Kinetic and Potential Energy Calculator
Compute total mechanical energy E = ½mv² + mgh from an object's mass, velocity, and height — combining kinetic energy from motion and gravitational potential energy from elevation. Useful for conservation-of-energy problems in physics, engineering, and any system where energy converts between motion and position.
Last updated: May 2026
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About this calculator
The calculator returns the total mechanical energy of an object that is both moving and elevated: E = ½·m·v² + m·g·h, where m is mass in kilograms, v is velocity in metres per second, g is the local gravitational field in m/s², and h is height above a chosen reference level in metres. The result is in joules (J). The first term is kinetic energy (KE = ½mv²), the energy carried by motion — note that velocity is squared, so doubling speed quadruples KE. The second term is gravitational potential energy (GPE = mgh), the energy stored by being above the reference level. Total mechanical energy is conserved in any closed system with only conservative forces; in the absence of friction or air resistance, KE and GPE trade off as an object falls or climbs. The gravitational field defaults to Earth (9.81 m/s²) but the calculator also includes presets for the Moon (1.62), Mars (3.71), and Jupiter (24.79) — useful for planetary science work. Edge cases: setting v = 0 gives pure GPE; setting h = 0 gives pure KE; setting both to zero gives 0 (trivial rest state). Negative heights are accepted and represent positions below the reference — the GPE term becomes negative, which is physically valid. The model ignores rotational kinetic energy, elastic potential energy, and relativistic effects; for objects approaching the speed of light (v > ~0.1c) the classical KE formula significantly underestimates the true energy.
How to use
Example 1 — Roller coaster at the bottom of a drop. A 500 kg car has just reached the bottom of a 40 m hill at 28 m/s on Earth. Enter mass = 500, velocity = 28, height = 0, gravity = 9.81. E = 0.5 × 500 × 28² + 500 × 9.81 × 0 = 0.5 × 500 × 784 + 0 = 196,000 J ≈ 196 kJ. ✓ Verify: at the top of the hill the same car had E = 0 + 500 × 9.81 × 40 = 196,200 J. Within rounding, energy is conserved — the 200 J difference reflects gravity’s 9.81 being a 3-sig-fig approximation. Example 2 — Astronaut on the Moon dropping a tool. A 2 kg wrench is held 1.5 m above the lunar surface and released from rest. Enter mass = 2, velocity = 0, height = 1.5, gravity = 1.62 (Moon preset). E = 0.5 × 2 × 0² + 2 × 1.62 × 1.5 = 0 + 4.86 = 4.86 J. ✓ When the wrench hits the surface, all 4.86 J converts to KE: ½ × 2 × v² = 4.86 → v ≈ 2.20 m/s. The same wrench dropped from the same height on Earth would land at about 5.4 m/s — Moon’s weaker gravity gives noticeably less impact speed despite identical mass and starting height.
Frequently asked questions
Why does velocity get squared in the kinetic energy formula?
The square comes from the work-energy theorem: when you accelerate an object from rest to velocity v with a constant force F, the work done is W = F·d, and substituting F = ma and d = ½at² gives W = ½mv². The squared term is not a convention or fitting parameter — it is a direct consequence of how force, distance, and time combine. Practically, kinetic energy scales much faster than speed: doubling speed quadruples KE, tripling produces 9× the energy. This is why high-speed crashes are dramatically more dangerous than low-speed ones — a car at 60 mph carries four times the KE of the same car at 30 mph, not double. The same principle governs why projectiles at higher velocities deliver disproportionately more impact, why faster-moving fluids carry orders of magnitude more energy, and why kinetic-energy weapons trade speed for destruction more efficiently than mass.
What is gravitational potential energy actually storing?
GPE represents the work that was done against gravity to lift an object to its current height, and equivalently the energy that gravity will return if it falls back down. Strictly speaking, it is energy stored in the configuration of the Earth-object system — not in the object itself — but treating it as “the object’s potential energy” is fine for everyday problems. The reference level (where h = 0) is arbitrary: ground level, a tabletop, sea level, or the centre of the Earth all work — what matters physically is only the change in GPE between two states. This is why GPE values can be negative if you pick a high reference. The formula mgh assumes constant gravity g, which is a good approximation as long as the height change is small compared to Earth’s radius (~6,370 km); for satellites and astrophysics you need the more general GPE = −GMm/r.
What is the difference between mechanical energy and total energy?
Mechanical energy includes only kinetic energy (motion) and potential energy (position in a force field) — typically GPE plus elastic PE in springs. Total energy is much broader: it adds thermal energy (random molecular motion), chemical energy (bonds), electrical energy, nuclear binding energy, and electromagnetic radiation. In a real system with friction, mechanical energy is NOT conserved — it leaks as heat — but total energy always is. This is what the first law of thermodynamics formalises. A car braking from 30 m/s to a stop loses all its KE; that energy doesn’t vanish, it warms the brake pads. Mechanical-energy conservation is an idealisation that works well in physics problems with no friction or only conservative forces, and breaks down in any system involving heat, light, or chemical change.
What are the most common mistakes people make with KE and GPE problems?
The first is mixing units: entering mass in grams while expecting joules out, or velocity in km/h instead of m/s. Always work in SI (kg, m/s, m) or convert at the end. The second is forgetting to square velocity in the KE term — writing E = ½mv instead of ½mv² is a classic mistake. The third is treating GPE as an absolute quantity rather than a relative one; only changes in GPE are physically meaningful, so the reference level choice can shift the absolute number freely. The fourth is applying mgh at heights large enough that gravity is no longer constant (orbital problems, ballistic missiles) — use the general −GMm/r form. The fifth is using classical KE at relativistic speeds: above v ≈ 0.1c, the relativistic KE = (γ − 1)mc² becomes meaningfully different from ½mv². Finally, people forget that mechanical energy is only conserved without friction — for any real system, account for energy lost to heat and air resistance before assuming KE₀ + GPE₀ = KE_final + GPE_final.
When should I not use this calculator?
Skip it for objects moving at relativistic speeds (above about 10% of the speed of light) — classical ½mv² systematically underestimates the true kinetic energy, and you need E_rel = (γ − 1)mc² where γ = 1/√(1−v²/c²). Avoid it for orbital and astrophysical problems where the height change is comparable to Earth’s radius — mgh assumes constant g, which only holds for low altitudes (under maybe 100 km); for satellites use the general gravitational potential −GMm/r. Skip it for rotational systems where most of the kinetic energy is in rotation (gyroscopes, flywheels, spinning balls) — that requires the rotational KE = ½Iω² term. Do not use it for elastic potential energy in springs or compressed gas — that requires a separate ½kx² calculation. And for any problem with significant friction or drag, the simple mechanical-energy formula gives an upper bound on energy, not the actual value; account for thermal losses separately.