Gravitational Potential Energy Calculator

Energy and Work Guide

What should I know about Energy Equations?

Gravitational PE: Ep = mgh. Where m = mass (kg), g = gravitational field strength (9.81 m/s² on Earth), h = height above reference level (m). Kinetic energy: Ek = ½mv². Where v = speed (m/s). Work done: W = Fd cosθ. Where F = force (N), d = distance (m), θ = angle between force and direction of motion. If force and motion in same direction: W = Fd (θ = 0, cos 0 = 1). Work done against gravity = increase in GPE = mgh. Power: P = W/t = Fv.

What do I need to know about Conservation of Energy?

In the absence of friction and air resistance: total mechanical energy is conserved. At the top: Ep = mgh, Ek = 0 (if starting from rest). At the bottom: Ep = 0, Ek = ½mv². Therefore: mgh = ½mv². v = √(2gh). The mass cancels — the speed at the bottom is independent of mass. Example: ball dropped from 10m. v = √(2 × 9.81 × 10) = √196.2 = 14.0 m/s. Pendulum: swings to same height on both sides (conservation). Roller coaster: speed at any point can be calculated from height difference.

What should I know about Work-Energy Theorem?

Net work done on an object = change in kinetic energy. W_net = ΔEk = ½mv² − ½mu². If a 5kg object is pushed by 50N over 3m (from rest): W = 50 × 3 = 150J. v = √(2×150/5) = √60 = 7.75 m/s. Work done against friction = friction force × distance = heat generated. Energy dissipated by friction: if an object slides down a slope and is slower than the frictionless case, the difference is heat: Q_friction = mgh − ½mv².

What should I know about Elastic and Gravitational PE?

Elastic PE: Ee = ½kx². Where k = spring constant (N/m), x = extension (m). A spring with k = 500 N/m extended by 0.1m: Ee = ½ × 500 × 0.01 = 2.5J. At maximum extension (zero velocity): all KE → elastic PE. At natural length: all elastic PE → KE. Combined conservation: if a ball compresses a spring: KE lost = elastic PE gained. mgh = ½kx² (ball on spring from height h). Types of energy: kinetic, gravitational PE, elastic PE, thermal (heat), chemical, nuclear, electrical, sound, light. Energy is always conserved overall, even as it converts between these different forms — this is the basis of the work-energy theorem used throughout mechanics.

What should I know about the GPE Formula and Reference Points?

Gravitational potential energy near Earth's surface is GPE = mgh, where m is mass (kg), g is 9.8 m/s², and h is height above a chosen reference point (metres). Crucially, GPE is always relative — there's no absolute zero height in nature, only a chosen reference (ground floor, sea level, the base of a drop). For a 2kg object raised 3m: GPE = 2×9.8×3 = 58.8 J relative to its starting point. If you instead measured height from a point 5m below the starting point, the same object would have GPE = 2×9.8×8 = 156.8J — a different number, but the CHANGE in GPE as the object moves is always the same (58.8J) regardless of which reference point you chose, which is the only thing that actually matters physically, since only energy differences do real work.

What do I need to know about Converting Between GPE and Kinetic Energy?

In the absence of air resistance or friction, total mechanical energy (GPE+KE) is conserved: energy lost from one form exactly reappears in the other. For an object dropped from height h, all GPE converts to KE by the moment it reaches the ground: mgh = ½mv², which rearranges to v=√(2gh) — the standard formula for impact speed. For a 5m drop: v=√(2×9.8×5)=√98≈9.9 m/s. This conservation principle is why a pendulum, roller coaster, or ball on a ramp can be analysed without needing to track forces moment by moment — comparing GPE at the highest point to KE at the lowest point directly gives the speed, as long as friction and air resistance are small enough to be ignored (or separately accounted for as an energy loss term).

What do I need to know about the General Form of Gravitational PE?

The familiar GPE = mgh is only an approximation valid near a planet's surface, where g is roughly constant. The true, general formula is U = −GMm/r, which is negative (by convention, zero energy is defined at infinite separation) and becomes less negative — i.e., increases — as r grows, exactly as mgh predicts for small height changes. This general form is essential for anything leaving a planet's immediate surface: calculating escape velocity, orbital energy, or the energy needed to send a probe between planets all require U = −GMm/r rather than the surface-level mgh approximation.

GPE Reference Table for Common Heights

Using GPE = mgh near Earth's surface (g = 9.8 m/s²):

Comparing the person climbing stairs (2,058 J) to a thrown baseball's kinetic energy (about 116 J at 40 m/s) shows that everyday height changes often store far more energy than everyday motion does — which is part of why regenerative systems (like elevators with counterweights, or electric vehicles' regenerative braking) focus so heavily on recovering gravitational and kinetic energy rather than treating it as automatically wasted.

Worked Example: Work Done Pumping Water to a Rooftop Tank

A building's water system needs to pump 2,000 liters (2,000 kg, since 1 liter of water ≈ 1 kg) of water from ground level up to a rooftop tank 25 m above the pump, against gravity. Find the minimum work (energy) required, ignoring friction and pipe losses. W = ΔGPE = mgh = 2,000 kg × 9.8 m/s² × 25 m = 2,000 × 245 = 490,000 J = 490 kJ. If the pump operates at 1,500 W (1,500 J/s) and is 80% efficient (since real pumps lose some energy to friction and heat), the electrical energy required is 490,000 J / 0.80 = 612,500 J, and the minimum time needed is 612,500 J / 1,500 W ≈ 408 seconds (about 6.8 minutes) running at full power. This is exactly the calculation water utilities and building engineers perform to size pumps and estimate electricity costs for moving water vertically — the energy cost scales directly with both the mass of water and the height it's lifted, which is why buildings with tall water towers or many floors require correspondingly larger pumps.

Gravitational Potential Energy on Other Planets

Since GPE = mgh depends directly on local gravitational acceleration, the same mass lifted to the same height stores very different amounts of potential energy depending on where you are:

Lifting the same object the same height on the Moon stores only about 16.5% of the energy it would on Earth (81/490 ≈ 0.165), directly reflecting the Moon's much weaker gravity (1.62/9.8 ≈ 0.165) — the ratio of GPE values exactly matches the ratio of gravitational accelerations, since mass and height are unchanged. This is a key consideration in planning lunar or Martian construction and vehicle design: less energy is needed to lift materials against weaker gravity, but conversely, falling objects also release less kinetic energy on impact, which affects everything from excavation equipment design to astronaut safety margins when working at height. It also means a given battery or fuel supply can lift far more total mass against lunar gravity than against Earth's for a fixed amount of stored energy, which is a major factor in planning crane and lander payload capacities for lunar and Martian missions. Engineers designing lunar landers and rovers rely on exactly this GPE-to-g ratio when budgeting fuel for descent and ascent, since every kilogram of propellant saved by weaker gravity is a kilogram that can instead be allocated to scientific payload or return cargo. The same scaling applies in reverse during landing: a spacecraft touching down on the Moon must shed far less kinetic and gravitational energy than one landing on Earth from the same height and speed, which simplifies braking and descent engine requirements considerably.