Physics I: Mechanics › Work & energy › full formula sheet

KEi + Ui = KEf + Uf

Say it: “when only conservative forces do work, kinetic plus potential energy stays constant”

Conservation of mechanical energy

The universe's favorite accounting trick: energy changes form — motion to height to spring — but the total never changes.

KE is kinetic energy (½mv²), U is potential (gravitational mgh, spring ½kx², or their sum). Subscripts i/f = initial/final. All in joules.

Before this lesson: Kinetic energy, Gravitational potential, Spring potential, Work–energy theorem

Where it comes from

Drop a ball and it speeds up as it falls: kinetic rises while gravitational potential falls. Stretch-release a spring and the coil's energy becomes the mass's speed. In each case one form becomes another — and the total never changes.

Before reading on: a 2 kg ball drops from 10 m. Ignoring air, how fast is it moving at the ground — and does the answer depend on the ball's mass? Guess both.

The speed is 14 m/s, independent of mass: mgh = ½mv² gives v = √(2gh), and m cancels. A bowling ball and a marble fall side by side — Galileo's old result, now an energy sentence.

E = KE + U = constantenergy changes costume — motion, height, spring — but the total is conservedSay it: “kinetic plus potential stays constant”

This is the work–energy theorem with the conservative forces' work rewritten as potentials — the most-used equation in all of mechanics.

Derivation

Start from Wnet = ΔKE and split the work by force type:

Wnet
=
Wcons + Wnoncons = ΔKE
Step 1 — split the work. Conservative forces (gravity, springs) vs nonconservative (friction, drag, applied pushes).
Wcons
=
−ΔU
Step 2 — potentials. By definition of potential energy, a conservative force's work is minus the potential change (see Ug, Us).
−ΔU + Wnoncons
=
ΔKE
Step 3 — substitute. Rearrange: Wnoncons = ΔKE + ΔU = Δ(KE + U).
Wnoncons = 0
⇒
ΔE = 0
Step 4 — conserve. With no nonconservative work, Δ(KE + U) = 0: E is constant. Friction's work is exactly the leak. ∎

How to use it

The procedure, every time:

  1. Check the leak. Friction, drag, motors, pushes doing work? Then E is not conserved — use Wnoncons = ΔE instead. Only pure gravity/spring coasts conserve.
  2. Pick two snapshots: initial (i) and final (f) states.
  3. Write Ei = Ef. KEi + Ugi + Usi = KEf + Ugf + Usf. Zero out the terms that vanish (rest → KE = 0; reference height → Ug = 0; relaxed spring → Us = 0).
  4. Solve for the unknown. Usually one speed or one height.
  5. Sanity check. Falling → faster; rising → slower. If your numbers say otherwise, recheck.

When it breaks

Friction converts mechanical energy to heat: Ef = Ei + Wfriction (negative). The total energy of the universe is still conserved — mechanical energy just isn't the whole story anymore.

Common mistake: applying Ei = Ef across a rough patch. If friction does work, the conserved quantity leaks — account for Wnoncons or the answer is fiction.

Worked examples

Four problems, easiest first. Two snapshots, one conserved total.

Example 1 — dropped ball: m = 2 kg from h = 10 m, find v at ground

  1. Snapshots. i: top (rest, h = 10). f: ground (h = 0, speed v).
  2. Energies. Ei = 0 + 2·9.8·10 = 196 J. Ef = ½·2·v² + 0 = v².
  3. Conserve. v² = 196, v = 14 m/s.
  4. Note. Mass canceled — v = √(2gh) for any mass. ✓
Common mistake: setting mgh = mv² (dropping the ½) and getting v ≈ 19.8 m/s. The kinetic side always carries its half.
Your turn — m = 1 kg dropped from h = 5 m. v at ground?

Answer: ≈ 9.9 m/s. v = √(2·9.8·5) = √98 ≈ 9.9 m/s (mass cancels).

Example 2 — how high? Launched upward at v₀ = 6 m/s

  1. Snapshots. i: ground (v = 6, h = 0). f: top (v = 0, h = ?).
  2. Energies. Ei = ½mv₀² + 0; Ef = 0 + mgh.
  3. Conserve. ½·36 = 9.8h → h = 18/9.8 ≈ 1.84 m.
Common mistake: h = v²/g = 3.67 m — forgetting the ½ again (it's v²/(2g)). The half haunts every energy equation.
Your turn — launched at 4 m/s. Max height?

