Gravitational, elastic, and electric PE with solve-for-any and step-by-step solutions
Potential energy is the stored energy an object possesses due to its position, configuration, or state. Unlike kinetic energy, which is the energy of motion, potential energy represents energy that has not yet been released — it is energy held in reserve, ready to do work when the system changes state. Understanding and calculating potential energy is fundamental to physics, engineering, and countless real-world applications, from designing roller coasters and buildings to analyzing molecular bonds and electrical circuits. This Potential Energy Calculator covers the three main types of potential energy you will encounter in physics: gravitational potential energy, elastic potential energy, and electric potential energy. Each type follows its own formula and applies to a different class of physical systems. Our calculator lets you switch between these three modes using the tabs at the top, and within each mode, you can choose which variable to solve for — a powerful feature for physics students working through homework problems or engineers verifying designs. Gravitational potential energy (GPE) is the most familiar form. It arises whenever an object is raised above a reference height against the pull of gravity. The formula PE = m × g × h tells you that the stored energy equals the object's mass times the local gravitational acceleration times the height above the reference point. On Earth, g = 9.81 m/s², but our calculator includes presets for all solar-system bodies — the Moon (1.62 m/s²), Mars (3.71 m/s²), Jupiter (24.79 m/s²), and others — so you can explore how gravity affects energy storage on different worlds. This is useful for astronomy enthusiasts and space-mission planners alike. Elastic potential energy (EPE) is the energy stored in a deformed elastic object, such as a stretched spring, a compressed bumper, or a bent bow. The formula PE = ½ × k × x² shows that the energy depends on the spring constant k (measured in N/m, a measure of stiffness) and the square of the displacement x from the equilibrium position. Because x is squared, elastic PE is always positive regardless of whether the spring is compressed or extended. Typical applications include car suspension systems, bungee cords, pogo sticks, shock absorbers, and trampoline springs. Electric potential energy (EPE) describes the energy stored in a system of two charged particles separated by a distance. Governed by Coulomb's Law, the formula PE = k_e × q₁ × q₂ / r uses Coulomb's constant k_e = 8.9875 × 10⁹ N·m²/C². When the charges have the same sign, the electric PE is positive, indicating a repulsive system where work must be done to push the charges together. When the charges have opposite signs, the PE is negative, indicating an attractive, bound system — like the electron and proton in a hydrogen atom — where energy must be supplied to pull them apart. Our calculator goes beyond simple computation. After calculating any form of potential energy, you receive the result expressed in ten different energy units simultaneously: joules, kilojoules, megajoules, calories, kilocalories, watt-hours, kilowatt-hours, BTUs, foot-pounds, and electronvolts. This multi-unit display makes it easy to compare energy values across different fields of study. A step-by-step solution panel shows the exact formula, the values substituted, and the final answer — perfect for checking your physics homework or understanding the derivation. A visual bar chart compares your result against real-world reference energies, giving you instant intuition for whether the energy is tiny (like lifting a paperclip) or enormous (like a car crash). Whether you are a high school student studying Hooke's Law, a university physics major exploring Coulomb's Law, or an engineer sizing a spring for a mechanical system, this Potential Energy Calculator gives you the precision and flexibility you need. All calculations run instantly in your browser with no sign-up required.
Understanding Potential Energy
What Is Potential Energy?
Potential energy is stored energy that an object has because of its position in a force field or its physical configuration. The term 'potential' refers to the capacity to do work — energy that exists in reserve. A book on a shelf has gravitational PE because gravity could pull it down and convert that stored energy into kinetic energy. A compressed spring has elastic PE because its restrained shape stores the work done to compress it. Two like charges held close together have electric PE because the repulsive force between them represents stored energy. In all cases, potential energy is defined relative to a chosen reference point. For gravitational PE, the reference is usually the ground (h = 0). For electric PE, the reference is infinity (charges infinitely far apart have zero electric PE). The SI unit for all forms of energy, including potential energy, is the joule (J), defined as one newton of force applied over one meter of distance.
How Is Potential Energy Calculated?
Each type of potential energy follows its own formula derived from first principles of physics. Gravitational PE uses PE = m × g × h, where m is mass in kilograms, g is gravitational acceleration in m/s² (9.81 on Earth's surface), and h is height in meters above the reference level. This formula can be rearranged to solve for any of the four variables. Elastic PE follows Hooke's Law: PE = ½ × k × x², where k is the spring constant in N/m and x is the displacement in meters. The factor of ½ arises from integrating the linear force F = kx over the displacement from 0 to x. Electric PE is computed with PE = k_e × q₁ × q₂ / r, using Coulomb's constant k_e = 8.9875 × 10⁹ N·m²/C², where q₁ and q₂ are charges in coulombs and r is the separation in meters. Unit conversions are applied before calculation: mass can be entered in kg, g, lb, or other units; height in m, cm, ft; charges in C, mC, µC, or nC; displacements in m, cm, mm, ft, or inches.
