Conservation of gravitational mechanical energy
When only conservative gravity does work, the sum of translational kinetic energy and gravitational potential energy remains constant. This calculator equates two states using ½mv² + mgh.
The model assumes constant gravity, a point-like body, and no drag, friction, rolling energy, rotation, propulsion, impact, or other energy transfer. Negative velocities are not represented because the calculator solves speed magnitude.
How to use the energy conservation calculator
- Choose unit system: Use kilograms, meters, and m/s or pound-force, feet, and ft/s as labeled.
- Choose the unknown: Select initial or final speed or height.
- Enter the other state values: Use heights referenced to the same zero level.
- Calculate and assess assumptions: Review the energy breakdown and determine whether nonconservative work is negligible.
Formula and variables
Solve the two-state balance for one speed or height. Mass cancels when only these two energy terms are involved, although it is retained to display energy values.
½mvᵢ² + mghᵢ = ½mv𝒇² + mgh𝒇- m — Mass
- Object mass (kg)
- v — Speed
- Magnitude of translational velocity (m/s)
- g — Gravity
- Constant gravitational acceleration (m/s²)
- h — Height
- Vertical position relative to one common reference (m)
Fall from twenty meters
A 10 kg object starts from rest at 20 m and reaches 0 m with no losses.
- Initial state
- h = 20 m, v = 0
- Final height
- 0 m
- v𝒇² = vᵢ² + 2g(hᵢ − h𝒇)
- v𝒇 = √(2 × 9.80665 × 20)
Result: Final speed is approximately 19.81 m/s.
The ideal final kinetic energy equals the initial gravitational potential energy.
Understanding your results
Compare both energy states
Initial and final totals should agree within displayed rounding.
- Height zero is arbitrary but must be consistent.
- Mass cancels from solved speed or height in this restricted model.
- An unreachable-state error means the requested kinetic energy would be negative.
Assumptions
- Gravity is constant over the height range.
- Only translational kinetic and gravitational potential energy change.
- No nonconservative force performs work.
Limitations
- Does not include drag, friction, springs, rotation, rolling, propulsion, heat, deformation, relativistic effects, or variable gravity.
- Solves nonnegative speed magnitude rather than signed velocity.
- Imperial mode interprets the entered lbf value as weight under standard gravity.
Common mistakes
- Using inconsistent height reference levels.
- Applying mechanical-energy conservation while ignoring friction or drag.
- Treating speed as a signed direction.
- Entering pound-mass when the field is labeled pound-force.
Practical use cases
Ideal falling and rising motion
Estimate speed changes from vertical position changes without drag.
Mechanical-energy education
Inspect the conversion between kinetic and gravitational potential energy.
Frequently asked questions
Why does mass cancel from the speed result?
Both kinetic and gravitational potential energy are proportional to mass in this model.
Can the calculator include friction?
No. Work by friction must be included as a nonconservative energy-transfer term.
Can height be negative?
Yes, if all heights use the same reference; only height differences affect the energy balance.
Sources and review
- Conservation of Energy — OpenStax University Physics Volume 1. Accessed 2026-07-13.
Reviewed 2026-07-13.