MATH A251 Course NotebookCalculus I · Fall 2026
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Experiment · Stage A / B / C

Can two clocks measure one instant?

Put down two gates and run the ball. Change the gap before you read the explanation below it.

Experiment 01 / Average velocity

THE BALL + THE GATES

Drag either gate, or focus it and use arrow keys. Shift + arrow moves one metre.

0 mTrack / 100 m100 m

Try these in order, or choose your own

The gate positions determine Δs before the run. The crossing times — and so the measured value of Δt — are not shown until the ball reaches each gate.

The delay between the two stamps represents the interval whose value is tB − tA.

Both positions are measured from the same reference point. Δs is the difference between them.

Because the ball moves only forward here, its average speed and average velocity have the same value.

Active Calculus §1.1 writes this without the Δ, as AV[a,b] = (s(b) − s(a)) / (b − a). Same subtraction over the same subtraction.

Run the ball once, then move Gate B toward Gate A and compare the log.

Run log

newest at bottom

No runs yet. Start wide, then squeeze the gates.

The experiment is ready. Gate A is at 25 metres and Gate B is at 100 metres.

The currently configured run has Gate A at 25 metres and Gate B at 100 metres. The change in position is 75 metres. Crossing times are reported after the ball reaches each gate.

  1. A · MEASURE ONE AVERAGE

    Start at 25 → 100. The answer describes the whole interval, honestly, but not any one moment.

  2. B · SQUEEZE THE GATES

    Try 25 → 36, then 25 → 30.25. Every gap gives a different average.

  3. C · CLOSE IN

    Keep narrowing. The gates never meet, but the readout settles near one number.

What you just discovered

These measurements already contain the derivative.

Call the time t and the position s(t) — the same letter Active Calculus uses for a position function. Gate A is crossed at time t and gate B at t+h, so

Δs=s(t+h)−s(t),Δt=h,

and the readout is

ΔsΔt=s(t+h)−s(t)h.

Squeezing the gates is making h small. At the 25-metre mark, the averages approach 10 m/s even though the two gates never occupy the same place:

s′(t)=limh→0s(t+h)−s(t)h.

A position function is one example of a function, so from Section 1.3 the book writes the same limit for any f:

f′(a)=limh→0f(a+h)−f(a)h.

Nothing changes but the letter. When the moving object is gone, s becomes f and the derivative is whatever that function's rate of change happens to measure.

The notation describes the same measurement process in mathematical terms. Here, the limit is the value approached as the distance between the gates tends to zero.

Active Calculus §1.1 writes this without the Δ, as AV[a,b]=s(b)−s(a)b−a. It is the same subtraction over the same subtraction. Δ is only a name for the difference.

This is the opening idea in

Active Calculus §1.1, “How do we measure velocity?”

Keep going

The book begins with this question too.

Read Active Calculus §1.1, “How do we measure velocity?” for the same move with position functions, tables, and graphs.