Physics I: Mechanics › Kinematics › full formula sheet

ac = v²/rSay it: centripetal acceleration equals speed squared over radius — always pointing toward the center

Centripetal acceleration

You can accelerate without speeding up or slowing down — changing direction is acceleration too, and it always points inward.

ac is centripetal (“center-seeking”) acceleration (m/s²), v the (constant) speed, r the circle’s radius. Direction: toward the center, perpendicular to v. Uniform circular motion only.

Before this lesson: Velocity

Where it comes from

A car takes a curve at a steady 20 m/s — speedometer frozen, yet the passengers feel shoved sideways. A satellite orbits at constant speed, yet it is perpetually falling. Both are accelerating, because acceleration is any change in velocity — and velocity includes direction. Turning the steering wheel is accelerating, even with the cruise control on.

Before reading on: the car doubles its speed through the same curve. Does the inward acceleration double — or do something steeper? Guess the scaling before the formula.
ac = v²/rSay it: centripetal acceleration equals speed squared over radiusdouble the speed → 4× the acceleration — the square strikes again

Two levers: faster (v² — violent scaling) and tighter (1/r — sharper curves bite harder). The direction is the part everyone gets wrong at first: it points inward, toward the center — never outward.

Derivation

Track the velocity vector as the object moves along the arc. In time Δt it sweeps angle Δθ; the velocity vector rotates by the same Δθ. Two similar isosceles triangles — one in space, one in velocity — do the rest:

Δs / r
=
Δv / v
Step 1 — similar triangles. The position triangle (sides r, r, arc Δs) and the velocity triangle (sides v, v, Δv) share the vertex angle Δθ, so corresponding sides are proportional: Δs/r = Δv/v.
Δv
=
v · Δs / r
Step 2 — solve for Δv. Cross-multiply. The velocity change grows with the distance traveled along the arc.
ac
=
Δv/Δt = v · (Δs/Δt) / r = v²/r
Step 3 — divide by Δt. Δs/Δt is the speed v. Take Δt → 0 and Δv points toward the center (the chord closes inward). ∎

Why inward? The velocity vectors at two nearby points differ by a vector pointing from the later tip back toward the earlier — which, as Δt → 0, aims at the circle’s center. The “outward fling” you feel in a turning car is your body’s inertia resisting the inward acceleration — a frame effect, not a force on the car.

How to use it

The procedure, every time:

  1. Confirm uniform circular motion. Constant speed, fixed radius. (Speeding up while turning adds a tangential component — see the tangential-quantities lesson.)
  2. Square the speed, divide by the radius. ac = v²/r.
  3. Point it at the center. Perpendicular to the velocity, inward. Draw the arrow before you compute — the direction is half the marks.
  4. Units check: (m/s)²/m = m/s² ✓.

The angular form

ac = ω²rwith ω = v/r — same formula, spinning viewpointSay it: centripetal acceleration equals omega squared times radius

A merry-go-round at ω = 0.5 rad/s, r = 40 m: ac = 0.25 · 40 = 10 m/s² — about 1 g, inward.

Common mistake: pointing ac outward (“centrifugal”). Centrifugal is the apparent effect in the rotating frame; the actual acceleration of the object is inward. On a free-body diagram in an inertial frame, the arrow goes to the center.
Common mistake: ac = v/r or ac = v²r — misplacing the radius. Units referee: v/r is 1/s (not acceleration); v²r is m³/s² (not acceleration). Only v²/r gives m/s².

Worked examples

Four problems, easiest first. In each one, read every step — the why of each move is the lesson.

Example 1 — the basic move: car at 20 m/s, curve r = 80 m

  1. List givens. v = 20 m/s (constant), r = 80 m.
  2. Square and divide. ac = 20²/80 = 400/80 = 5 m/s², toward the curve’s center.
  3. Feel it. 5/9.8 ≈ 0.5 g sideways — a firm but comfortable highway curve.
  4. Units. (m/s)²/m = m/s² ✓
Common mistake: ac = 20/80 = 0.25 m/s² — forgetting to square. At highway speeds the square is everything: 20² = 400, not 20.
Your turn — car at 15 m/s through r = 50 m. Centripetal acceleration?

Answer: 4.5 m/s² inward. 15²/50 = 225/50 = 4.5 m/s², toward the center.

