AtlasCalcsUtility Calculation Suite
2026-08-21 7 min

Potential Energy Calculator

Compute gravitational potential energy from mass, height, and gravity. Converts results to joules, kJ, kcal, BTU, and eV with step-by-step breakdown and equivalent drop height.

Calculator

Mass(kg) Help
kg

Mass of the object in kilograms

Height(m) Help
m

Vertical height above reference level in meters

Gravity(m/s²) Help
m/s²

Gravitational acceleration (Earth ≈ 9.81, Moon ≈ 1.62, Mars ≈ 3.72)

Decimal Precision
Results update live as you type

Results

Normal

Potential Energy

490.50J

Gravitational potential energy in joules

Potential Energy

0.49kJ

Gravitational potential energy in kilojoules

Potential Energy

0.12kcal

Gravitational potential energy in kilocalories

Potential Energy

0.46BTU

Gravitational potential energy in British thermal units

Potential Energy

3.06e+21eV

Gravitational potential energy in electron-volts

Weight (mg)

98.10N

Gravitational force acting on the mass

Visualizations

Results Overview

490.5JPotential Energy
0.5JPotential Energy
0.1JPotential Energy
0.5JPotential Energy
3,060,000,000,000,000,000,000JPotential Energy
98.1JWeight

Formula

Gravitational potential energy is the energy stored in an object due to its position above a reference level in a gravitational field. The formula PE = mgh multiplies mass by gravitational acceleration and height. The reference level (usually ground) is where h = 0 and PE = 0. The work done to lift the object equals the potential energy gained.

PE = mgh

Alternative Form

PE = Wh (where W = mg is weight)

Variables

SymbolNameUnitDescription
PEPotential EnergyJStored gravitational energy, measured in joules
mMasskgMass of the object being lifted
gGravitational Accelerationm/s²Acceleration due to gravity (Earth ≈ 9.81 m/s²)
hHeightmVertical height above the reference level
WWeightNGravitational force: W = m × g

Explanation

The gravitational potential energy formula PE = mgh is derived from the work done against gravity. To lift a mass m through height h, a force equal to its weight (mg) must be applied over distance h. Work = Force × Distance = mg × h = mgh. By the work-energy theorem, this work is stored as potential energy. The formula assumes a uniform gravitational field (valid near Earth's surface). For large height changes, use PE = mgh only as an approximation.

Derivation

Work done against gravity: W = F × d. The force needed to lift an object at constant velocity equals its weight: F = mg. The displacement is the height h. Therefore W = mg × h = mgh. By conservation of energy, this work is stored as gravitational potential energy: PE = mgh. The SI unit is the joule (1 J = 1 N·m = 1 kg·m²/s²).

Step-by-Step Calculation

Follow each step to understand how the result is calculated from your inputs.

Step 1

Identify the given values from the problem.

Mass (m):10.00 kg
Height (h):5.00 m
Gravity (g):9.81 m/s²
Result: m = 10.00 kg, h = 5.00 m, g = 9.81 m/s²
Step 2

Calculate the weight (gravitational force) of the object.

W = m × g
m:10.00 kg
g:9.81 m/s²
Intermediate: W = 10.00 × 9.81 = 98.10 N
Result: W = 98.10 N
Step 3

Multiply weight by height to get potential energy in joules.

PE = m × g × h
m:10.00 kg
g:9.81 m/s²
h:5.00 m
Intermediate: PE = 10.00 × 9.81 × 5.00 = 490.50 J
Result: PE = 490.50 J
Step 4

Convert potential energy to other energy units.

kJ = J/1000, kcal = J/4184, BTU = J/1055.06, eV = J/1.602×10⁻¹⁹
PE (J):490.50 J
Intermediate: 490.50 J → 0.49 kJ → 0.12 kcal → 0.46 BTU → 3.06e+21 eV
Result: 0.49 kJ = 0.12 kcal = 0.46 BTU = 3.06e+21 eV
Step 5

Compute the equivalent drop height for reference.

h_equivalent = PE / (mg)
PE:490.50 J
m × g:98.10 N
Intermediate: h = 490.50 / 98.10 = 5.00 m
Result: Equivalent drop height = 5.00 m
Final Answer
490.50

Worked Examples

See how the calculator works with different scenarios.

