Conservation of Energy Calculator
Calculate the total mechanical energy, maximum height, and velocity at different points using conservation of energy.
Enter mass, height, and velocity.
When only conservative forces act (no friction, no air resistance), the total mechanical energy stays constant. That energy splits between kinetic (motion) and potential (height):
E_total = KE + PE = (1/2)mv² + mgh
Where m is mass in kg, v is speed in m/s, g = 9.81 m/s², and h is height in meters.
Because E_total is conserved, knowing the state at any one point tells you the state at any other point.
Maximum height (when v = 0 at the top): h_max = E_total / (mg) = h + v²/(2g)
Maximum speed (when h = 0 at the bottom): v_max = √(2 × E_total / m) = √(v² + 2gh)
Example: a 2 kg ball thrown upward at 10 m/s from a height of 3 m. KE = 0.5 × 2 × 100 = 100 J PE = 2 × 9.81 × 3 = 58.86 J E_total = 158.86 J Max height = 158.86 / (2 × 9.81) = 8.10 m above the ground Max speed at ground = √(2 × 158.86 / 2) = 12.60 m/s
The principle explains why roller coasters work: the highest hill stores maximum potential energy, and every descent converts it to speed. No engine needed beyond the initial lift.
This calculator assumes no friction. In real systems, some energy always converts to heat, so the actual max height will be less than predicted and the actual max speed will be less too. The discrepancy gives you a measure of frictional losses.
The chart shows how kinetic and potential energy contribute to the total at the entered conditions.