Instantaneous vs Average Speed

Average speed is the ratio of the distance traveled to the time taken to travel it. Instantaneous speed is the speed at a given moment, equivalent to the average speed over an infinitely small time interval. This fundamental distinction is part of the middle and high school curriculum.

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How to measure it in class

With the FizziQ app, it is possible to measure and compare these two types of speed.

Steps:

Average speed with GPS: use the “GPS Speed” instrument for a journey
Also calculate total distance / total time and compare
Instantaneous speed by chronophotography: film a motion with FizziQ
Use video tracking to measure successive positions
Calculate the speed between each frame (Δx/Δt)
Compare instantaneous speed (between two frames) and average speed (over the entire path)

Scientific activities on this topic

Possible extensions with FizziQ: running and speed analysis.

Learn more

Average speed:

Average speed gives no information about speed variations during the journey. A runner who covers 100 m in 10 s has an average speed of 10 m/s, whether they ran steadily or accelerated then slowed down.

Instantaneous speed:

It is the limit of average speed as the time interval approaches zero. In practice, with FizziQ, it is approximated by calculating the speed over a very short interval (between two successive frames of a video).

Graphical representation:

On a position-time graph, average speed is the slope of the chord connecting two points
Instantaneous speed is the slope of the tangent to the curve at a point

The case of GPS:

The smartphone GPS calculates speed from position variations. It gives a quasi-instantaneous speed, but averaged over a few seconds to filter out noise.

Formula

Average speed: v_avg = Δd / Δt = (d₂ - d₁) / (t₂ - t₁)

Instantaneous speed: v = lim(Δt→0) Δd/Δt = dd/dt

where:

v_avg: average speed (m/s)
Δd: distance traveled (m)
Δt: duration of the journey (s)
v: instantaneous speed (m/s)

Application examples

A TGV from Paris to Lyon (450 km in 2h): v_avg = 225 km/h, but v_inst varies from 0 to 320 km/h
A 100 m sprint in athletics: v_avg ≈ 10 m/s, but maximum v_inst ≈ 12 m/s around 60 m
A car speedometer displays instantaneous speed
Google Maps gives the estimated time based on average speeds

FAQ

Q: How do you measure instantaneous speed with FizziQ? A: In video tracking, measure the position on two successive frames and divide by the time interval between frames (1/30 s for a 30 fps video).

Q: Is GPS speed instantaneous or average? A: It is quasi-instantaneous, calculated over the last few seconds of position data. For slow or stationary movement, it may fluctuate due to GPS inaccuracies.

Q: Can you have zero instantaneous speed with a non-zero average speed? A: Yes, if you stop during the journey. For example, stopping at a red light gives v_inst = 0, but the v_avg over the entire trip remains non-zero.

Kinematics - Linear Acceleration - Uniform Linear Motion - GPS speed - Chronophotography - Reference Frame