Scenario

A reservoir holds 1 million m³ of water at an average height of 50 m above the dam's turbine. Water density is 1000 kg/m³. Calculate the total gravitational potential energy stored.

Inputs

Mass (m):1 × 10⁹ kg (1 million m³ × 1000 kg/m³)
Height (h):50 m
Gravity (g):9.81 m/s²

Calculation

PE = mgh = 1 × 10⁹ × 9.81 × 50 = 4.905 × 10¹¹ J

Result

4.905 × 10¹¹ J = 490,500 kJ = 490.5 GJ ≈ 136,250 kWh

Interpretation

The reservoir stores about 490 GJ of potential energy — enough to power approximately 13,600 average US homes for one day (at 10 kW each). Hydroelectric dams convert this stored PE into electricity by channeling water through turbines. The actual electrical output is less due to turbine efficiency (~85-90%) and friction losses.

Understanding Your Result

Your Result: Potential Energy: 490.50 J

Typical range: Lifting a 1 kg book 1 m: ~10 J. Person (75 kg) on a 3 m diving board: ~2200 J. Water in a small pond (1000 m³, 5 m head): ~49 MJ. Large hydroelectric dam (1 km³, 100 m head): ~10¹⁵ J (1 PJ).

What This Means

The result tells you how much energy is stored in an object due to its height in a gravitational field. A result of 490 J means you would need to do 490 joules of work to lift the object to that height — and that energy is stored, ready to be converted to kinetic energy if the object falls. The higher the mass, gravity, or height, the more energy is stored.

Why It Matters

Gravitational potential energy is fundamental to hydroelectric power generation, roller coaster design, structural engineering (energy of falling objects), sports science, and understanding energy conservation. Engineers use PE calculations to size dams, design safety barriers, calculate impact energies, and analyze any system where objects change elevation.

Professional Interpretation

Potential energy quantifies the work available from gravity. In dam design, PE determines power output capacity. In construction, falling-object PE determines guard rail requirements. In sports, PE conversion to KE determines ball trajectory. The formula assumes uniform gravity — for satellite orbits or very large height differences, the integral form PE = −GMm/r must be used.

Recommendations

  • •Always use consistent SI units: mass in kg, height in m, gravity in m/s² for results in joules
  • •Remember PE is relative to your chosen reference level — moving the reference changes the value
  • •For objects at rest, total mechanical energy = PE (KE = 0); for falling objects, PE converts to KE
  • •At heights above a few km, Earth's gravity decreases — use g(h) = g₀ × (R/(R+h))² for accuracy
  • •For energy recovery systems (regenerative braking, pumped hydro), PE represents the maximum recoverable energy

Educational Content

Introduction

Gravitational potential energy is one of the most intuitive forms of energy in physics. It represents the energy stored in an object due to its position in a gravitational field — specifically, its height above a reference level. The concept is central to understanding energy conservation, power generation, and countless engineering applications. The formula PE = mgh was formalized in the 19th century, though the concept of "energy of position" dates back centuries earlier.

Concept Explanation

Gravitational potential energy is defined as PE = mgh, where m is mass (kg), g is gravitational acceleration (m/s²), and h is height above the reference level (m). The SI unit is the joule (1 J = 1 N·m = 1 kg·m²/s²). Key properties: (1) PE depends on the choice of reference level — only differences in PE are physically meaningful. (2) PE increases linearly with m, g, and h. (3) PE is a scalar, not a vector. (4) In a closed system, PE converts to KE and vice versa (conservation of mechanical energy). (5) The work done against gravity equals the PE gained: W = mgh.