Answer: ≈ 0.82 m. h = 16/(2·9.8) = 8/9.8 ≈ 0.82 m.

Example 3 — spring launch: m = 0.5 kg, k = 200 N/m, x = 0.2 m

  1. Snapshots. i: spring compressed (rest, Us = ½·200·0.04 = 4 J). f: spring relaxed, mass moving at v.
  2. Conserve. 4 = ½ × 0.5 × v² = 0.25v².
  3. Solve. v² = 16, v = 4 m/s.
Common mistake: mixing Ug in when heights don't change — or forgetting that at max compression the mass is instantaneously at rest (KEi = 0).
Your turn — k = 100 N/m, x = 0.3 m, m = 0.5 kg. Launch speed?

Answer: ≈ 4.24 m/s. Us = 4.5 J; v² = 2·4.5/0.5 = 18; v ≈ 4.24 m/s.

Example 4 — roller coaster: hill h₁ = 30 m to h₂ = 10 m, starts from rest

  1. Snapshots. i: top of first hill (rest, h = 30). f: second hilltop (h = 10, speed v).
  2. Energies (per unit mass). Ei/m = 9.8·30 = 294; Ef/m = v²/2 + 9.8·10 = v²/2 + 98.
  3. Conserve. v²/2 = 196, v² = 392, v ≈ 19.8 m/s.
  4. Shortcut seen: v = √(2gΔh) with Δh = 20 m — only the drop matters.
Before moving on: would a heavier car be faster at the bottom? The mass canceled in every step above — predict, then check against Example 1's note.
Common mistake: using h = 30 m in √(2gh) instead of the drop Δh = 20 m. Energy cares about the change between your two snapshots, not the absolute height.
Your turn — from rest at 20 m to a point at 5 m. v = ?

Answer: ≈ 17.1 m/s. Δh = 15 m; v = √(2·9.8·15) = √294 ≈ 17.1 m/s.

Memorization tips

  • Say it aloud: “E initial equals E final — kinetic plus potential, both snapshots.”
  • Two snapshots: every conservation problem is 'state A' vs 'state B'. Write both E's fully, then cross out the zeros.
  • Zero the easy terms: at rest → KE = 0; at reference height → Ug = 0; relaxed spring → Us = 0. Most problems collapse to two terms.
  • Mass often cancels. Pure gravity coasts give v = √(2gΔh) — no m. If your answer depends on mass where it shouldn't, recheck.
  • Friction is the leak: Ef = Ei + Wfriction. Never conserve across a rough patch without the leak term.
  • Only the drop matters: for coasters and slides, v = √(2gΔh) — the vertical drop between snapshots, not the path length.

Final challenge

Five mixed questions — snapshots, leaks, and mass cancellation. Score 5/5 and conservation is yours.

← Back to the Physics I formula sheet

How to learn a formula here

  1. Read each section in order — every section ends with a short quiz. Take it before moving on; the questions test exactly what you just read.
  2. Work the examples with the answers covered, then uncover one step at a time and compare.
  3. Finish with the final challenge — five mixed questions including the classic traps.
  4. Retake what you miss — every quiz reshuffles each attempt, and every answer explains itself.

Frequently asked questions

When is mechanical energy actually conserved?

When only conservative forces (gravity, ideal springs) do work. Friction, drag, engines, or pushes add/remove mechanical energy via Wnoncons = ΔE. 'Smooth', 'frictionless', and 'ignore air resistance' are the magic words.

Is energy conservation the same as the work-energy theorem?

It's the theorem reorganized: Wnet = ΔKE with conservative work rewritten as −ΔU gives Δ(KE+U) = Wnoncons. Conservation is the Wnoncons = 0 special case.

Why does mass cancel in free fall?

Both sides scale with m: mgh = ½mv² → gh = v²/2. Gravity pulls harder on bigger masses but there's proportionally more mass to accelerate — the two effects cancel exactly.

Can I mix gravitational and spring potential in one problem?

Yes — U is the sum of all potentials: E = KE + mgh + ½kx². A mass bouncing on a spring uses all three terms across its snapshots.

Does conservation work with nonconservative forces present?

Not as Ei = Ef — but the grown-up form Wnoncons = ΔE still works. Friction's (negative) work is exactly the mechanical energy lost to heat.

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