Why Does Potential Energy Matter?
Potential energy is central to energy conservation — one of the most powerful principles in all of physics. The law of conservation of energy states that in a closed system, total mechanical energy (kinetic + potential) remains constant. This principle explains why a roller coaster car accelerates at the bottom of a hill (PE converts to KE), why a pendulum swings rhythmically (PE and KE exchange back and forth), and why hydroelectric dams generate electricity (GPE of water converts to electrical energy). In engineering, calculating elastic PE helps designers size springs for safety margins — an automotive suspension spring stores several hundred joules to cushion road impacts. In chemistry and atomic physics, electric PE explains chemical bonding: atoms bond when the total electric PE of the system decreases. Understanding electric PE also underpins capacitor design, electrostatic precipitators, and the behavior of charged particle beams in accelerators.
Limitations and Caveats
This calculator uses idealized models appropriate for introductory and intermediate physics. The gravitational formula PE = mgh assumes a uniform gravitational field and is accurate near Earth's surface. For very large heights (orbital mechanics) or precise geodetic work, the full gravitational potential formula PE = -GMm/r should be used instead. The elastic PE formula assumes Hooke's Law holds, meaning the material deforms linearly and within its elastic limit — real springs can become non-linear under large deformations or when stretched beyond their yield point. For the electric PE formula, the model assumes point charges in a vacuum. In real materials, the permittivity of the medium reduces the effective Coulomb constant. The formula also treats charges as dimensionless points; for extended charge distributions, integration over the charge density is required. Finally, signs matter in electric PE: a negative result indicates a bound system (attractive charges), while a positive result indicates a free system (repulsive charges). These distinctions are noted in the results panel.
Formulas
The energy stored by an object of mass m (kg) at height h (m) in a gravitational field with acceleration g (m/s²). On Earth, g = 9.81 m/s². Valid for uniform gravitational fields near a planet's surface.
The energy stored in a spring or elastic material with spring constant k (N/m) displaced by x (m) from its equilibrium position. Derived from integrating Hooke's Law F = kx over the displacement.
The energy stored between two point charges q₁ and q₂ (coulombs) separated by distance r (m). Coulomb's constant kₑ = 8.9875 × 10⁹ N·m²/C². Positive PE = repulsive; negative PE = attractive (bound) system.
Reference Tables
Surface Gravity of Solar System Bodies
| Body | g (m/s²) | Relative to Earth |
|---|---|---|
| Sun | 274.0 | 27.9× |
| Mercury | 3.70 | 0.38× |
| Venus | 8.87 | 0.90× |
| Earth | 9.81 | 1.00× |
| Moon | 1.62 | 0.17× |
| Mars | 3.71 | 0.38× |
| Jupiter | 24.79 | 2.53× |
| Saturn | 10.44 | 1.06× |
| Uranus | 8.87 | 0.90× |
| Neptune | 11.15 | 1.14× |
| Pluto | 0.62 | 0.06× |
Common Spring Constants
| Object | Typical k (N/m) | Typical Use |
|---|---|---|
| Rubber band | 1 – 10 | Office, crafts |
| Screen door spring | 50 – 100 | Household |
| Bungee cord | 40 – 100 | Recreation |
| Trampoline spring | 2,000 – 5,000 | Sports, recreation |
| Car suspension spring | 20,000 – 30,000 | Automotive |
| Industrial die spring | 50,000 – 200,000 | Manufacturing |
| Railroad buffer spring | 500,000 – 1,000,000 | Rail transport |
Worked Examples
Gravitational PE — Person on a Diving Board
Identify values: m = 75 kg, g = 9.81 m/s², h = 10 m
Apply formula: PE = m × g × h
Substitute: PE = 75 × 9.81 × 10
Calculate: PE = 7,357.5 J = 7.358 kJ
Elastic PE — Car Suspension Spring
Identify values: k = 25,000 N/m, x = 0.08 m
Apply formula: PE = ½ × k × x²
Substitute: PE = ½ × 25,000 × (0.08)²
Calculate: PE = 0.5 × 25,000 × 0.0064 = 80 J
Electric PE — Two Protons in a Helium Nucleus
Identify values: kₑ = 8.9875 × 10⁹ N·m²/C², q₁ = q₂ = 1.602 × 10⁻¹⁹ C, r = 2.5 × 10⁻¹⁵ m
Apply formula: PE = kₑ × q₁ × q₂ / r
Substitute: PE = 8.9875 × 10⁹ × (1.602 × 10⁻¹⁹)² / (2.5 × 10⁻¹⁵)
Calculate numerator: 8.9875 × 10⁹ × 2.566 × 10⁻³⁸ = 2.307 × 10⁻²⁸
Divide: PE = 2.307 × 10⁻²⁸ / 2.5 × 10⁻¹⁵ = 9.23 × 10⁻¹⁴ J
Convert to eV: 9.23 × 10⁻¹⁴ / 1.602 × 10⁻¹⁹ ≈ 576 keV
How to Use the Potential Energy Calculator
Choose the PE Type
Select the tab that matches your problem: Gravitational (mgh) for objects lifted in a gravity field, Elastic (½kx²) for springs and elastic materials, or Electric (k_e q₁q₂/r) for charged particle systems. Each tab shows only the inputs relevant to that formula.
