Mechanical Energy Calculator
KE + PE at one state, or conservation of energy between two states with one unknown.
What is Mechanical Energy?
Mechanical energy is the sum E = KE + PE, and in the ideal frictionless world that sum is conserved: E₁ = E₂ between any two points of the motion. That single equation is why a coaster's speed at the bottom depends only on the height it fell from, not on the path. The page runs it both ways — one state's total, or the full two-state comparison with exactly one unknown at state 2.
How it works
Each state takes KE and PE directly or as m + v and m + h, with g editable and 9.8 m/s² pre-filled. The two-state mode needs state 1 complete and exactly one blank at state 2 — two blanks are refused, because one equation cannot fix two unknowns. The unknown is solved from E₁ − (the other term), and a square root that goes negative is reported as an unreachable state rather than returned as an imaginary speed. The before/after table shows both totals agreeing.
- Enter what you know. Every field recalculates as you type — there is no submit button. Leave the quantity you want blank and the page solves for it.
- Read the answer and the working. The result appears above the fold; underneath it is the formula with your numbers substituted, so the arithmetic can be checked rather than trusted.
- Check the units and the size. Units are shown on every value and selectable per field, so a result in kJ, kPa or km/h is converted rather than assumed. Sanity-check the magnitude against a known landmark such as 1 g/cm³ for water.
Examples
A textbook problem with a known answer
Every tool here is checked against standard example problems — the molar mass of C6H12O6 is 180.16 g/mol, and free fall from 20 m takes about 2.02 s at g = 9.8 m/s².
A result you want to reason backwards from
Leave the quantity you already know blank instead: given the final answer, most tools recover the starting value — the mass needed for a target concentration, or the height a fall must have had.
Common mistakes
Forgetting that a formula is only as good as its subscripts
H2O and H2O2 are different compounds. The formula field shows its parse — element by element — so a mis-typed subscript is visible before it reaches the arithmetic.
Mixing units in the middle of a calculation
The most common source of a wrong answer is a value in grams next to a formula expecting kilograms, or Celsius next to a gas constant expecting kelvin. This category converts on the way in and labels what it used.
Frequently asked questions
When does mechanical energy stay constant?
When nothing non-conservative acts: no friction, no air resistance, no motor. A pendulum on a string, a coaster on steel rails, a ball in free fall — all close. Add drag or a motor and the total stops being conserved, which is why the page names the model rather than implying it is universal.
Why does the speed at the bottom not depend on the path?
Because PE depends only on height. Any frictionless route from height h delivers the same mgh to KE, so the speed is the same for the steep drop and the long slide — energy is a state function, which is the deepest reason the two-state comparison works at all.
It says my state is unreachable — what does that mean?
State 2 asks for more energy than state 1 has: a higher point, or a faster speed, than the total allows. The negative under the square root is the physics saying so, and the page reports it rather than printing an imaginary answer.