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Average Acceleration Formula

Kinematics formulasVectorClass 11Updated 1 Nov 2026

aavg=ΔvΔt=vf−vitf−tia_{avg} = \frac{\Delta v}{\Delta t} = \frac{v_f - v_i}{t_f - t_i}
SI unit
m/s²
Dimensions
[M⁰ L¹ T⁻²]
where
  • a_avg average acceleration m/s²
  • v_f final velocity m/s
  • v_i initial velocity m/s
  • Δt time interval s

The average acceleration formula is aavg=ΔvΔta_{avg} = \frac{\Delta v}{\Delta t}, the change in velocity over a time interval divided by the length of that interval. It looks only at the velocity at the start and at the end. Whatever happened in between, smooth or jerky, is folded into one number.

Indian books often write it as (v − u)/t, with u for the initial velocity and v for the final one. US books write v_f and v_i, and this page uses that spelling. When the acceleration is constant, the average and the instantaneous values are the same, which is why many school problems never separate them. The distinction matters as soon as the acceleration varies, or the direction of motion changes.

Quick facts

  • Formula

    aavg=vf−viΔta_{avg} = \frac{v_f - v_i}{\Delta t}

  • SI unit

    m/s²

  • Dimensional formula

    [M⁰ L¹ T⁻²]

  • Type

    Vector, in the direction of Δv

  • Graph meaning

    Slope of the chord on a velocity time graph

  • Equals instantaneous value

    Only when acceleration is constant

Formula of average acceleration, symbol by symbol

Terms in a_avg = Δv / Δt
SymbolMeaningSI unit
aavga_{avg}Average acceleration over the intervalm/s²
vfv_fVelocity when the clock stopsm/s
viv_iVelocity when the clock startsm/s
Δt\Delta tLength of the interval, end time minus start times

Since velocity is a vector, Δv\Delta v is a vector subtraction. In one dimension that just means keeping signs. In two dimensions, such as motion around a bend, you subtract arrows, and the result can be large even if the speed has not changed at all.

Average acceleration formula physics examples

Example 1: a sprinter leaving the blocks

A sprinter goes from rest to 9.0 m/s in 3.0 s. Find the average acceleration.

aavg=9.0−03.0=3.0 m/s2a_{avg} = \frac{9.0 - 0}{3.0} = 3.0\ \text{m/s}^2

Example 2: slowing down, km/h to m/s

A car slows from 54 km/h to 18 km/h in 4.0 s. Find the average acceleration.

In SI units the speeds are 15 m/s (from 54 km/h) and 5.0 m/s (from 18 km/h).

aavg=5.0−154.0=−2.5 m/s2a_{avg} = \frac{5.0 - 15}{4.0} = -2.5\ \text{m/s}^2

The car has a retardation of 2.5 m/s².

Example 3: a ball bouncing off a wall

A tennis ball hits a wall at 20 m/s and rebounds at 15 m/s along the same line. Contact lasts 0.050 s. Find the average acceleration.

Take toward the wall as positive: v_i = +20 m/s, v_f = −15 m/s.

aavg=−15−200.050=−350.050=−700 m/s2a_{avg} = \frac{-15 - 20}{0.050} = \frac{-35}{0.050} = -700\ \text{m/s}^2

The size is 700 m/s², directed away from the wall. Writing 15 − 20 here is the most common error.

Whenever the direction of motion reverses, the change in velocity is the sum of the two speeds, not their difference.

Turning at constant speed

A car drives half way around a circular track at a steady 10 m/s, taking 6.0 s. Its speed never changes, yet its average acceleration is not zero. At the start it moves at 10 m/s one way, and at the end at 10 m/s the opposite way. The change in velocity has size 20 m/s.

aavg=206.0≈3.3 m/s2a_{avg} = \frac{20}{6.0} \approx 3.3\ \text{m/s}^2

This vector view is how circular motion leads to the centripetal acceleration result.

Average and instantaneous acceleration

On a velocity time graph, the average acceleration between two times equals the gradient of a straight line drawn between those two points. The instantaneous acceleration at one time is the slope of the tangent there. If the graph is a straight line, chord and tangent coincide and the two values agree, which is the case of uniform acceleration covered on the constant acceleration formula page.

For the basic definition with u and v, see the main acceleration formula page. For the velocity change on its own, see the change in velocity formula.

Mistakes to avoid

  • Subtracting speeds instead of velocities when the object reverses or turns.
  • Swapping initial and final values, which flips the sign.
  • Using total distance anywhere in this formula. Only velocities and time appear.
  • Assuming zero average means zero acceleration all along. A car that speeds up and then slows back to its starting velocity has zero average acceleration but was accelerating the whole time.

The kinematics formulas hub collects the rest of the straight line motion toolkit in one table.

Frequently asked questions

What is the formula for average acceleration?

Average acceleration = change in velocity / time interval, a_avg = (v_f − v_i) / Δt. Its SI unit is m/s².

Is average acceleration a vector?

Yes. It points in the direction of the change in velocity, which is not always the direction of motion.

Can average acceleration be zero while the object is accelerating?

Yes. If the final velocity equals the initial velocity, the average is zero, even though the object may have sped up and slowed down along the way.

What is the difference between average acceleration and acceleration?

Average acceleration covers a whole time interval. Instantaneous acceleration is the value at one moment. They match only when the acceleration is constant.

How is average acceleration found from a velocity time graph?

Join the two points on the graph with a straight line and find its slope: rise in velocity divided by run in time.