Applications

▸Hydroelectric power generation — water PE drives turbines
▸Roller coaster and amusement ride design — PE ↔ KE conversion
▸Structural engineering — energy of falling objects for safety barriers
▸Pumped-storage hydroelectricity — storing energy by lifting water
▸Sports science — projectile motion, diving, skiing
▸Spacecraft orbital mechanics — PE in gravitational fields
▸Elevator and crane engineering — PE gained by lifted loads
▸Seismic energy analysis — PE of displaced ground masses

Benefits

  • •Directly quantifies the energy available from gravity-driven systems
  • •Simple linear formula — easy to calculate and intuitively understood
  • •Conservation of PE + KE enables analysis without tracking forces
  • •Universal applicability — from lifting a book to planetary orbits
  • •Foundation for understanding energy storage and power generation

Limitations

  • •Assumes constant gravity — inaccurate for large height differences (>10 km)
  • •Reference level is arbitrary — only PE differences matter
  • •Does not account for air resistance or other energy losses
  • •Classical formula — breaks down for relativistic speeds or extreme gravitational fields

Best Practices

  • •Always define your reference level (usually ground, h = 0) before calculating
  • •Convert all units to SI before using the formula: kg, m, m/s²
  • •Use g = 9.81 m/s² for Earth, 1.62 m/s² for Moon, 3.72 m/s² for Mars
  • •For falling objects, PE lost = KE gained: mgh = ½mv², so v = √(2gh)
  • •In multi-object systems, calculate PE for each mass separately and sum
  • •For rotating systems with height changes, consider both PE and rotational KE

Common Misconceptions

Myth: Potential energy only exists when an object is moving.
Truth: PE is stored energy — it exists even when the object is stationary. A book on a shelf has PE even though it is not moving. It only becomes KE when the object falls.
Myth: PE depends on the path taken to reach the height.
Truth: PE is path-independent — only the vertical height matters. Lifting a box straight up 5 m gives the same PE as carrying it up a ramp to 5 m height.
Myth: Heavier objects fall faster because they have more PE.
Truth: All objects fall at the same rate in a vacuum regardless of mass. PE = mgh is larger for heavier objects, but KE = ½mv² gained is also proportionally larger, so velocity is the same: v = √(2gh).
Myth: PE is always positive.
Truth: PE can be negative if the object is below the reference level. The formula PE = mgh gives negative values when h < 0. Only differences in PE are physically meaningful.
Myth: PE = mgh works everywhere in the universe.
Truth: The formula assumes uniform gravity. For large distances (satellites, interplanetary travel), use PE = −GMm/r. Near Earth's surface, mgh is an excellent approximation.

Practical Tips

  • •Quick mental estimate: lifting 1 kg by 1 m stores about 10 J of PE (using g ≈ 10)
  • •A 70 kg person on a 3 m diving board has about 2060 J of PE — roughly equivalent to a 1-watt bulb running for 34 minutes
  • •Pumped hydro storage: lifting 1 m³ of water 100 m stores about 272 Wh of energy
  • •The KE gained by a falling object equals the PE lost: ½mv² = mgh, so v = √(2gh)
  • •For quick conversion: 1 kWh = 3.6 MJ — useful for comparing PE to electricity bills
  • •A 50 kg object on a 10 m roof has about 4.9 kJ of PE — equivalent to a small food snack (1.2 kcal)
  • •Moon gravity is about 1/6 of Earth's — the same object at the same height has 1/6 the PE on the Moon

Common Mistakes

Using weight (in newtons) instead of mass (in kg)
The formula uses mass in kg, not weight in N. If you have weight, divide by g to get mass: m = W/g. Using weight directly gives results g² times too large.
Using centimeters or feet instead of meters
Convert to meters before using the formula. 100 cm = 1 m. 1 ft = 0.3048 m. Using cm gives results 10,000× too small; using ft gives results ~10× too small.
Forgetting that PE is relative to the reference level
PE depends on where you set h = 0. A ball on a table has positive PE relative to the floor but zero PE relative to the table. Always define your reference.
Using 9.81 m/s² for non-Earth gravity
Different celestial bodies have different gravity. Moon: 1.62 m/s². Mars: 3.72 m/s². Jupiter: 24.79 m/s². Check the local gravitational acceleration.
Assuming PE is conserved in all real-world scenarios
PE is conserved only in ideal systems without friction, air resistance, or other losses. In reality, some PE converts to heat, sound, and deformation energy during motion.
Confusing PE with work
PE = mgh is the energy stored; work = force × distance is the process of storing it. They are equal in magnitude when lifting at constant velocity, but they are different concepts.

