potential energy can transform into kinetic energy when a system moves. For example, a raised object can fall and gain speed, while a stretched spring can pull an object into motion.

Starting Situation	Stored Energy	Possible Energy Change
Object lifted above a reference level	Gravitational potential energy	Can become kinetic energy as the object falls
Spring stretched or compressed	Elastic potential energy	Can become kinetic energy as the spring moves back toward equilibrium
System with friction or damping	Stored mechanical energy	Some energy may become heat, sound, or internal energy

In ideal textbook examples, mechanical energy may be conserved. In real systems, friction, air resistance, damping, deformation, and heat can reduce the amount of potential energy converted into useful motion.

Gravitational PE vs Elastic PE

Feature	Gravitational PE	Elastic PE
Main formula	PE = mgh	PE = ½kx²
Depends on	Mass, gravity, and height	Spring constant and displacement
Reference point	Chosen height zero level	Unstretched equilibrium position
Can be negative?	Yes, if height is below the chosen reference	No, because displacement is squared
Best for	Height and lifting problems	Spring stretch/compression problems

When This Calculator Is Useful

This calculator is useful for simple potential-energy problems where the required inputs are known and the assumptions match the situation.

• Physics homework and exam practice
• Finding gravitational energy from mass, gravity, and height
• Finding spring energy from spring constant and displacement
• Comparing energy stored at different heights
• Comparing energy stored in different springs
• Understanding energy conversion between potential and kinetic energy
• Converting energy values between J, kJ, Wh, kWh, and ft·lbf
• Learning how reference levels affect gravitational PE

When You May Need More Than This Calculator

A simple potential energy calculator may not be enough when the system is non-ideal, safety-critical, or outside the range of introductory formulas.

Use a more detailed method when working with:

• large gravitational distances or orbital mechanics
• gravity that changes significantly with height
• non-linear springs
• springs beyond their elastic limit
• damped or friction-heavy systems
• impact, fall safety, or injury risk
• machinery, lifting systems, or spring-loaded devices
• structural or mechanical design
• energy losses from heat, sound, or deformation
• engineering safety factors and regulatory compliance

Common Mistakes to Avoid

• Using the wrong formula: use PE = mgh for gravitational height problems and PE = ½kx² for spring problems.
• Forgetting the reference level: gravitational potential energy depends on the chosen zero-height reference.
• Using weight instead of mass: gravitational PE uses mass and gravity, not weight entered as if it were mass.
• Mixing height and displacement: height belongs to gravitational PE, while spring displacement belongs to elastic PE.
• Forgetting to square spring displacement: elastic PE depends on x².
• Assuming negative spring displacement gives negative energy: elastic PE remains non-negative because displacement is squared.
• Using the ideal spring formula outside the elastic range: damaged, overstretched, or non-linear springs may not follow PE = ½kx².
• Ignoring energy losses: friction, damping, heat, and deformation can reduce useful energy transfer.
• Using mgh for very large height changes: large-scale gravitational problems may require a more complete gravitational model.

Important Assumptions and Limitations

• This calculator uses standard introductory-physics formulas.
• Gravitational mode uses PE = mgh.
• Elastic mode uses PE = ½kx².
• The gravitational formula assumes gravity is approximately constant over the selected height range.
• The height value is measured relative to a chosen reference level.
• The elastic formula assumes an ideal Hooke’s-law spring.
• The spring displacement is measured from the unstretched equilibrium position.
• The calculator does not model friction, damping, air resistance, heat, deformation, or energy loss.
• The calculator does not determine whether a spring is safe, damaged, or beyond its elastic limit.
• For engineering, safety, machinery, structural, or large-scale gravitational problems, use appropriate professional methods.

Practical Uses of a Potential Energy Calculator

• Calculate energy stored by lifting an object
• Estimate energy stored in a compressed or stretched spring
• Compare gravitational energy at different heights
• Compare spring energy for different spring constants
• Convert energy values into J, kJ, Wh, kWh, and ft·lbf
• Understand conservation of mechanical energy examples
• Support physics homework and classroom demonstrations
• Prepare for related topics such as kinetic energy, work, power, and oscillations

References

1. OpenStax College Physics 2e: Gravitational Potential Energy
2. OpenStax University Physics Volume 1: Potential Energy of a System
3. OpenStax College Physics 2e: Energy and the Simple Harmonic Oscillator
4. OpenStax Physics: Mechanical Energy and Conservation of Energy
5. NIST Guide to the SI: Energy Conversion Factors

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• Kinetic Energy Calculator
• Work Calculator
• Work and Power Calculator
• Velocity Calculator
• Acceleration Calculator

Frequently Asked Questions

What is the formula for gravitational potential energy?

The gravitational potential energy formula is PE = mgh. This means potential energy equals mass times gravitational acceleration times height.

What is the formula for elastic potential energy?

The elastic potential energy formula is PE = ½kx². This means elastic energy equals one-half times the spring constant times displacement squared.

What does height mean in gravitational potential energy?

Height is measured relative to a chosen zero-reference level. For example, you might choose the floor, ground, table surface, or another point as zero height.

Can gravitational potential energy be negative?

Yes. Gravitational potential energy can be negative if the selected height is below your chosen zero-reference level. The sign depends on the reference level, not on whether the formula is broken.

Can elastic potential energy be negative?

No. Elastic potential energy from PE = ½kx² is zero or positive because displacement is squared. Stretching and compressing a spring can both store energy.

Why is spring displacement squared?

Spring force changes with displacement, so the stored energy increases with the square of displacement. Doubling the stretch or compression makes elastic potential energy four times larger for the same spring constant.

What units does potential energy use?

In SI units, potential energy is measured in joules (J). The calculator can also convert the result into kJ, Wh, kWh, and ft·lbf.

Does this calculator include friction or energy loss?

No. This calculator estimates stored potential energy using ideal formulas. It does not include friction, air resistance, damping, heat loss, deformation, or other real-world losses.

Does PE = mgh work everywhere?

PE = mgh works best near a planet’s surface when gravity is approximately constant across the height range. For large distances, orbital problems, or changing gravitational fields, a more complete gravitational model is needed.

Does PE = ½kx² work for every spring?

No. PE = ½kx² assumes an ideal spring that follows Hooke’s law. Real springs may become non-linear, damaged, or unsafe if stretched or compressed beyond their elastic range.

Disclaimer: This Potential Energy Calculator provides educational estimates for gravitational potential energy and elastic spring potential energy. Gravitational mode uses PE = mgh and assumes a nearly constant gravitational acceleration over the selected height range, so it is best for near-surface height problems rather than large-scale orbital or space calculations. Elastic mode uses PE = ½kx² and assumes an ideal spring that follows Hooke’s law within its elastic range. The result does not include friction, air resistance, heat loss, damping, spring damage, material failure, non-linear spring behavior, or full system energy transfer. Height depends on the reference level you choose, and elastic displacement is measured from the spring’s unstretched equilibrium position. Use this calculator for homework, learning, and general comparisons, and use a full physics or engineering method for safety-critical springs, lifts, falls, machinery, structures, or large gravitational-field problems.