Select What to Solve For
Use the 'Solve for' dropdown to choose your unknown variable. For Gravitational PE you can solve for energy, mass, height, or gravity. For Elastic PE you can solve for energy, spring constant k, or displacement x. For Electric PE you can solve for energy, either charge, or the separation distance r.
Enter Known Values with Units
Type your known values into the visible input fields. Use the unit dropdowns beside each field to match your measurement system — mass can be in kg, g, lb, or oz; height in m, cm, ft, or inches. For gravitational calculations, click a planet button to auto-fill the correct gravity for Earth, Moon, Mars, Jupiter, and more.
Read the Results and Context
Your answer appears instantly in the result card. Expand 'Step-by-Step Solution' to see the full worked derivation. Scroll down to the Energy Context chart to compare your result against everyday energy references — from lifting an apple (≈1 J) to consuming 1 kWh of electricity (3,600,000 J). All ten energy units are shown simultaneously in the conversions table.
Frequently Asked Questions
What is the formula for gravitational potential energy?
The formula for gravitational potential energy is PE = m × g × h, where m is the mass in kilograms, g is the gravitational acceleration in meters per second squared (9.81 m/s² on Earth), and h is the height above the chosen reference point in meters. The result is in joules. This formula assumes a uniform gravitational field, which is valid for heights well below Earth's orbital altitudes. On other planets, simply substitute the appropriate surface gravity — for example, g = 1.62 m/s² on the Moon, giving objects far less stored energy at the same height than on Earth.
What does a negative electric potential energy mean?
A negative electric potential energy means the two charges have opposite signs — one positive and one negative — and are in a bound, attractive system. In physics, the reference point for electric PE is infinity: two charges infinitely far apart have zero PE. When opposite charges are brought closer, the system loses energy (work is done by the attractive force), so the PE becomes negative. A negative electric PE indicates that energy must be supplied to separate the charges to infinity. This is analogous to a gravitational system where an object below the reference height has negative gravitational PE. The hydrogen atom, with an electron bound to a proton, is the classic example of a negative electric PE system.
What is the difference between potential energy and kinetic energy?
Potential energy is stored energy due to position or configuration — it is energy 'waiting to be released.' Kinetic energy is the energy of motion, given by KE = ½mv². In a conservative system (one with no friction or air resistance), total mechanical energy E = KE + PE remains constant. This is the law of conservation of energy. When a ball falls from a height, its gravitational PE decreases while its kinetic energy increases by the same amount. At the bottom of the fall, just before impact, all of the initial PE has converted to KE. Roller coasters, pendulums, and satellite orbits are all governed by this interplay. Springs and elastic bands store PE when deformed, releasing it as KE when they snap back.
How does the spring constant affect elastic potential energy?
The spring constant k (also called the stiffness coefficient) measures how resistant a spring is to deformation. A higher k means a stiffer spring that requires more force to compress or extend by the same distance. Because elastic PE = ½kx², a stiffer spring stores more energy for the same displacement. For example, a car suspension spring with k = 25,000 N/m compressed by 10 cm (0.1 m) stores PE = ½ × 25,000 × 0.01 = 125 J. A soft bungee cord with k = 60 N/m stretched by 10 m stores PE = ½ × 60 × 100 = 3,000 J — much more energy due to the large displacement, despite a much lower spring constant.
Why does gravitational acceleration differ on other planets?
Gravitational acceleration at a planet's surface is determined by g = GM/R², where G is the universal gravitational constant, M is the planet's mass, and R is its radius. Larger planets have stronger gravity not simply because they are bigger, but because of the ratio of mass to radius squared. Jupiter, with a mass 318 times Earth's but a radius only 11 times larger, has a surface gravity of about 24.79 m/s² — roughly 2.5 times Earth's. The Moon, being both much less massive and much smaller than Earth, has g = 1.62 m/s², about one-sixth of Earth's gravity. This means the same object at the same height stores about six times less gravitational PE on the Moon than on Earth.
How do I convert joules to kilocalories (food calories)?
One food calorie (kcal, the 'Calorie' on nutrition labels) equals exactly 4,184 joules. To convert a potential energy result from joules to kilocalories, divide by 4,184. For example, a 70 kg person climbing a 3-meter ladder stores approximately 2,060 J of gravitational PE, which equals 2,060 / 4,184 ≈ 0.49 kcal — less than half a food calorie. This illustrates why physical activity burns only small amounts of the stored food energy: most of the metabolic energy goes into heat, and the actual mechanical work performed is a small fraction. Our calculator automatically shows the kcal conversion alongside all other energy units in the conversions table.