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

Wnet = ΔKE = KEf − KEi

Say it: “the net work done on an object equals its change in kinetic energy”

The work–energy theorem

The master equation of mechanics: every push, pull, and drag is bookkept as a change in motion — no forces to juggle individually.

Wnet is the work of the net force (add every force's work, signs included), ΔKE the final minus initial kinetic energy. Both in joules.

Before this lesson: Work (constant force), Kinetic energy

Where it comes from

A crate slides with friction pulling back while you push forward. Newton's laws demand you track every force, find the net, then grind through kinematics. There is a better ledger: add up the work of all forces, and that total equals the change in kinetic energy — one equation, no acceleration needed.

Before reading on: you push with +200 J of work while friction does −60 J. The crate started at rest. Is its final KE 200 J, 140 J, 260 J, or 60 J?

140 J. Work is a team total: +200 − 60 = 140 J of net work becomes 140 J of kinetic energy. Forces that oppose steal from the total before it becomes motion.

Wnet = ΔKEevery force's work, signed, added up — equals the change in motionSay it: “the net work equals the change in kinetic energy”

This is why the theorem is a shortcut: kinematics needs acceleration and time; energy needs only the total work and the two speeds. When a problem gives forces and distances but no time, reach for energy.

Derivation

Newton's second law plus the time-free kinematic equation — nothing else:

Wnet
=
Fnet·d
Step 1 — definition. Net work is the net force's work over the displacement. (Equivalently, the sum of each force's work.)
=
(ma)·d
Step 2 — Newton. Fnet = ma. Now it's a kinematics problem in disguise.
=
m · (v² − v₀²)/2
Step 3 — eliminate a and d. From v² = v₀² + 2ad (see time-free kinematics): ad = (v² − v₀²)/2.
=
½mv² − ½mv₀² = KEf − KEi
Step 4 — recognize KE. Each term is a kinetic energy. Net work is the change in kinetic energy. ∎

Why “net” is load-bearing: Step 2 used Fnet = ma. A single force's work (say, just your push) equals ΔKE only if it's the only force. Miss a friction term and the ledger won't balance.

How to use it

The procedure, every time:

  1. List every force doing work. Push, pull, friction, gravity (if height changes), spring — each contributes signed work. Perpendicular forces contribute zero.
  2. Sum with signs. Wnet = W₁ + W₂ + … Opposing forces enter negative. This sum is the whole left side.
  3. Write the KE change. ΔKE = ½mvf² − ½mvi². If it starts from rest, KEi = 0.
  4. Set equal, solve. One equation, one unknown — usually a speed, a distance, or a force.
  5. Sanity-check the sign. Positive net work → speeds up. Negative → slows down. If the story says “braking” and your Wnet is positive, recheck.

When energy beats forces

Reach for Wnet = ΔKE when the problem gives forces and distances but no time — braking distances, stopping forces, speeds after a push. Stick with Newton's laws + kinematics when it asks for time or acceleration directly.

Common mistake: setting one force's work equal to ΔKE while ignoring friction. The theorem balances only when every force's work is on the left side.

Worked examples

Four problems, easiest first. Keep the ledger balanced: every force accounted for.

Example 1 — push from rest: Fnet = 12 N, d = 6 m, m = 4 kg

  1. Net work. Wnet = 12 × 6 = 72 J.
  2. Energy equation. 72 = ½ × 4 × v² − 0, so 2v² = 72.
  3. Solve. v² = 36, v = 6 m/s.
  4. Check. Positive work → sped up from rest. ✓
Common mistake: v = 36 m/s — forgetting the square root and the ½m. Unpack ½mv² = 72 one operation at a time.
Your turn — Fnet = 20 N, d = 10 m, m = 5 kg, from rest. v = ?

Answer: ≈ 8.94 m/s. W = 200 J; v² = 2·200/5 = 80; v = √80 ≈ 8.94 m/s.

Example 2 — braking distance: 1200 kg car at 25 m/s, braking force −3000 N

  1. Energy to kill. KEi = ½ × 1200 × 625 = 375,000 J; KEf = 0.
  2. Theorem. Wnet = −3000 × d = 0 − 375,000 = −375,000 J.
  3. Solve. d = 375,000 / 3000 = 125 m.
  4. Check. Negative work drained all the KE — the car stops. ✓
Common mistake: dropping the minus on the braking force and getting d = −125 m. A negative distance is the equation telling you the signs are inconsistent — the work must be negative here.
Your turn — 1000 kg car at 20 m/s, braking force −2500 N. Stopping distance?

