Speed: Average & Instantaneous

A road trip's summary versus what the speedometer says right now.

12 minUpdated 21 October 2026average speedinstantaneous speedslope of s–tarea under v–t

Lesson at a glanceShow the key points

Average speed = distance travelled / time taken; it belongs to an interval.
Instantaneous speed v = ds/dt; it belongs to a single instant.
On a distance–time graph, the slope of a chord is the average speed and the slope of the tangent is the instantaneous speed.
Distance travelled s = ∫ v dt, the area under the speed–time graph.
For equal distances at speeds v₁ and v₂ the average speed is 2v₁v₂/(v₁ + v₂), not the arithmetic mean.

“We averaged 60 km/h” summarises a whole road trip in one number. The speedometer needle tells a different, moment-by-moment story. Both are speed, measured over different stretches of time.

Average speed

If a particle covers a distance $s$ between times $t_1$ and $t_2$ , its average speed over that interval is

Average speed(3.1)

v_{av} = \frac{s}{t_2 - t_1}

Average speed captures the overall quickness of the trip. Think of a 300 km drive that takes 5 hours: you averaged 60 km/h, though you probably crawled through towns, sped on the highway and stopped for chai. The average hides all that detail.

Instantaneous speed

A sprinter leaves the blocks slowly and builds up. To capture how fast she is going at one particular moment $t$ , take a short interval from $t$ to $t + \Delta t$ in which she covers $\Delta s$ , compute $\Delta s/\Delta t$ , and then let $\Delta t$ shrink toward zero. Both $\Delta s$ and $\Delta t$ vanish, but their ratio approaches a definite limit, just as in the lesson on derivatives:

Instantaneous speed(3.2)

v = \lim_{\Delta t \to 0}\frac{\Delta s}{\Delta t} = \frac{ds}{dt}

Average speed belongs to an interval; instantaneous speed belongs to an instant. When people just say “speed,” they mean the instantaneous kind.

Interactive

Average speed versus instantaneous speed

The particle covers s = (1.5 m/s²)t². The slope of the chord AB is the average speed over the interval; shrink the interval and it becomes the slope of the tangent at A, the instantaneous speed.

\Delta t

3.000s

v_{av} = \Delta s/\Delta t

10.50m/s

v = ds/dt

6.00m/s

Interval

\Delta t

3.000s

Worked example

Speeding up steadily

A particle covers distance $s = (1.5\text{ m/s}^2)\,t^2$ . Find (a) its average speed from $t = 0$ to $t = 4.0\text{ s}$ and (b) its speed at the instant $t = 4.0\text{ s}$ .

Going backwards: distance from speed

If the speed at time $t$ is $v$ , then in a tiny interval $dt$ the particle covers $ds = v\,dt$ . Add up all those little pieces from $t_1$ to $t_2$ :

Distance from speed(3.3)

s = \int_{t_1}^{t_2} v\,dt

On a speed–time graph that integral is simply the area under the curve between the two times.

Interactive

Distance hides under the speed graph

Drag the round handles to reshape how speed changes with time. Drag the two markers on the time axis to choose an interval. The shaded area is the distance covered.

t_1

0.00s

t_2

4.00s

area = distance16.00m

Worked example

Area of a triangle

A particle starts from rest and its speed rises steadily to 8 m/s in 4 s. How far does it travel in those 4 s?

Worked example

A trap in averaging

A cyclist rides 10 km at 30 km/h, then another 10 km at 60 km/h. What is the average speed for the whole 20 km?

The dimension of speed is $[LT^{-1}]$ and its SI unit is the metre per second (m/s).

What to remember

1Average speed = total distance / total time; it belongs to an interval.
2Instantaneous speed $v = ds/dt$ ; it belongs to an instant.
3Chord slope on an s–t graph = average speed; tangent slope = instantaneous speed.
4Area under a v–t graph = distance travelled.

Check yourself

On a distance–time graph, a horizontal stretch means the object is…

A car drives 1 hour at 40 km/h and then 1 hour at 80 km/h. Its average speed is…

Keep exploring

Hand-picked videos, simulations and readings that explain this idea from a different angle.