Assumptions

This calculator makes the following assumptions to provide estimates:

Uniform Gravitational Field: The formula assumes gravity is constant over the height range. This is valid near Earth's surface (h << 6,371 km, Earth's radius). For heights above ~10 km, gravity decreases slightly and PE = mgh becomes an approximation.
Constant Mass: The object's mass does not change during the process. This excludes scenarios like rocket propulsion where mass decreases as fuel is consumed.
No Air Resistance: The calculation gives the total PE stored. It does not account for energy lost to air resistance during lifting or falling.
Rigid Body: The object is treated as a rigid point mass. Deformation energy and internal strain energy are not included in the PE calculation.

Limitations

Be aware of these limitations when using this calculator:

Non-Relativistic Only: The classical PE = mgh formula is invalid for objects moving near the speed of light or in extremely strong gravitational fields (near black holes, neutron stars).
No Rotational Component: This calculator computes translational gravitational PE only. Rotational effects (e.g., a spinning object changing height) require additional analysis.
Reference Level Dependency: The absolute value of PE depends on the chosen reference level. Only PE differences (ΔPE) are physically meaningful and measurable.
No Energy Loss Modeling: The result is the total PE at the given instant. It does not account for energy dissipated to friction, drag, or other losses during lifting or falling.

Frequently Asked Questions

Answers to common questions about this calculator.

