Mechanical energy
Sum of translational kinetic and gravitational potential energies.
Physics · Mechanics · Motion & Energy
Mechanical energy — balance and conservation
Add translational kinetic energy and gravitational potential energy for one state, or use their conserved sum to find the speed at a second height. Both heights use the same zero reference.
Created v1.0
By Thibaut Grzelak, Mechanical Analysis Engineer
02
Results
Kinetic energy
Translational kinetic energy at the entered speed.
Potential energy
Relative to the chosen zero-height level.
Final speed
Nonnegative speed at the reachable final height.
Final kinetic energy
Translational kinetic energy in the final state.
Final potential energy
Final gravitational energy relative to the common zero.
Initial mechanical energy
½mv_i² + mgh_i
Conserved mechanical energy
Eₘ,f = Eₘ,i
Relation and numerical substitution
Eₘ = Eₖ + Eₚ = ½mv² + mgh—
Point mass or translation only; uniform constant gravity. Conservation mode assumes no friction, drag or other nonconservative work. Both states share one height reference.
01Formulas and symbols
Formulas used
Eₖ
Eₖ = ½ × m × v²Translational kinetic energy at the entered state.Eₚ
Eₚ = m × g × hGravitational potential energy at signed height h.Eₘ
Eₘ = Eₖ + EₚMechanical energy is the sum of kinetic and potential energies.Eₘ,i
Eₘ,i = ½ × m × v_i² + m × g × h_iInitial mechanical energy available to reach the final state.Eₘ,f
Eₘ,f = Eₘ,iWithout dissipative work, initial and final mechanical energy are equal.v_f
v_f = √(v_i² + 2 × g × (h_i − h_f))Positive square root gives the final speed, not a velocity direction.Eₖ,f
Eₖ,f = ½ × m × v_f²Kinetic energy at the final state.Eₚ,f
Eₚ,f = m × g × h_fPotential energy at the final state.02Assumptions and limits
Scope of validity
- Point mass or translation only; uniform constant gravity. Conservation mode assumes no friction, drag or other nonconservative work. Both states share one height reference.
03Validation example
Reference numerical case
- Balance: m = 2 kg, v = 3 m/s, h = 4 m, g = 10 m/s² gives Eₖ = 9 J, Eₚ = 80 J and Eₘ = 89 J.
- Conservation: m = 2 kg, v_i = 0, h_i = 5 m, h_f = 0 and g = 10 m/s² gives v_f = √100 = 10 m/s, Eₖ,f = 100 J and Eₚ,f = 0. Total energy remains 100 J.
04References
Technical references
FAQ
Can potential or mechanical energy be negative?
Yes. Heights below the chosen zero give negative mgh. This is a reference choice, not negative mass or negative kinetic energy. Only energy differences determine the final speed.
What does an inaccessible height mean?
The requested state would require negative kinetic energy. The calculator reports an error instead of a real speed. A zero radicand is valid and gives zero speed at a turning point.
Why does mass not change the final speed?
The same positive mass multiplies both sides of energy conservation and cancels. Mass is still needed to calculate the energy values in joules.
05Two states, one energy budget
In conservation mode, ½mv_i² + mgh_i = ½mv_f² + mgh_f. Moving the potential-energy difference to the kinetic term gives v_f² = v_i² + 2g(h_i − h_f). The diagram shows states and reference levels, not a trajectory.
06When conservation does not apply
Friction, drag or external work can change mechanical energy. This calculator excludes springs and rotational kinetic energy, and does not model a path, travel time or constraints that may prevent reaching the final state.
07Reference and gravity
Keep the same zero level for both heights and use the local value of g when precision matters. Changing the common zero changes energy values but not the conserved speed result. Display rounding does not enter the numerical calculation.