Physics I: Mechanics › Kinematics › full formula sheet

ā = Δv / ΔtSay it: average acceleration equals change in velocity over change in time

Average acceleration

How fast velocity itself is changing — and the one equation that turns a stopwatch reading into a velocity prediction.

The bar on ā means average over the whole interval. Δv = v − v₀ (final minus initial, sign included), Δt the elapsed time. Units: m/s² — “meters per second, per second.”

Where it comes from

A speedometer answers one question: how fast right now? But a different question shows up constantly — how fast is the velocity itself changing? A sports car jumping 0→100 km/h in 4 seconds feels violent; a bus doing it in 40 seconds feels lazy. Same change in velocity, different rate of change. That rate has a name:

Before reading on: a car goes from rest to 30 m/s in 5 s. Guess its average acceleration — then check it against the formula below.
ā = Δv / Δt = (30 − 0) / 5 = 6 m/s²Say it: average acceleration equals change in velocity over change in time6 m/s² means “6 more meters per second, every second”

Why this matters beyond word problems: acceleration is what forces produce (F = mā). The crash dummy, the rocket launch, the sprinter’s start — every one is a story about Δv/Δt. The average here is deliberate: it asks for no details about how the push was delivered, only what it accomplished. Watch what that simplicity buys us — a formula that predicts the future.

Derivation

There is almost nothing to derive — and that is the point. The definition is the formula; rearranging it gives the first equation of constant-acceleration motion. Watch:

ā
=
Δv / Δt = (v − v₀) / t
Step 1 — the definition. Average acceleration is change in velocity (Δv = v − v₀) divided by elapsed time (Δt = t). If the motion starts at t = 0, Δt is just t.
āt
=
v − v₀
Step 2 — multiply by t. Clearing the denominator moves the time interval to the left side.
v
=
v₀ + āt
Step 3 — add v₀. Final velocity = starting velocity + everything acceleration added along the way. When acceleration is constant, ā = a and this is the famous v = v₀ + at.

Average vs. instantaneous: if the acceleration changes during the interval (a jerky driver), ā is only the net result. The pages on velocity and free fall meet the instantaneous version — where the interval shrinks to a single instant.

How to use it

The procedure, every time:

  1. Pick a positive direction and write both velocities with signs. Moving “backwards” is negative velocity, not zero — the sign is the physics.
  2. Compute Δv = v − v₀ (final minus initial, always in that order).
  3. Divide by Δt. The answer is in m/s².
  4. Sanity-check the sign. Negative ā means velocity decreased in your positive direction — usually “slowing down,” but read the geometry, not the word.

Flipping it: predict the future

v = v₀ + ātSay it: final velocity equals initial velocity plus average acceleration times timegiven the start, the push, and the time — know the end

Know any three of v, v₀, ā, t and the fourth is yours. A train holds ā = 1.5 m/s² for 20 s from rest: v = 0 + 1.5·20 = 30 m/s.

Common mistake: dividing distance by time — that gives average speed (a different animal). Acceleration divides a change in velocity by time. “100 m in 10 s” → 10 m/s is speed, not acceleration.
Common mistake: writing Δv as v₀ − v (initial minus final). The delta is always final minus initial: it answers “how much did it change to?”

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: a car, 0→28 m/s in 7 s

  1. List givens. v₀ = 0 m/s, v = 28 m/s, Δt = 7 s. Positive direction = the car’s motion.
  2. Δv = v − v₀ = 28 − 0 = 28 m/s.
  3. Divide. ā = 28 / 7 = 4 m/s².
  4. Read it back. Every second, the car gains 4 m/s of velocity. After 7 s: 4·7 = 28 m/s ✓
Common mistake: writing ā = 28/7 = 4 but with units “m/s” — a unit check catches it instantly. You divided velocity (m/s) by time (s), so the unit is (m/s)/s = m/s².
Your turn — a car goes 0→24 m/s in 6 s. Average acceleration?

Answer: 4 m/s². Δv = 24 − 0 = 24 m/s; ā = 24/6 = 4 m/s².