What is gravitational potential energy?
Gravitational potential energy (PE) is the energy stored in an object due to its position above a reference level in a gravitational field. It is calculated as PE = mgh, where m is mass in kg, g is gravitational acceleration in m/s², and h is height in meters. The SI unit is the joule (J).
What is the potential energy formula?
The formula is PE = mgh (mass × gravity × height). An equivalent form is PE = Wh, where W = mg is the object's weight in newtons. Both formulas give the same result and are useful in different situations.
Why does PE depend on height?
Height represents the distance against the gravitational force. Work done against gravity equals force × distance = mg × h. This work is stored as potential energy. The higher the object, the more work was required to lift it, and the more energy is stored.
What are the units of potential energy?
The SI unit is the joule (J = kg·m²/s²). Common conversions: 1 kJ = 1000 J, 1 kcal = 4184 J, 1 BTU = 1055.06 J, 1 eV = 1.602 × 10⁻¹⁹ J, 1 kWh = 3.6 MJ.
Can potential energy be negative?
Yes. If the object is below the reference level (h < 0), PE = mgh is negative. This means the object has less PE than the reference point. Only differences in PE are physically meaningful — the absolute value depends on where you set h = 0.
What is the reference level for PE?
The reference level is the height where h = 0 and PE = 0. It is arbitrary — you can choose any level. Common choices are the ground, the bottom of a container, or the lowest point of a pendulum swing. Only PE differences matter.
How does PE relate to work?
The work done to lift an object at constant velocity equals the PE gained: W = F × d = mg × h = PE. By the work-energy theorem, this work is stored as potential energy. If you lift a 10 kg box 2 m, you do about 196 J of work, and the box gains 196 J of PE.
What is the difference between PE and KE?
PE (potential energy) is stored energy due to position; KE (kinetic energy) is energy of motion. PE = mgh, KE = ½mv². In a closed system, PE converts to KE and vice versa: as an object falls, PE decreases while KE increases, keeping total energy constant.
How does PE convert to KE?
When an object falls freely (no air resistance), its PE converts to KE. The conservation equation is: PE_lost = KE_gained, so mgh = ½mv², giving v = √(2gh). After falling 10 m: v = √(2 × 9.81 × 10) ≈ 14 m/s, regardless of mass.
What is the PE of water behind a dam?
Water PE = mgh, where m is the total water mass, g = 9.81 m/s², and h is the average height above the turbine. A reservoir with 1 million m³ of water at 50 m head stores about 490 GJ (136,000 kWh) of PE, enough to power thousands of homes for a day.
How does PE work on the Moon?
The formula is the same (PE = mgh), but g_moon = 1.62 m/s² instead of 9.81 m/s². The same object at the same height has about 16.5% of its Earth PE on the Moon. This is why astronauts can jump higher on the Moon — the energy barrier is lower.
What is the PE of a pendulum?
At the maximum swing angle θ, a pendulum of length L has PE = mgL(1 − cos θ) relative to the lowest point. As it swings down, all PE converts to KE. At the bottom: ½mv² = mgL(1 − cos θ), so v = √(2gL(1 − cos θ)).
How much PE does a person have on a diving board?
A 70 kg person on a 3 m diving board has PE = 70 × 9.81 × 3 ≈ 2060 J (2.06 kJ). This is enough energy to lift the same person about 3 m — which is exactly what the diving board accomplished. The energy converts to KE during the dive.
What is the relationship between PE and altitude?
PE increases linearly with height near Earth's surface. For large altitudes (aircraft, satellites), gravity decreases with distance: g(h) = g₀ × (R/(R+h))². For satellite orbits, PE = −GMm/r (negative, with r from Earth's center).
Can PE be used to generate electricity?
Yes. Hydroelectric dams convert water PE to electricity via turbines. Pumped hydro stores electricity by pumping water uphill (converting electrical PE to gravitational PE) and releases it by letting water flow downhill through turbines. This is the most common large-scale energy storage method.
What is the PE of a compressed spring?
A compressed spring stores elastic potential energy: PE = ½kx², where k is the spring constant (N/m) and x is the compression (m). This is different from gravitational PE. A spring with k = 500 N/m compressed 0.1 m stores ½ × 500 × 0.1² = 2.5 J.
How does PE relate to energy conservation?
In a closed system without friction, total mechanical energy (PE + KE) is constant. As an object falls, PE decreases and KE increases by the same amount. This conservation law allows solving problems without tracking forces — just equate PE at one point to KE at another.
What is the PE of the International Space Station?
The ISS (420,000 kg) at 408 km altitude has PE = mgh ≈ 420,000 × 9.81 × 408,000 ≈ 1.68 × 10¹² J (1.68 TJ) relative to Earth's surface. However, at orbital altitudes, the constant-g approximation breaks down — the correct formula gives a somewhat different value.
How is PE used in roller coaster design?
Roller coasters use PE ↔ KE conversion. At the top of the first hill (maximum h), PE is maximum and KE is minimum. As the coaster descends, PE converts to KE, giving maximum speed at the bottom. Designers ensure sufficient initial PE to complete the entire track without additional energy input.
What is the PE of a raised weight in a clock?
Grandfather clocks use falling weights to store PE. A 2 kg weight descending 1 m stores PE = 2 × 9.81 × 1 = 19.62 J. This energy drives the clock mechanism for about 8 hours. The weight is then manually raised to "recharge" the PE.
How does PE affect structural engineering?
In construction, falling objects (tools, materials) gain KE from PE. A 5 kg tool falling 20 m has PE = 981 J — enough to cause serious injury. This determines minimum guard rail heights, safety net specifications, and personal protective equipment requirements.
What is gravitational PE in a non-uniform field?
Near Earth's surface, g is nearly constant so PE = mgh works well. For satellites or interplanetary travel, gravity varies with distance: PE = −GMm/r, where G is the gravitational constant, M is the central body mass, and r is the distance from the center. PE is negative and approaches zero at infinite distance.
How do I calculate PE for multiple objects?
Calculate PE for each object individually using PE = mgh, then sum them. For example, 5 boxes of 10 kg each at 3 m height: total PE = 5 × (10 × 9.81 × 3) = 1471.5 J. Alternatively, total mass = 50 kg, so PE = 50 × 9.81 × 3 = 1471.5 J.
What is the PE of a satellite in orbit?
A satellite of mass m at distance r from Earth's center has PE = −GMm/r (negative). The total orbital energy is E = PE + KE = −GMm/(2r). A 1000 kg satellite at 400 km altitude has total energy ≈ −3.07 × 10¹⁰ J. Adding energy (making it less negative) raises the orbit.
How does PE change with latitude?
Gravity varies slightly with latitude due to Earth's shape (oblate spheroid) and rotation. g ≈ 9.78 m/s² at the equator and ≈ 9.83 m/s² at the poles. For a 1000 kg mass at 100 m height, PE differs by about 50 J between equator and poles — negligible for most engineering purposes.
What is the PE of tides?
Tidal energy comes from the gravitational PE difference between water levels. A tidal barrage with a 5 m head difference and 10⁸ m³ of water stores PE = 10⁸ × 1000 × 9.81 × 5 / 2 ≈ 2.45 × 10¹² J per tidal cycle. This is converted to electricity via turbines.
How is PE measured experimentally?
PE cannot be measured directly — only PE differences are measurable. Experimentally, measure the work done against gravity (W = mgh) by lifting an object at constant velocity with a force sensor. Alternatively, measure the KE gained by a falling object (KE = ½mv²) using photogates, which equals the PE lost.
What is the PE of a mountain?
A mountain's PE is the integral of dm × g × h over its volume. For a simplified cone (height 3000 m, base radius 5000 m, density 2500 kg/m³): mass ≈ 3.9 × 10¹³ kg, average height ≈ 1000 m, PE ≈ 3.9 × 10¹⁷ J (390 PJ). This is enormous but spread over geological time scales.
Can PE be negative in practical situations?
Yes. If you set the reference level at a ceiling and an object is on the floor, its PE is negative relative to the ceiling. In orbital mechanics, PE is always negative (reference at infinity). Negative PE means the object is bound — you must add energy to escape.
How does PE apply to energy storage?
PE is used in pumped hydro storage (lifting water uphill), gravitational energy storage (lifting heavy blocks), and flywheel energy storage (rotational KE, related to PE). These systems convert excess electricity to PE during low demand and release it during peak demand.
What is the maximum height for PE = mgh?
The formula works well up to about 10-20 km altitude. Beyond that, Earth's gravity decreases noticeably. At 100 km (Kármán line), g ≈ 9.5 m/s² (3% less). At satellite altitude (400 km), g ≈ 8.7 m/s² (11% less). Use the inverse-square law for greater accuracy at high altitudes.