Example 2 — from the period: merry-go-round, r = 4 m, T = 5 s

  1. Speed from the period. One lap = 2πr = 8π m in 5 s: v = 8π/5 ≈ 5.027 m/s.
  2. Square and divide. ac = 5.027²/4 = 25.27/4 ≈ 6.32 m/s², toward the hub.
  3. Feel it. ~0.64 g — a fast playground spinner.
  4. Cross-check via ω: ω = 2π/5 = 1.2566 rad/s; ω²r = 1.579·4 = 6.32 m/s² ✓
Common mistake: v = r/T = 4/5 = 0.8 m/s — forgetting the 2π. One lap is the full circumference 2πr, not the radius.
Your turn — r = 3 m, T = 6 s. Centripetal acceleration?

Answer: 3.29 m/s² inward. v = 2π · 3/6 = π ≈ 3.142 m/s; a = 9.870/3 ≈ 3.29 m/s².

Example 3 — the angular form: ω = 0.5 rad/s, r = 40 m

  1. List givens. ω = 0.5 rad/s, r = 40 m.
  2. Apply ac = ω²r. = 0.25 · 40 = 10 m/s², inward.
  3. Cross-check via v: v = ωr = 20 m/s; v²/r = 400/40 = 10 m/s² ✓.
  4. When to use which: given ω → ω²r; given v → v²/r. Same physics, pick the shorter arithmetic.
Common mistake: ac = ωr = 20 m/s² — forgetting to square ω. The square is non-negotiable in every form of this formula.
Your turn — ω = 1.2 rad/s, r = 5 m. Centripetal acceleration?

Answer: 7.2 m/s² inward. 1.2² · 5 = 1.44 · 5 = 7.2 m/s².

Before reading on: a car threads a curve of r = 25 m at 10 m/s, then takes it again at 20 m/s. The acceleration at 10 m/s is 4 m/s² — predict the 20 m/s value before computing. Linear or steeper?

Example 4 — the scaling law: 10→20 m/s at r = 25 m

  1. At 10 m/s: ac = 100/25 = 4 m/s².
  2. At 20 m/s: ac = 400/25 = 16 m/s².
  3. Ratio: 4×. Doubling speed quadruples the inward acceleration — the v² at work.
  4. Why curves have speed limits. 16 m/s² ≈ 1.6 g — beyond what tires typically hold. Speed limits on curves aren’t about the straightaway; they’re about v²/r.
Common mistake: “twice as fast, twice the acceleration.” The square law punishes speed: 2× speed needs 4× the grip. This is the single most safety-relevant scaling in the unit.
Your turn — r = 10 m; speed 6→12 m/s. Both accelerations? Ratio?

Answer: 3.6 m/s² → 14.4 m/s² — ratio 4. 36/10 = 3.6; 144/10 = 14.4.

Memorization tips

  • Say it aloud: “v-squared over r, toward the center.” Formula and direction in one breath.
  • Constant speed ≠ no acceleration. Turning changes velocity’s direction — that is acceleration. Cruise control doesn’t cancel it.
  • Inward, always. The arrow points at the center. “Centrifugal” outward is the feeling, not the physics (in an inertial frame).
  • Square-law reflex: ×2 speed → ×4 ac; ×2 radius → ×½ ac. Wide and slow is gentle; tight and fast is violent.
  • Two forms, one formula: v²/r when you know speed, ω²r when you know spin rate. Convert with v = ωr.
  • Units referee: only v²/r yields m/s². If your expression doesn’t, the radius is misplaced.

Final challenge

Five mixed questions — basics, applications, and the traps, all in one. Score 5/5 and Centripetal acceleration 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 there acceleration if the speed is constant?

Because velocity includes direction. In circular motion the velocity vector keeps rotating, so it keeps changing — and changing velocity is the definition of acceleration. The speedometer can sit still while the accelerometers scream.

Which way does centripetal acceleration point?

Toward the center of the circle, perpendicular to the velocity. 'Centripetal' literally means 'center-seeking'. The outward fling you feel is inertia (your body resisting the inward turn), not the acceleration's direction.

What is the difference between centripetal and centrifugal?

Centripetal is the real inward acceleration of an object moving in a circle (a = v^2/r). 'Centrifugal' is the apparent outward effect felt inside the rotating frame — useful as a feeling, but not a force on the object in an inertial frame.

Why does doubling the speed quadruple the acceleration?

Because a_c = v^2/r: (2v)^2 = 4v^2. Faster motion both covers the curve's angle quicker and carries more velocity to redirect — the two effects multiply. This is why curves have speed limits.

Does centripetal acceleration do work?

No — it's always perpendicular to the velocity, so F dot v = 0. That's why speed stays constant in uniform circular motion: the inward force bends the path without adding energy.

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