About the Conservation of Energy Calculator
This conservation of energy calculator solves classic mechanics problems where kinetic energy (½mv²) and gravitational potential energy (mgh) trade places. Enter the mass, starting speed and height, then either the final height to get the final speed, or a final speed (zero for the top of the motion) to get the height the object reaches. It also lists the kinetic, potential and total mechanical energy at both points.
Use it for falling objects, pendulums, roller coasters, skateboard ramps, thrown balls or sledges on a hill. If some energy is lost to friction or air resistance, enter that loss in joules and the calculator subtracts it from the mechanical energy available at the second point.
The method assumes that only gravity does work (apart from the loss you enter), that g is constant over the height range, and that the path shape does not matter — only the start and end heights. Heights can be measured from any reference level as long as both use the same one. You can switch gravity to the Moon, Mars or Jupiter.
How to use the conservation of energy calculator
- 1Choose whether you want the final speed or the height reached.
- 2Enter the mass, initial speed and initial height.
- 3Enter the final height (or the final speed, 0 for the top of the motion).
- 4Add any energy lost to friction, and choose the planet’s gravity.
- 5Read the answer and compare energies at both points on the chart.
Formula and method
When only conservative forces such as gravity do work, the total mechanical energy — kinetic energy ½mv² plus gravitational potential energy mgh — stays constant. Any energy dissipated by friction or air resistance is added on the right-hand side as E_loss.
Solving for the final speed gives v₂ = √(v₁² + 2g(h₁ − h₂) − 2E_loss/m). Solving for the height reached at a chosen speed gives h₂ = (½mv₁² + mgh₁ − E_loss − ½mv₂²) ÷ mg; with v₂ = 0 this is the maximum height. Potential energy uses a constant g, which is accurate for heights that are small compared with the planet’s radius.
- m
- Mass (kg)
- v₁, v₂
- Initial and final speed (m/s)
- h₁, h₂
- Initial and final height (m)
- g
- Gravitational acceleration (m/s²)
- E_loss
- Energy dissipated by friction or drag (J)
Worked examples
70 kg person dropping 20 m from rest
All 70 × 9.80665 × 20 = 13,729 J of potential energy becomes kinetic energy. v₂ = √(2 × 9.80665 × 20) = 19.8 m/s, about 71 km/h, regardless of the mass.
Roller coaster car with friction
The car starts with ½ × 500 × 5² + 500 × 9.80665 × 40 = 202,383 J. After losing 20,000 J to friction and keeping 49,033 J of potential energy at 10 m, 133,350 J is kinetic, so v₂ = √(2 × 133,350 / 500) = 23.1 m/s.
Baseball thrown straight up at 15 m/s
At the top the speed is zero, so all kinetic energy becomes potential energy: the ball rises v₁²/2g = 225/19.613 = 11.47 m above the release point, reaching 12.97 m above the ground.
Frequently asked questions
What is the law of conservation of energy?+
Energy cannot be created or destroyed, only converted between forms. In mechanics, this means kinetic plus potential energy stays constant unless work is done by friction, drag or an external force.
Does mass affect the final speed?+
Not when friction is ignored. Mass appears in both ½mv² and mgh and cancels, so v = √(2gh) for any mass. Mass only matters once you include a fixed energy loss such as a friction value in joules.
Does the shape of the slope matter?+
For a frictionless path, no — only the difference in height matters. A steep drop and a gentle ramp of the same height give the same final speed, though they take different times.
Where should I put the zero of height?+
Anywhere you like. Potential energy values change with the reference level, but the difference between two heights — and therefore the speed — does not. Use the ground or the lowest point for convenience.
How do I account for friction?+
Estimate the work done by friction as friction force × distance travelled (or µmg cos θ × distance on a slope) and enter it as the energy lost. The calculator removes it from the energy available at the end point.