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

KE = ½mv²

Say it: “kinetic energy equals one-half mass times velocity squared”

Kinetic energy

The energy of motion itself — why doubling your speed quadruples the crash, and where the ½ comes from.

m is mass in kilograms, v the speed in m/s (the square kills the sign — kinetic energy is never negative). KE is in joules.

Before this lesson: Work (constant force)

Where it comes from

Drop a 1 kg rock from 1 m and it lands with a thud. Drop it from 4 m and it hits four times as hard — not twice. The energy of motion grows with the square of speed, and that squaring has consequences: highway braking distances quadruple when speed doubles.

Before reading on: two identical cars, one at 20 m/s and one at 40 m/s. How many times more kinetic energy does the faster car carry — 2×, 3×, or 4×? Commit to a number.

The answer is 4×. And the ½ is not decoration either — it falls straight out of the work it takes to get something moving. Push a mass m from rest with a constant net force F = ma over a distance d:

W
=
Fd = (ma)d
Work by the net force — the energy delivered to the mass.
=
m · (v² − v₀²)/2
Time-free kinematics: v² = v₀² + 2ad, so ad = (v² − v₀²)/2. (See the time-free equation.)
=
½mv² − ½mv₀²
Starting from rest (v₀ = 0), the work delivered is ½mv². We define kinetic energy as this quantity. ∎
KE = ½mv²the work needed to accelerate mass m from rest to speed vSay it: “kinetic energy equals one-half m v squared”

Derivation

The derivation above is the whole story, but let's see exactly where the ½ is born — it is the same ½ as the triangle area in variable-force work:

a
=
(v − 0)/t = v/t
Step 1 — constant acceleration. From rest to speed v in time t, the (constant) acceleration is v/t.
d
=
vavg·t = (v/2)·t
Step 2 — distance covered. Average speed while ramping 0 → v is v/2. This is the ½: speed grows linearly, so its average is half the final.
W
=
F·d = (m·v/t)·(v·t/2)
Step 3 — work. Force × distance. The t cancels — time was just the middleman.
=
½mv²
Step 4 — kinetic energy. The delivered work, defined as KE. The square comes from v appearing twice (in F via a, and in d via vavg); the ½ from averaging the ramp. ∎

How to use it

The procedure, every time:

  1. Use speed, not velocity. v² kills the sign — a car at −20 m/s has the same KE as one at +20 m/s. Direction is irrelevant to kinetic energy.
  2. Square first, then halve, then multiply by m. ½mv² — squaring after halving gives the wrong answer. Order: v² → × m → ÷ 2.
  3. Respect the square. Doubling v quadruples KE; tripling gives 9×. When a problem changes speed, scale by the square of the ratio.
  4. Watch the units. kg · (m/s)² = J. Grams must become kilograms before you compute.
  5. KE is never negative. m > 0 and v² ≥ 0, always. A negative KE means an arithmetic error — full stop.
Common mistake: writing KE = mv² (dropping the ½) or KE = ½mv (forgetting to square). Both are unit-correct-looking but wrong — the derivation shows the ½ comes from averaging and the square from v appearing twice.

Worked examples

Four problems, easiest first. Feel the square in each one.

Example 1 — basic: m = 2 kg, v = 3 m/s

  1. Square the speed. v² = 9.
  2. Assemble. KE = ½ × 2 × 9 = 9 J.
  3. Sanity check. Positive, joules, small numbers → small energy. ✓
Common mistake: ½ × 2 × 3 = 3 J — forgetting to square v. The square is the most-dropped operation in the chapter.
Your turn — m = 4 kg, v = 5 m/s. KE = ?

Answer: 50 J. ½ × 4 × 25 = 50 J.

Example 2 — the square bites: 1000 kg car at 20 m/s vs 40 m/s

  1. At 20 m/s. KE = ½ × 1000 × 400 = 200,000 J.
  2. At 40 m/s. KE = ½ × 1000 × 1600 = 800,000 J.
  3. Ratio. 800,000 / 200,000 = 4. Twice the speed, four times the energy — and roughly four times the braking distance.
Common mistake: guessing 2× (“twice as fast, twice the energy”). Linear intuition fails wherever a square lives — always scale by the square of the speed ratio.
Your turn — a 1500 kg car at 10 m/s. KE = ?

Answer: 75,000 J. ½ × 1500 × 100 = 75,000 J.

Example 3 — solve for speed: KE = 200 J, m = 8 kg

  1. Rearrange. v² = 2·KE/m.
  2. Compute. v² = 400/8 = 50, so v = √50 ≈ 7.07 m/s.
  3. Check. ½ × 8 × 50 = 200 J. ✓ (Speed is ±7.07 m/s — KE can't tell you the direction.)
Common mistake: forgetting the square root and answering v = 50 m/s. The formula gives v²; the question asks for v. Units catch it: m²/s² ≠ m/s.
Your turn — KE = 100 J, m = 2 kg. v = ?

Answer: 10 m/s. v² = 200/2 = 100, v = 10 m/s.

Example 4 — real object: baseball, m = 0.145 kg at 40 m/s

  1. Convert. 0.145 kg is already in kilograms — no conversion needed. (If it were 145 g, divide by 1000 first.)
  2. Compute. KE = ½ × 0.145 × 1600 = 0.0725 × 1600 = 116 J.
  3. Feel it. 116 J is roughly the energy of a hard slap — plausible for a fastball.
Common mistake: using 145 (grams) directly: ½ × 145 × 1600 = 116,000 J — a thousand times too big. Grams-to-kilograms is the silent killer of KE problems.
Your turn — m = 0.5 kg, v = 6 m/s. KE = ?

Answer: 9 J. ½ × 0.5 × 36 = 9 J.

Memorization tips

  • Say it aloud: “kinetic energy equals one-half m v squared.” Say “squared” with emphasis — it's the part brains drop.
  • The doubling rule: 2× speed → 4× energy, 3× → 9×. This one fact answers a whole family of ratio questions.
  • Speed, not velocity: the square erases direction. If a problem gives v = −15 m/s, just use 15.
  • Grams first: convert g → kg before anything else. A 1000× error hides in every gram.
  • Never negative: KE ≥ 0 always. A minus sign means you misplaced something — hunt it down.
  • The ½ origin: it's the average of a 0→v ramp. If you blank on the formula, re-derive: W = (mv/t)(vt/2).

Final challenge

Five mixed questions — the square, the half, and unit traps. Score 5/5 and kinetic energy 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

Why is kinetic energy ½mv² and not just mv²?

The ½ comes from averaging: accelerating 0 → v at constant a, the average speed is v/2, so the distance covered is (v/2)t while the force is m(v/t). Multiply: (mv/t)(vt/2) = ½mv². It's the same ½ as the triangle area under a ramping force.

Does direction matter for kinetic energy?

No. v² erases the sign, so +20 m/s and −20 m/s give identical KE. Kinetic energy measures how much motion there is, not where it's headed.

Can kinetic energy be negative?

Never. Mass is positive and v² is non-negative, so KE ≥ 0 always. A negative result is an arithmetic error, not physics.

Why does doubling speed quadruple the energy?

Because v appears twice in the derivation — once in the force (via acceleration) and once in the distance (via average speed). Each doubling contributes a factor of 2: 2 × 2 = 4.

How is kinetic energy related to work?

Net work becomes kinetic energy: Wnet = ΔKE. That's the work-energy theorem — kinetic energy is literally the running total of net work done on the object.

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