What is potential energy?

Gravitational potential energy is the energy stored in an object due to its position above a reference point. The formula is PE = mgh, where m is mass (kg), g is gravity (9.81 m/s²), and h is height (m). Result is in joules.

How are potential and kinetic energy related?

As an object falls, potential energy converts to kinetic energy. At the top: PE max, KE zero. At the bottom: KE max, PE zero (relative to bottom). Total mechanical energy is conserved (in absence of friction). v = √(2gh) at bottom.

What's the reference point for potential energy?

You choose. PE is relative — only differences matter physically. Common choices: ground level, lowest point in problem, Earth's center (for orbital problems). The same object has different PE values depending on the chosen reference. Physics works out the same regardless.

How is gravitational PE used in power generation?

Hydroelectric dams convert water's PE to electricity. Reservoir water at height h has PE = mgh. Released through turbines, this becomes mechanical and then electrical energy. Pumped hydro stores excess electricity by lifting water; releases it when needed.

Why does PE = mgh fail at large altitudes?

PE = mgh assumes constant g. At higher altitudes, g decreases (Earth's gravity weakens with distance). For low altitudes (< 10 km), the approximation is accurate to ~0.3%. For satellites or interplanetary travel, use full formula U = −GMm/r.

Why is PE = −GMm/r negative?

It's a convention. U is defined as zero at infinity (r → ∞). Any closer position has negative PE because work must be added (positive energy) to move the object away to infinity. This negative convention makes orbital mechanics math cleaner.

How much PE does climbing a mountain give you?

For a 70 kg person climbing 1,000 m: PE = 70 × 9.81 × 1,000 = 686,700 J ≈ 165 kcal. That's a lower bound — real metabolic cost is 3-5× higher due to inefficient muscle work, heat loss, and breathing. Still, climbing mountains burns serious calories.

Can potential energy be negative?

In one sense yes (PE relative to a higher reference). In another sense, for the U = −GMm/r formula, U is negative by convention. The physical effects (work done, kinetic energy released) depend on PE differences, which are always meaningful regardless of sign.

Potential Energy Calculator

Q: Why is PE = −GMm/r negative?

It's a convention. U is defined as zero at infinity (r → ∞). Any closer position has negative PE because work must be added (positive energy) to move the object away to infinity. This negative convention makes orbital mechanics math cleaner.

Q: How much PE does climbing a mountain give you?

For a 70 kg person climbing 1,000 m: PE = 70 × 9.81 × 1,000 = 686,700 J ≈ 165 kcal. That's a lower bound — real metabolic cost is 3-5× higher due to inefficient muscle work, heat loss, and breathing. Still, climbing mountains burns serious calories.

Q: Can potential energy be negative?

In one sense yes (PE relative to a higher reference). In another sense, for the U = −GMm/r formula, U is negative by convention. The physical effects (work done, kinetic energy released) depend on PE differences, which are always meaningful regardless of sign.

Find the gravitational potential energy of an object at a given height. Uses the formula PE = mgh where m is mass, g is gravitational acceleration, and h is height above a reference point.

Gravitational potential energy (PE) is the energy stored in an object because of its height above a reference point. Lift a box up, and you've added energy to it — energy that can be released as kinetic energy when the box falls. The formula PE = mgh captures this simply: heavier objects have more PE, higher objects have more PE, and stronger gravity (a different planet) gives more PE.

PE is a relative quantity — it depends on what reference height you choose. A 1 kg book on a 1 m shelf has 9.81 J of PE relative to the floor, but 88 J of PE relative to the basement 8 m below. Physics works regardless of reference choice; only differences in PE (which produce work and kinetic energy) matter.

Together with kinetic energy, gravitational PE forms the basis of mechanical energy conservation in classical mechanics. Drop the book and PE converts to KE as it falls; the impact velocity follows from ½mv² = mgh → v = √(2gh). This back-and-forth between PE and KE explains pendulums (high PE at the swing's top, max KE at the bottom), roller coasters (PE at the top of the first hill provides KE for the whole ride), hydroelectric dams (water's PE drives turbines), and countless other systems.

The PE = mgh formula is a near-Earth approximation. At larger scales, gravitational PE is U = −GMm/r — negative because work is needed to separate the object from the central body. Only at small heights relative to Earth's radius does this simplify to the familiar linear form.

Common applications: roller coaster design, dam and hydroelectric calculations, falling-object analysis, energy storage (pumped hydro, lifted weights), drop-test engineering, and any physics problem involving height changes.