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Glossary

Technical terms used in mortgage calculations explained in plain language.

Gravitational Potential Energy
Energy stored in an object due to its position in a gravitational field: PE = mgh. Scalar quantity measured in joules (J).
Joule (J)
The SI unit of energy. 1 J = 1 kg·m²/s² = 1 N·m = 1 W·s. Named after James Prescott Joule.
Mass
A measure of the amount of matter in an object, measured in kilograms (kg). In PE = mgh, mass scales energy linearly.
Gravitational Acceleration
The acceleration due to gravity: g ≈ 9.81 m/s² on Earth's surface, 1.62 m/s² on the Moon, 3.72 m/s² on Mars.
Height
Vertical distance above a reference level, measured in meters (m). PE is zero at the reference level (h = 0).
Reference Level
The height where h = 0 and PE = 0. It is arbitrary and can be chosen for convenience. Only PE differences matter.
Weight
Gravitational force on an object: W = mg. Measured in newtons (N). Related to PE by PE = Wh.
Conservation of Energy
Energy cannot be created or destroyed, only converted between forms. Total energy in an isolated system is constant.
Work-Energy Theorem
The net work done on an object equals its change in kinetic energy: W_net = ΔKE. When lifting at constant velocity, W = ΔPE.
Kilocalorie (kcal)
Heat energy to raise 1 kg of water by 1°C. 1 kcal = 4,184 J. Commonly called a "food calorie."
British Thermal Unit (BTU)
Heat energy to raise 1 lb of water by 1°F. 1 BTU = 1,055.06 J. Used in HVAC and heating.
Electron-Volt (eV)
Energy gained by an electron through 1 volt of potential. 1 eV = 1.602 × 10⁻¹⁹ J. Used in particle physics.
Hydroelectric Power
Electricity generated from the gravitational potential energy of water flowing downhill through turbines.
Pumped Hydro Storage
Energy storage by pumping water uphill (storing PE) and releasing it downhill through turbines (generating electricity).
Mechanical Energy
Total mechanical energy = PE + KE. In a closed system without friction, mechanical energy is conserved.

Authoritative References

Trusted sources and standards referenced by this calculator.

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Last Updated:

2026-08-21

Version:

2.0.0

License:

Free to use for educational and professional purposes.