Potential Energy Calculator

Gravitational PE = mgh, where m is mass (kg), g is gravitational acceleration (9.81 m/s² on Earth), and h is height above the reference point (meters). The reference point is arbitrary — what matters is the change in height.

A 1,000 kg car at the top of a 50-meter hill has PE = 1,000 × 9.81 × 50 = 490,500 J (490.5 kJ) relative to the bottom. This energy converts to kinetic energy as the car descends — at the bottom (assuming no friction), it reaches v = √(2gh) = √(2 × 9.81 × 50) = 31.3 m/s (70 mph).

Hydroelectric dams exploit this principle: water falling through a height (head) drives turbines. The Hoover Dam's 180-meter head generates enormous energy per cubic meter of water: PE = 1,000 × 9.81 × 180 = 1,765,800 J per m³.

Elastic PE = ½kx², where k is the spring constant (N/m or lb/in) and x is the displacement from equilibrium. This relationship is quadratic — doubling the compression quadruples the stored energy.

A spring with k = 500 N/m compressed 0.1 m stores PE = ½ × 500 × 0.01 = 2.5 J. A car suspension spring (k ≈ 25,000 N/m) compressed 5 cm by hitting a bump stores PE = ½ × 25,000 × 0.0025 = 31.25 J per spring. Bows and crossbows store elastic PE — a compound bow with k ≈ 2,500 N/m at 0.7m draw stores approximately 612 J, enough to launch an arrow at 90+ m/s.

The spring constant determines the force-displacement relationship: F = kx (Hooke's law) — a stiffer spring requires more force to compress but stores more energy per unit displacement.

The conservation of energy principle states that total energy (kinetic + potential) remains constant in an isolated system (ignoring friction and air resistance). At the top of a roller coaster hill, energy is mostly potential; at the bottom, it is mostly kinetic.

This principle enables simple calculations: a ball dropped from 5 meters reaches v = √(2 × 9.81 × 5) = 9.9 m/s at ground level regardless of mass. With friction, some energy converts to heat — a real roller coaster at the bottom of a 30-meter drop reaches approximately 85-90% of the theoretical speed due to friction and air resistance losses.

In engineering, potential energy analysis is critical for:

• sizing pumps (must overcome gravitational PE of water being lifted)
• designing safety systems (energy absorbed by crash barriers)
• energy storage (pumped hydro stores energy as gravitational PE by pumping water uphill during low-demand periods)

To calculate gravitational potential energy, multiply mass by gravitational acceleration by height: PE = m × g × h. Mass m is in kilograms, g is the local gravitational acceleration (9.81 m/s² near Earth's surface), and h is height in metres above your chosen reference point. The result is in joules (J).

For example, a 2 kg book placed on a 3 m shelf stores PE = 2 × 9.81 × 3 = 58.86 J relative to the floor. Because the formula is linear in each variable, doubling the height doubles the stored energy.

HyperPhysics (Georgia State University) notes that mgh is a near-surface approximation where g is treated as constant.

The SI unit of potential energy is the joule (J), defined by BIPM as one kilogram-metre-squared per second-squared (1 J = 1 kg·m²/s²), equivalent to one newton-metre. In the PE = mgh formula the units combine as kg × (m/s²) × m = kg·m²/s² = J, which confirms the equation is dimensionally consistent. NIST maintains the joule as the coherent SI energy unit.

Common conversions include:

• 1 J = 0.7376 ft·lb
• 1 calorie = 4.184 J
• 1 kilowatt-hour = 3.6 × 10⁶ J
• 1 electron-volt = 1.602 × 10⁻¹⁹ J

Always keep mass in kilograms and length in metres so the answer emerges directly in joules without extra conversion factors.

Rearranging PE = mgh lets you solve for any single unknown.

• To find height, divide energy by the product of mass and gravity: h = PE / (m × g). For instance, a 3 kg object holding 147.15 J on Earth sits at h = 147.15 / (3 × 9.81) = 5 m.
• To find mass, use m = PE / (g × h).
• To find local gravity, use g = PE / (m × h).

Because each variable appears to the first power, these inversions are simple division. Khan Academy recommends checking units after rearranging — the answer for height should carry metres, and for mass kilograms — to catch algebra errors before trusting the number.

When an object falls freely, gravitational potential energy converts into kinetic energy while total mechanical energy stays constant. Setting the energies equal, mgh = ½mv², the mass cancels and the impact speed becomes v = √(2gh) — independent of mass.

A stone dropped from 20 m on Earth reaches v = √(2 × 9.81 × 20) = √392.4 ≈ 19.81 m/s (about 71 km/h) at the ground, ignoring air resistance.

Encyclopaedia Britannica frames this as the conservation of mechanical energy: energy is transferred between forms rather than created or destroyed. In reality, drag and friction dissipate some energy as heat and sound, so measured speeds fall slightly below the ideal value.

Gravitational potential energy scales directly with local surface gravity g, so the same mass at the same height stores different energy on each world. NIST and planetary references list g ≈ 1.62 m/s² on the Moon, 3.72 m/s² on Mars, 9.81 m/s² on Earth, and 24.79 m/s² on Jupiter.

A 5 kg object raised 10 m therefore stores:

• 81 J on the Moon
• 186 J on Mars
• 490.5 J on Earth
• 1,239.5 J on Jupiter

The Moon value is roughly 16.5% of Earth's because 1.62 / 9.81 ≈ 0.165. This is why astronauts can lift heavy loads to great heights on the Moon using far less energy, and why the same fall causes gentler impacts.

Elastic potential energy is calculated with PE = ½ × k × x², where k is the spring constant in newtons per metre (N/m) and x is the displacement from the equilibrium position in metres. Because energy depends on x², stretching or compressing a spring twice as far stores four times the energy.

A spring with k = 800 N/m compressed 0.05 m stores PE = ½ × 800 × (0.05)² = ½ × 800 × 0.0025 = 1.0 J. The related restoring force follows Hooke's law, F = kx, as described by HyperPhysics.

Elastic PE is always positive whether the spring is stretched or compressed, since x² is never negative — a point Khan Academy emphasises for beginners.

Potential energy analysis underpins many engineering systems:

• Pumped-storage hydropower stores gravitational PE by pumping water uphill during low demand and releasing it through turbines at peak times — a reservoir head of 300 m gives each cubic metre PE = 1,000 × 9.81 × 300 = 2,943,000 J.
• Roller-coaster designers use the chain-lift hill as a PE reservoir that feeds every drop and loop.
• Spring-based systems store elastic PE in vehicle suspensions, watch mainsprings, archery bows, and mechanical clamps.
• Crash barriers and packaging foams are engineered to absorb kinetic energy by converting it into deformation.

Britannica and IEEE resources note that battery and capacitor systems store energy electrically, complementing these mechanical potential-energy stores in modern grids.

• The most frequent error is mixing units — using grams instead of kilograms, or centimetres instead of metres — which throws the joule result off by factors of thousands. Convert to kilograms and metres before applying PE = mgh.

• A second mistake is using g = 9.81 m/s² for non-Earth problems; the Moon and Mars need their own values.

• For elastic PE, forgetting to square the displacement, or squaring only part of it, is common: PE = ½kx² requires x² in metres-squared.

• Another slip is treating the reference height as fixed rather than a chosen zero point — only the change in height matters.

• Finally, the v = √(2gh) impact speed assumes no air resistance, so real measured speeds are lower.