Inputs

Mass (kg)

Height (m)

Gravity (m/s²)

Earth: 9.81, Moon: 1.62, Mars: 3.72

Results

Potential Energy

490.5 J

Energy (kJ)

0.491 kJ

Impact Velocity

9.90 m/s

Potential Energy Results

Parameter	Value
Mass	10.00 kg (22.05 lbs)
Height	5.00 m (16.40 ft)
Gravity	9.81 m/s²
Potential Energy	490.5 J
Energy (kJ)	0.4905 kJ
Energy (calories)	117.23 cal
Velocity at Ground	9.90 m/s (if dropped)
Formula	PE = mgh

Last updated: May 29, 2026

Formula

**Gravitational potential energy (near surface):** PE = m × g × h Where: - PE = potential energy (J) - m = mass (kg) - g = gravitational acceleration (m/s²) - h = height above reference point (m) **Worked example: 5 kg book on a 2 m shelf** PE = 5 × 9.81 × 2 = 98.1 J If the book falls, it converts that 98.1 J to kinetic energy (and ultimately sound, heat, deformation on impact). **Velocity from height (energy conservation):** If PE_top = KE_bottom (frictionless): mgh = ½mv² v = √(2gh) For a 2 m fall: v = √(2 × 9.81 × 2) = √(39.24) ≈ 6.26 m/s. For a 10 m fall: v = √(196.2) ≈ 14.0 m/s. **Common gravitational PE values:** | Situation | PE | |---|---| | 1 kg book at 1 m | 9.81 J | | 1 kg book at 10 m | 98.1 J | | 70 kg person on 1 m chair | 687 J | | 70 kg person on 100 m building | 68,700 J = 68.7 kJ | | 1,500 kg car on 50 m hill | 735,750 J ≈ 736 kJ | | 1 cubic meter water 100 m up | 9.81 × 10⁵ J ≈ 1 MJ | | Hoover Dam reservoir (lifted) | ~10¹⁴ J | **General form (large heights or other planets):** U = −G × M × m / r Where: - G = 6.674 × 10⁻¹¹ N·m²/kg² - M = central body mass (kg) - r = distance from center (m) Negative because zero PE is defined at r → ∞. Near Earth's surface, with r = R_Earth + h and h << R_Earth: U ≈ −GMm/R + (GMm/R²) × h = constant + mgh (The constant depends only on choice of reference; mgh is the part that varies with height.) **Other forms of potential energy:** | Type | Formula | Notes | |---|---|---| | Gravitational (near surface) | mgh | linear in h | | Gravitational (general) | −GMm/r | inverse in r | | Elastic (spring) | ½kx² | quadratic in displacement | | Electric | qV | charge × potential | | Coulombic (two charges) | kq₁q₂/r | inverse in r | | Chemical | varies | bonds, fuels, batteries | | Nuclear | varies | binding energy of nuclei | **Mechanical energy conservation:** Total mechanical energy = KE + PE = constant (no friction). ½mv² + mgh = constant A roller coaster: - Top of first hill: PE max, KE = 0 (slow at top). - Bottom of valley: KE max, PE = 0. - Half-height: KE = PE. **Work done against gravity:** To lift mass m by height h: W = F × d = mg × h = PE gained So mgh = work needed = energy stored. **Pumped hydro storage:** Lift water up at off-peak times → store PE → release through turbine at peak demand. Example: 1,000 m³ water (10⁶ kg) lifted 200 m: PE = 10⁶ × 9.81 × 200 = 1.96 × 10⁹ J ≈ 545 kWh Round-trip efficiency: ~70-85%. World's largest battery — and the cheapest grid-scale energy storage. **Worked example: roller coaster** Coaster starts at 50 m height, then drops to 5 m valley. Speed at valley (assuming frictionless)? ΔPE = mgΔh = m × 9.81 × 45 ½mv² = 9.81 × 45 × m (mass cancels) v = √(2 × 9.81 × 45) = √883 ≈ 29.7 m/s ≈ 107 km/h In reality, 70-90% achieved due to friction. **Hydroelectric power:** Power = (ρ × g × h × Q) × η Where Q = flow rate (m³/s), η = turbine efficiency. Example: Hoover Dam. h = 180 m, Q = ~600 m³/s, η = 0.9. P = 1000 × 9.81 × 180 × 600 × 0.9 ≈ 953 MW Real average: ~2 GW peak. Provides ~4 TWh/year. **PE relative to different references:** For 1 kg book on 1 m shelf in a 3-story building: - Relative to shelf: 0 J. - Relative to floor: 9.81 J. - Relative to basement (4 m below floor): 49.05 J. - Relative to Earth's center (~6,378 km below): 6.26 × 10⁷ J. Differences in PE are what matter physically; absolute values depend on chosen zero. **Spring PE:** PE_spring = ½kx² Quadratic in displacement. A spring stretched 1 cm stores PE; stretched 2 cm stores 4× the PE. **Electrical PE:** For a charge q at potential V: PE = qV For two point charges: PE = kq₁q₂/r Same inverse-r form as gravity, but can be positive or negative depending on signs.

How to use this calculator

Enter mass in kg.
Enter height above reference point in meters.
Enter gravity (default 9.81 m/s² for Earth surface).
Calculator returns potential energy in joules.
For drops, this PE equals KE at the bottom (ignoring friction).
Convert: 1 kJ = 1,000 J; 1 kWh = 3.6 × 10⁶ J.

