Speed vs. Velocity: Average & Instantaneous

Guiding Question

A speedometer reads 0 m/s, yet the car had an average speed of 12 m/s on the trip. A runner finishes a 400 m lap with an average velocity of zero. How can these both be true?

Use the simulation to tell apart speed and velocity, and the average and instantaneous versions of each. Choose a motion, pick a time interval (t₁ to t₂) and an instant (t), and watch how the five quantities are found on the motion diagram, the position–time graph, and the velocity–time graph.

Speed vs. Velocity: the big idea

Speed scalar

How fast something moves. Magnitude only, never negative. Built from distance (d), the total length of the path traveled.

Velocity vector

How fast and in which direction. The sign (+ or −) gives the direction along the x-axis. Built from displacement (Δx = xf − xi), the straight-line change in position.

Each one comes in two versions: average (over a time interval Δt) and instantaneous (at a single moment, the limit as Δt → 0).

Motion diagram (+x to the right)
t = 3.00 s
The object drops a breadcrumb every 0.5 s as it moves. Breadcrumbs in the upper lane were moving in the +x direction, those in the lower lane in the −x direction. Orange breadcrumbs fall inside the interval t₁ → t₂ and carry their instantaneous velocity vector (longer arrow = faster). Press Play to watch them appear.
displacement Δx path traveled (distance d) instantaneous velocity (object and breadcrumbs)
Time interval
Δt = t₂ − t₁
Displacement vector
Δx = x(t₂) − x(t₁)
Distance scalar
d = total path length
always ≥ |Δx|
Average velocity vector
v̄ = Δx / Δt
Average speed scalar
average speed = d / Δt
no direction
Instantaneous velocity vector
v = Δx/Δt as Δt → 0
Instantaneous speed scalar
|v| (the speedometer reading)
no direction
Position vs. time
t = 3.00 s
Average velocity = slope of the secant line through the two interval points. Instantaneous velocity = slope of the tangent line at t. The dotted line from t = 0 to t shows the average velocity so far.
Velocity vs. time
t = 3.00 s
v(t) can be negative; speed |v| cannot. The shaded area under the speed curve equals the distance d.
average velocity average speed

Summary: five quantities, side by side

QuantityTypeDefinitionOn a position–time graphOn a velocity–time graph
Speed / Velocityscalar / vectorSpeed = how fast. Velocity = how fast and which way.Steepness of the curve / slope with its signHeight of |v| curve / height of v curve
Average velocityvectorv̄ = Δx / ΔtSlope of the secant line between t₁ and t₂Height of a rectangle over Δt with the same net area as under v(t)
Average speedscalard / ΔtNot a single slope; total vertical travel of the curve ÷ ΔtAverage height of the |v| curve over the interval
Instantaneous velocityvectorLimit of Δx / Δt as Δt → 0Slope of the tangent line at tValue of v(t) at that instant
Instantaneous speedscalar|v| at one instantMagnitude of the tangent slope (ignore sign)Value of |v(t)| at that instant
Key asymmetry: Instantaneous speed is always the magnitude of instantaneous velocity. But average speed is not generally the magnitude of average velocity. They match only when the object never reverses direction during the interval.

Check your understanding

Predict first, then use the simulation to check.

1. In “Walk out, pause, walk back,” what are the average velocity and average speed for the whole trip (0–10 s)?

Δx = 4 m − 0 m = +4 m and d = 12 m + 8 m = 20 m. Average velocity = +4 m ÷ 10 s = +0.40 m/s; average speed = 20 m ÷ 10 s = 2.0 m/s. The walker reversed direction, so average speed is larger than |average velocity|.

2. In “Ball rolled up a ramp,” when is the instantaneous speed zero? What is the average velocity from 0 to 10 s?

The speed is zero at t = 5 s, the top of the ramp, where the tangent line on the x–t graph is horizontal. The ball returns to x = 0, so Δx = 0 and average velocity = 0 m/s, even though average speed = 30 m ÷ 10 s = 3.0 m/s.

3. In “Constant velocity cart,” why are all four speed/velocity readouts the same (ignoring sign)?

The x–t graph is a straight line, so every secant and every tangent has the same slope, and the cart never reverses, so d = |Δx|.

4. Can average speed ever be less than the magnitude of average velocity?

No. Distance traveled can never be less than the straight-line displacement, so d/Δt ≥ |Δx|/Δt.

5. Pick any motion and press “Shrink Δt → 0.” What happens to the secant line and to the average velocity?

The secant line rotates until it lies on top of the tangent line, and the average velocity approaches the instantaneous velocity at t. That limit is the definition of instantaneous velocity.

6. Does a car’s speedometer show average speed, average velocity, instantaneous speed, or instantaneous velocity?

Instantaneous speed: it reads how fast the car is going right now and has no direction (it reads the same driving north or south, and it would not go negative in reverse).