Conservation of energy calculator

A conservation of energy calculator turns the ideal mechanical-energy identity into a practical speed-from-height solver for homework, labs, and quick engineering checks. Enter the height and speed of a body at one point, the height at a second point, and optionally gravitational acceleration g, then receive the non-negative speed at the second height together with the radicand, a substituted equation, and ordered step strings you can paste into worksheets or automated tests. The underlying relation is v2 equals the square root of v1 squared plus two g times the height drop from point one to point two, which follows when kinetic plus gravitational potential energy is conserved under constant g with no friction or other non-conservative work. Mass is never required because it cancels for a single particle.

What mechanical energy conservation means and the speed formula

Under constant gravitational acceleration and with no non-conservative work, the sum of kinetic energy and gravitational potential energy is the same at every point along a path. For a particle of mass m that is kinetic energy one-half m v squared plus potential energy m g h, writing the equality at point one and point two and dividing through by m yields one-half v1 squared plus g h1 equals one-half v2 squared plus g h2.

Solving for the unknown speed at the second height produces the textbook square-root formula: v2 equals the square root of v1 squared plus two g times the difference h1 minus h2. When the body descends, h1 is larger than h2, the radicand grows, and speed increases; when it climbs, the radicand shrinks, and if it would go negative the target height lies above the mechanical-energy ceiling for that launch state.

Heights are measured from any fixed reference you choose—sea level, the table top, or an arbitrary zero—as long as both h1 and h2 use the same zero. Speed v1 is a non-negative magnitude; direction is not recovered from energy alone, so the response always reports a non-negative v2. Gravitational acceleration g defaults to nine point eight one metres per second squared for common Earth textbook problems; override it for local gravity, planetary surfaces, or scaled numerical exercises.

This calculator does not model friction, air drag, springs, rotation, or multi-body systems: reduce those problems yourself or add non-conservative work terms before calling a pure conservation solver.

Practical tips, limits, and related searches

Use one consistent unit system end to end—for example metres, metres per second, and metres per second squared—before calling; this landing does not perform unit conversion. When teaching energy methods, treat this tool as the pure conservation step only: if kinetic friction does work, subtract that energy yourself or switch to a work-energy formulation.

This page targets people searching for a conservation of energy calculator, mechanical energy speed from height tool, or ideal freefall energy identity online; it is not a full dynamics suite that integrates trajectories, not a kinetic-energy-only calculator, and not a unit converter. Prefer this capability when you already know height and speed at one point and only need the speed at another height under constant g; if your source problem gives mass and you only need kinetic or potential energy separately, use the dedicated kinetic-energy or potential-energy capabilities instead.

Round-trip checks are easy: recompute v1 squared plus two g times height drop by hand and confirm the square root equals the published v2 within the same twelve-decimal rounding used in the response.