Worked examples

Roller coaster first hill

**Scenario:** A 500 kg roller coaster car at the top of a 60 m first hill. Energy and bottom speed (no friction)? **Calculation:** PE = 500 × 9.81 × 60 = 294,300 J ≈ 294 kJ. Speed at bottom: v = √(2 × 9.81 × 60) = √1,177 ≈ 34.3 m/s = 124 km/h (77 mph). **Result:** ~34 m/s at bottom, ideal. Real coasters lose ~15-20% to friction → ~30-32 m/s actual. First hill must be tallest because it provides all the energy; subsequent hills must be lower to ensure the train makes it over.

Pumped hydro storage

**Scenario:** A pumped hydro plant lifts 10,000 m³ of water 300 m. Energy stored? **Calculation:** PE = 10⁷ kg × 9.81 × 300 = 2.94 × 10¹⁰ J. Convert: 2.94 × 10¹⁰ / 3.6 × 10⁶ ≈ 8,175 kWh. **Result:** ~8.2 MWh storage — about 200-300 typical EV batteries. Real plants like Bath County (Virginia) store ~24 GWh — equivalent to ~600,000 EV batteries. Grid-scale efficiency: 70-85% round-trip.

Climbing potential

**Scenario:** 70 kg hiker climbs from 1,000 m to 2,500 m elevation. Minimum energy used? **Calculation:** PE = 70 × 9.81 × 1,500 = 1,030,050 J ≈ 1.03 MJ. Convert to nutritional calories: 1.03e6 / 4,184 ≈ 246 kcal. **Result:** ~246 calories of physical work just lifting yourself. Real metabolism much higher (~600-1,000 kcal/hr for moderate hiking) — most energy goes to maintaining body temperature, biomechanical inefficiency, and breathing in thin air. PE calculation gives a hard floor.

When to use this calculator

**Use gravitational PE for:**

- **Roller coaster and ride design**: hill heights and energy budgets. - **Hydroelectric power**: storage and generation calculations. - **Pumped storage**: grid-scale energy storage. - **Falling object analysis**: terminal velocity, impact energy. - **Mountain rescue**: rappelling, lifting weights, rope systems. - **Construction**: crane work, lifting calculations. - **Aerospace**: launch energy, orbital insertion. - **Geology**: landslide energy, glacier flow.

**When to use full formula (−GMm/r):**

- High altitudes (rockets, space). - Other planets at varying altitudes. - Trajectory calculations beyond Earth's vicinity. - When height is comparable to Earth's radius (~6,371 km).

For everyday heights (< ~10 km), PE = mgh is accurate enough.

**Energy conservation problems:**

Total mechanical energy E = KE + PE = constant (no friction).

For any two points in a frictionless system: KE₁ + PE₁ = KE₂ + PE₂ ½mv₁² + mgh₁ = ½mv₂² + mgh₂

This lets you find velocities at any point given heights, or vice versa.

**With friction:**

E_final = E_initial − W_friction

Friction does negative work, converting mechanical energy to heat.

**Common applications:**

- **Hydroelectric**: world's largest renewable source (~16% of global electricity). - **Pumped hydro storage**: 95% of grid storage capacity worldwide. - **Roller coasters**: design for safe energy return. - **Bungee jumping**: PE → KE → elastic PE → KE → ... - **Trebuchets**: counterweight PE → projectile KE. - **Clocks (pendulum)**: PE swings between bob heights. - **Lifts and elevators**: PE work to lift people.

**Energy storage comparison:**

| Method | Energy density | |---|---| | Lifted mass (mgh, 10 m) | 0.027 Wh/kg | | Lifted mass (1 km) | 2.7 Wh/kg | | Compressed air | ~5 Wh/kg | | Lithium-ion battery | 100-265 Wh/kg | | Gasoline | 12,800 Wh/kg | | Hydrogen (liquid) | 33,000 Wh/kg | | Uranium-235 (fission) | 22 million Wh/kg | | Matter-antimatter | 25 billion Wh/kg |

Gravitational PE is energy-poor per kg, but cheap (water, dirt). Why hydropower is cheap and abundant where geography allows.

**Software:**

- **AutoCAD Civil 3D**: cut/fill calculations. - **HEC-RAS**: hydraulic modeling. - **HOMER**: renewable energy planning. - **Python/MATLAB**: custom analysis.

**Pitfalls:**

- **Reference frame**: PE depends on chosen zero. Differences matter; absolutes don't. - **Using mgh at large altitudes**: error grows with h/R_Earth. - **Ignoring friction**: real systems lose energy. - **Forgetting that g changes**: 0.3% per 10 km altitude. - **Mixing units**: J vs kJ vs kWh vs cal vs Btu. - **Calculating sign**: positive when above reference, negative below. - **Confusing weight with mass**: weight depends on g, mass doesn't.

Common mistakes to avoid

Choosing inconsistent reference heights for different parts of a problem.
Using PE = mgh at altitudes where g varies significantly.
Ignoring frictional losses in energy conservation problems.
Mixing units (J vs kJ vs Wh vs cal).
Forgetting PE is relative — only differences are physically meaningful.
Confusing weight (force, mg) with mass (matter, kg).
Using wrong g for different planets or altitudes.
Adding PE to systems incorrectly (e.g., counting same height twice).

Frequently Asked Questions

Sources & further reading

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