Answer: 80 m. KE = ½×1000×400 = 200,000 J; d = 200,000/2500 = 80 m.

Example 3 — already moving: m = 3 kg, vi = 4 m/s, Wnet = 96 J

  1. Initial KE. KEi = ½ × 3 × 16 = 24 J.
  2. Add the work. KEf = 24 + 96 = 120 J.
  3. Solve. vf² = 2 × 120 / 3 = 80, vf ≈ 8.94 m/s.
Common mistake: solving ½mv² = 96 directly (v ≈ 8 m/s) — forgetting the object already had 24 J. ΔKE means change: final minus initial.
Your turn — m = 2 kg, vi = 3 m/s, Wnet = 91 J. vf = ?

Answer: 10 m/s. KEi = 9 J; KEf = 100 J; vf² = 100; vf = 10 m/s.

Example 4 — friction wins slowly: 0.2 kg puck at 8 m/s, friction −1 N over 4 m

  1. Initial KE. KEi = ½ × 0.2 × 64 = 6.4 J.
  2. Friction's work. W = −1 × 4 = −4 J.
  3. Final KE. KEf = 6.4 − 4 = 2.4 J; vf² = 2 × 2.4 / 0.2 = 24, vf ≈ 4.9 m/s.
  4. Check. KEf > 0 — the puck is still moving, just slower. If KEf had gone negative, the puck would have stopped before 4 m.
Common mistake: a negative KEf accepted without question. KE can't go negative — it means the object stopped mid-distance. Cap it at zero and solve for the stopping point instead.
Your turn — 0.5 kg block at 6 m/s, friction −2 N over 2 m. vf = ?

Answer: ≈ 4.47 m/s. KEi = 9 J; W = −4 J; KEf = 5 J; vf² = 20.

Memorization tips

  • Say it aloud: “the net work equals the change in kinetic energy.” Stress net — it's the word that breaks ledgers.
  • The ledger habit: draw a tiny table — one row per force, signed work in the right column, sum at the bottom. Missing rows are missing energy.
  • Rest is zero: “starts from rest” means KEi = 0; “comes to rest” means KEf = 0. Translate the English before computing.
  • Signs tell the story: Wnet > 0 speeds up, Wnet < 0 slows down. If the story and the sign disagree, the setup is wrong.
  • Energy vs forces: forces + distances + speeds → energy. Time or acceleration asked → Newton + kinematics. Pick the tool by what's given.
  • Negative KEf is a stop sign: it means the object halted before the full distance. Set KEf = 0 and solve for where it stopped.

Final challenge

Five mixed questions — ledgers, signs, and the stopping trap. Score 5/5 and the theorem 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

Is it W of each force, or W of the net force?

Both give the same number: Σ(Fid cosθi) = Fnetd. Summing each force's work is usually easier because you rarely need the net force vector itself.

Can I use the theorem when forces vary?

Yes — compute each force's work with the integral (or area) and sum. The theorem Wnet = ΔKE holds for any forces; only the work computation gets harder.

Why doesn't the theorem mention time?

Because work accumulates over distance, not time. Two identical pushes over the same distance give the same ΔKE whether they took one second or one hour — power differs, energy doesn't.

What if the object moves in a circle?

The theorem still holds: ΔKE over the trip equals the net work along the path. For uniform circular motion the net (centripetal) force is perpendicular to motion, so Wnet = 0 and speed never changes.

How does this connect to conservation of energy?

When the only forces doing work are conservative (gravity, springs), their work can be rewritten as −ΔU, and Wnet = ΔKE becomes KE + U = constant. See mechanical energy conserved.

More from the codex

Live

Physics I formula sheet

All the mechanics formulas — kinematics, forces, energy, momentum — printable and quiz-ready.

Open sheet → Live

Formula Sheet Builder

Mix and match any sections into your own printable sheet.

Open tool → Live

Prompt Simulator

Practice prompt engineering with deterministic scoring.

Open tool →