🪨 The instant in between
A speed that lasts no time at all
Drop a stone off a bridge. At the exact second it passes your hand, how fast is it falling? There’s no stretch of time left to measure, just one frozen moment. The answer turns out to be exact anyway.
Speed is easy, when you’ve got a stretch of time.
Start here, end there, divide. Done.
But freeze the clock at one exact instant…
…and there’s no stretch left to divide.
The plan
Catch a speed with nothing but two photographs
Here’s what you’re about to do: drop a stone off a bridge into the gorge below, and try to pin down its exact speed at the one-second mark, using nothing but photographs and a stopwatch. You’ll hit a wall almost straight away.
Good. That wall is the whole point.
The trap
One frozen photo. Can you get a speed from it?
Here’s the stone, one second into its fall. You know exactly how far it’s dropped: 5 metres. That’s the whole photograph: one moment, one position, frozen.
Your turn. You have exactly one photo, frozen at the one-second mark. Can you work out the stone’s exact speed at that instant from that one photo alone?
From this one frozen photo…
Not one photo.
Two.
Closer and closer together…
…until they’re practically the same instant.
The why
Zoom in until it stops changing
Keep the first photo fixed at the 1-second mark. Take a second photo a little later, and work out the average speed between the two: distance apart, divided by time apart. Then drag that second photo closer and closer to the first.
Before you try it: as the second photo gets closer, will that average keep changing forever, or settle down to one number?
Average speed between the two photos
15.00 m/s
Second photo: 1.00 s later.
Drag all the way right. Watch the line between the two photos rotate until it hugs the curve, and the number above stop moving.
| Gap to second photo | Average speed |
|---|---|
| 1.00 s later | 15.0 m/s |
| 0.50 s later | 12.5 m/s |
| 0.20 s later | 11.0 m/s |
| 0.05 s later | 10.25 m/s |
| 0.01 s later | 10.05 m/s |
| → gap shrinks to nothing | 10 m/s (the real answer) |
That settled number…
not an average.
Not a guess.
The exact speed, at that exact instant.
This isn’t just a trick for falling stones.
It works on anything that’s changing.
And it does something even stranger…
it can find the exact top of a hill, without ever seeing the whole hill.
Try it yourself
Catch the exact top of the toss
This time you toss the stone up off the bridge; it arcs over the gorge and drops back down. Scrub through the toss below. At every instant, the same zoom-in trick reads off the stone’s exact rate: rising, falling, or, for one instant, neither.
Exact rate, right now
+19.9 m/s
Still rising.
Find the exact instant where the rate reads as close to zero as you can get it. That instant is the very top of the arc, and you never had to look at the whole curve to find it.
A speedometer does this every second.
So does a fitness watch’s “current pace”.
Zooming in on a changing thing until the rate settles has a name.
Differentiation. One of the biggest ideas in maths, and you just did it with a stone and a stopwatch.
Where this hides
Anywhere something changes, this same trick can catch its exact rate: how fast a population is growing right now, how fast a battery is draining right now, the precise top of a thrown ball’s arc, the exact instant a rollercoaster stops rising and starts falling.
Next time someone says a speedometer or a pace watch is “just a guess”, you know exactly what’s happening under the glass: two photos, zoomed in until they’re practically the same instant.