ReadSpeed and velocityThe Physics Classroom ReadThe meaning of slope on a position–time graphThe Physics Classroom ReadArea on a velocity–time graphThe Physics Classroom ReadInstantaneous velocity and speedOpenStax University Physics §3.2 ReadVelocity and speedHyperPhysics

PreviousDistance vs Displacement Next Velocity & Position Vectors

Speed: Average & Instantaneous

A road trip's summary versus what the speedometer says right now.

12 minUpdated 21 October 2026average speedinstantaneous speedslope of s–tarea under v–t

Lesson at a glanceShow the key points

Average speed = distance travelled / time taken; it belongs to an interval.
Instantaneous speed v = ds/dt; it belongs to a single instant.
On a distance–time graph, the slope of a chord is the average speed and the slope of the tangent is the instantaneous speed.
Distance travelled s = ∫ v dt, the area under the speed–time graph.
For equal distances at speeds v₁ and v₂ the average speed is 2v₁v₂/(v₁ + v₂), not the arithmetic mean.

“We averaged 60 km/h” summarises a whole road trip in one number. The speedometer needle tells a different, moment-by-moment story. Both are speed, measured over different stretches of time.

Average speed

If a particle covers a distance $s$ between times $t_1$ and $t_2$ , its average speed over that interval is

Average speed(3.1)

v_{av} = \frac{s}{t_2 - t_1}

Instantaneous speed

Instantaneous speed(3.2)

v = \lim_{\Delta t \to 0}\frac{\Delta s}{\Delta t} = \frac{ds}{dt}

Average speed belongs to an interval; instantaneous speed belongs to an instant. When people just say “speed,” they mean the instantaneous kind.

Interactive

Average speed versus instantaneous speed

The particle covers s = (1.5 m/s²)t². The slope of the chord AB is the average speed over the interval; shrink the interval and it becomes the slope of the tangent at A, the instantaneous speed.

\Delta t

3.000s

v_{av} = \Delta s/\Delta t

10.50m/s

v = ds/dt

6.00m/s

Interval

\Delta t

3.000s

Worked example

Speeding up steadily

A particle covers distance $s = (1.5\text{ m/s}^2)\,t^2$ . Find (a) its average speed from $t = 0$ to $t = 4.0\text{ s}$ and (b) its speed at the instant $t = 4.0\text{ s}$ .

Going backwards: distance from speed

If the speed at time $t$ is $v$ , then in a tiny interval $dt$ the particle covers $ds = v\,dt$ . Add up all those little pieces from $t_1$ to $t_2$ :

Distance from speed(3.3)

s = \int_{t_1}^{t_2} v\,dt

On a speed–time graph that integral is simply the area under the curve between the two times.

Interactive

Distance hides under the speed graph

Drag the round handles to reshape how speed changes with time. Drag the two markers on the time axis to choose an interval. The shaded area is the distance covered.

t_1

0.00s

t_2

4.00s

area = distance16.00m

Worked example

Area of a triangle

A particle starts from rest and its speed rises steadily to 8 m/s in 4 s. How far does it travel in those 4 s?

Worked example

A trap in averaging

A cyclist rides 10 km at 30 km/h, then another 10 km at 60 km/h. What is the average speed for the whole 20 km?

The dimension of speed is $[LT^{-1}]$ and its SI unit is the metre per second (m/s).

What to remember

1Average speed = total distance / total time; it belongs to an interval.
2Instantaneous speed $v = ds/dt$ ; it belongs to an instant.
3Chord slope on an s–t graph = average speed; tangent slope = instantaneous speed.
4Area under a v–t graph = distance travelled.

Check yourself

On a distance–time graph, a horizontal stretch means the object is…

A car drives 1 hour at 40 km/h and then 1 hour at 80 km/h. Its average speed is…

Keep exploring

Hand-picked videos, simulations and readings that explain this idea from a different angle.

PreviousDistance vs Displacement Next Velocity & Position Vectors

Average speed#

Instantaneous speed#

Going backwards: distance from speed#

Check yourself

Keep exploring

Average speed#

Instantaneous speed#

Going backwards: distance from speed#

Check yourself

Keep exploring

Average speed

Instantaneous speed

Going backwards: distance from speed

Average speed

Instantaneous speed

Going backwards: distance from speed