Your speedometer reads $60\ \text{km h}^{-1}$ right now, that is instantaneous speed. The trip computer says you averaged $42\ \text{km h}^{-1}$, that is average speed. Both are useful, both are different, and confusing them is one of the most common errors in motion problems. This lesson pins down all four quantities and the single relationship that ties them together.
A bus covers $30\ \text{km}$ in $1\ \text{hour}$, but it stops at traffic lights and crawls through town for part of the trip.
Before reading: is the bus's speedometer reading the same as its average speed of $30\ \text{km h}^{-1}$ at every moment? Write yes or no and one sentence explaining why.
Same trip, two averages, the difference is distance versus displacement
Average speed asks "how much ground per second?" while average velocity asks "how far from the start per second, and in which direction?", so they share a time but use a different top line.
Average speed is total distance over total time (a scalar). Average velocity is net displacement over total time (a vector). The velocity relationship on the formula sheet is:
$$v_{av} = \frac{\Delta s}{\Delta t}$$
where $\Delta s$ is the change in displacement and $\Delta t$ the time interval. Average speed uses the same form but with total distance on top instead of displacement.
Average speed $= \dfrac{\text{total distance}}{\text{total time}}$ (scalar). Average velocity $v_{av} = \dfrac{\Delta s}{\Delta t}$ (vector, using net displacement). They are equal only for straight-line motion with no reversal.
Pause, copy the highlighted statement into your book before moving on.
Did you get this? True or false: a runner who completes one full lap of a track has an average velocity of zero but a non-zero average speed.
Putting the numbers into the relationship
We just saw that the two averages differ by distance versus displacement. That raises a question: how do those different top lines play out on a real trip? This card answers it → by working the same journey both ways.
The same time goes underneath both calculations; you just feed the correct top line, total distance for speed, net displacement for velocity.
A cyclist rides $300\ \text{m}$ east, then $100\ \text{m}$ west, taking $40\ \text{s}$ in total. Find the average speed and the average velocity.
To find each average, divide by the same total time: average speed uses total distance, average velocity uses $v_{av} = \dfrac{\Delta s}{\Delta t}$ with net displacement (and direction). When the path reverses, the velocity comes out smaller than the speed.
Add the highlighted worked method to your notes before the check below.
Quick check: A car has a net displacement of $120\ \text{m}$ north in $30\ \text{s}$. What is its average velocity?
Pattern A, Calculate then compare
A swimmer swims $50\ \text{m}$ to the far wall of a pool and back to the start in $80\ \text{s}$. In your book: (1) calculate the average speed; (2) calculate the average velocity; (3) explain in one sentence why one is zero.
Two truths, one lie: which statement about speed and velocity is the lie?
Dot point: Explain how instantaneous speed, average speed, instantaneous velocity and average velocity are measured and calculated
Zooming in to a single moment
We just learned to average over a whole trip. That raises a question: what is your speed at one exact moment, not over the whole journey? This card answers it → that is the instantaneous value, measured over a very short interval.
Instantaneous speed and velocity describe a single instant, the value you get by shrinking the time interval down toward zero, which is exactly what a speedometer estimates.
To measure an instantaneous value, you record displacement over a tiny time interval (for example, using a light gate or a motion sensor sampling every few milliseconds). The shorter the interval, the closer the average over that interval gets to the true instantaneous value. A speedometer continuously displays instantaneous speed; the trip computer's "average speed" is the whole-journey average.
An instantaneous value is the speed or velocity at one moment, found by measuring displacement over a very short time interval. A speedometer shows instantaneous speed; whole-trip averages are calculated over the full distance or displacement.
Pause, copy the highlighted distinction into your book before the check below.
Last check: Decide if the statement is true or false.
A car's speedometer reading at any moment is its instantaneous speed, not its average speed for the trip.
Odd one out: three of these give an instantaneous value. Which one gives an average?
Dot point: Solve problems using v_av = Δs/Δt (average speed/velocity relationships)
A fresh set drawn from this lesson's question bank, feedback shown immediately. +5 XP per correct · +25 XP all correct
Pick your answer, then rate your confidence, that tells the system what to drill next.