Example 2 — slowing down: 10→4 m/s in 3 s

  1. List givens. v₀ = 10 m/s, v = 4 m/s, Δt = 3 s.
  2. Δv = 4 − 10 = −6 m/s. The change is negative — the velocity shrank.
  3. Divide. ā = −6 / 3 = −2 m/s².
  4. Interpret. The minus says “losing 2 m/s of forward velocity every second.” Check: 10 − 2·3 = 4 m/s ✓
Common mistake: refusing the negative answer (“acceleration can’t be negative”). It can — the sign only says which way the velocity is changing relative to your chosen positive direction.
Your turn — a skater goes 12→3 m/s in 3 s. Average acceleration?

Answer: −3 m/s². Δv = 3 − 12 = −9 m/s; ā = −9/3 = −3 m/s² — slowing in the positive direction.

Example 3 — predict the future: train at ā = 1.5 m/s² for 20 s

  1. List givens. v₀ = 0 m/s, ā = 1.5 m/s², t = 20 s.
  2. Use v = v₀ + āt. = 0 + 1.5·20.
  3. Compute. = 30 m/s (about 108 km/h).
  4. Check the units. (m/s²)·s = m/s ✓ — and v₀ + āt = 0 means the whole velocity came from the acceleration, as it should.
Common mistake: multiplying 1.5 × 20 = 30 and reporting “30 m/s²”. The product āt has the unit of velocity — always confirm the unit matches what you solved for.
Your turn — a dragster accelerates from rest at 2.5 m/s² for 12 s. Final velocity?

Answer: 30 m/s. v = 0 + 2.5 × 12 = 30 m/s.

Example 4 — hard braking: 22→0 m/s in 4.4 s

  1. List givens. v₀ = 22 m/s, v = 0 m/s, Δt = 4.4 s.
  2. Δv = 0 − 22 = −22 m/s.
  3. Divide. ā = −22 / 4.4 = −5 m/s² — about half a g of braking.
  4. Reality check. 5 m/s² × 4.4 s = 22 m/s of velocity removed ✓.
Common mistake: setting v = 22 and v₀ = 0 (reading “22 to 0” backwards). Then Δv = +22 and the car apparently accelerates while stopping — the order final-minus-initial is load-bearing.
Your turn — a cyclist brakes 18→0 m/s in 3 s. Average acceleration?

Answer: −6 m/s². Δv = 0 − 18 = −18 m/s; ā = −18/3 = −6 m/s².

Memorization tips

  • Say it aloud: “average acceleration equals change in velocity over change in time.” The sentence and the formula are the same object.
  • Unit sanity: (m/s) ÷ s = m/s². If your answer’s units aren’t m/s², you divided the wrong things.
  • Negative is information, not error: it means velocity is decreasing along your positive direction. Always draw the arrow first.
  • Average ≠ instantaneous: ā tells you the net change, not how it happened. A rocket that throttles up and down can still have ā = 4 m/s².
  • The rearranged form v = v₀ + āt is a prediction machine: start + push × time = finish. Learn to read it as a sentence.
  • When v₀ = 0 (starts from rest), the formula collapses to ā = v/t — the simplest possible version.

Final challenge

Five mixed questions — basics, applications, and the traps, all in one. Score 5/5 and Average 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

What is average acceleration?

Average acceleration is the rate at which velocity changes: a-bar = delta-v / delta-t, in m/s^2. It tells you how much velocity is gained or lost per second over a time interval.

How is acceleration different from velocity?

Velocity is how fast position changes (m/s); acceleration is how fast velocity changes (m/s^2). A car can have high velocity and zero acceleration (cruising), or zero velocity and large acceleration (the instant a dragster launches).

Can average acceleration be negative?

Yes — the sign says which way velocity is changing relative to your chosen positive direction. Braking while moving forward gives negative average acceleration.

When is average acceleration different from instantaneous acceleration?

When the acceleration varies during the interval. Average acceleration is the net result: a-bar = (v - v0)/t. If a jerky driver speeds up and slows down but ends at the same pace, a-bar is zero even though instantaneous acceleration was nonzero throughout.

What is the SI unit of acceleration and why?

Meters per second squared, m/s^2. Velocity (m/s) divided by time (s) gives (m/s)/s = m/s^2 — read it as 'meters per second, per second': the velocity gained each second.

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