ApplyBand 5(4 marks) 1. A runner jogs $400\ \text{m}$ around a circular track in $100\ \text{s}$, returning to the starting line. Calculate the average speed and the average velocity, and explain why they differ.
1 mark: average speed $= 400/100$ · 1 mark: average speed $= 4.0\ \text{m s}^{-1}$ · 1 mark: average velocity $= 0\ \text{m s}^{-1}$ (zero displacement) · 1 mark: explanation that speed uses distance, velocity uses displacement
ApplyBand 5(4 marks) 2. A car travels $60\ \text{km}$ east in $1.0\ \text{h}$, then $60\ \text{km}$ west in $2.0\ \text{h}$. Calculate its average speed and its average velocity for the whole trip, and explain why they differ.
1 mark: average speed $= 120/3.0 = 40\ \text{km h}^{-1}$ · 1 mark: displacement $= 0$ · 1 mark: average velocity $= 0$ · 1 mark: explains speed uses distance and velocity uses displacement
Dot point: Solve problems using v_av = Δs/Δt (average speed/velocity relationships)
UnderstandBand 4(3 marks) 3. Distinguish between instantaneous velocity and average velocity, and describe how each is measured in practice.
1 mark: instantaneous is at a single moment, average is over an interval · 1 mark: instantaneous measured over a very short interval, or as the gradient of a tangent · 1 mark: average measured as total displacement over total time
Dot point: Explain how instantaneous speed, average speed, instantaneous velocity and average velocity are measured and calculated
MC answers and full explanations are shown inline as you complete each question. Use the retry button to attempt a fresh set from the lesson bank.
Q1 (4 marks): Average speed uses total distance: $\text{avg speed} = \dfrac{400\ \text{m}}{100\ \text{s}} = 4.0\ \text{m s}^{-1}$. Average velocity uses net displacement: because the runner finishes exactly where they started, the displacement is zero, so $v_{av} = \dfrac{\Delta s}{\Delta t} = \dfrac{0}{100} = 0\ \text{m s}^{-1}$. The two values differ because average speed is a scalar based on the total path length, while average velocity is a vector based on the straight-line change in position, which is nothing for a complete loop.
Q2 (4 marks): Average speed uses total distance over total time: $$v_{av} = \frac{120\ \text{km}}{3.0\ \text{h}} = 40\ \text{km h}^{-1}$$ Average velocity uses displacement over total time. The car finishes where it started, so the displacement is zero and $$\vec{v}_{av} = \frac{0}{3.0} = 0\ \text{km h}^{-1}$$ They differ because the two definitions use different numerators. Speed is built on distance, a scalar that only ever accumulates, while velocity is built on displacement, a vector that cancels when the motion reverses. The car was certainly moving, so the zero average velocity describes the net change in position, not the motion itself.
Q3 (3 marks): Instantaneous velocity is the velocity at one particular moment, whereas average velocity is the single value that describes a whole interval. In practice, instantaneous velocity is measured by taking the displacement over an interval short enough that the velocity does not appreciably change within it, for example a light gate over a few centimetres, or graphically as the gradient of the tangent to a displacement-time graph at that instant. Average velocity is measured by taking the total displacement between the start and finish points and dividing by the total time, which requires only two position measurements and a clock.
You can now calculate average speed and average velocity and tell them apart from instantaneous values. Next lesson extends velocity into two dimensions, combining perpendicular velocities into one resultant.
↩ Back to focus-area hubAt the start you considered a bus averaging $30\ \text{km h}^{-1}$ while stopping and crawling. The speedometer is not always reading $30\ \text{km h}^{-1}$: it shows instantaneous speed, which is zero at the lights and higher on open road, while $30\ \text{km h}^{-1}$ is only the whole-trip average. Revisit Card 3 if this surprised you.
A short spaced-retrieval set, mixing this lesson with earlier ones. It gets harder to guess and easier to remember.
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