🪨 The instant in between

A speed that lasts no time at all

Drop a stone off a bridge. How fast is it falling exactly one second after you let go? A single instant has no length of time to measure, but you can still find the exact answer.

Speed is easy to work out when you’ve got a stretch of time.

You divide the distance by the time it took.

But freeze the clock at one exact instant…

…and there’s no time left to divide by.

The plan

Catch a speed with nothing but two photographs

You’re going to drop a stone off a bridge into the gorge below and work out its exact speed one second after you let go, using only photographs and a stopwatch. You’ll get stuck almost straight away.

That’s fine. You’ll see how to get past it.

The trap

Can you get a speed from one frozen photo?

Here’s the stone one second after you let go. You know it has fallen exactly 5 metres, but the photo only shows you where it is at that one moment.

Your turn. Using only this photo, can you work out how fast the stone is going at that instant?

From this one frozen photo…

Instead of one photo…

…you take two…

…and move them 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, which is the distance between them divided by the time between them. Then drag the 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 the slider all the way to the right. The line between the two photos turns until it lies along the curve, and the number above stops changing.

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)

As the gap between the photos shrinks…

…the average speed settles on one number…

…10 metres per second…

…and that’s the exact speed at the one-second mark.

You can use this on more than falling stones.

It works on anything that’s changing.

You can also use it…

…to find the exact top of a hill without seeing the whole hill.

Try it yourself

Catch the exact top of the toss

This time you throw the stone up off the bridge, and it goes over the gorge in an arc and comes back down. Drag the slider through the throw below. The same zoom-in method gives you the stone’s exact rate at every instant, whether it’s going up or coming down. For one instant it’s doing neither.

Exact rate, right now

+19.9 m/s

It’s still going up.

Find the instant where the rate is as close to zero as you can get it. That’s the very top of the arc, and you’ve found it without needing the whole curve.

A speedometer does this every second.

So does the “current pace” on a fitness watch.

Zooming in on something that’s changing until the rate settles is called…

differentiation. It’s one of the biggest ideas in maths, and you’ve just done it with a stone and a stopwatch.

Where else it’s used

You can use the same method to find the exact rate of anything that changes, such as how fast a population is growing or a battery is draining right now. It also finds the top of a thrown ball’s arc, or the moment a rollercoaster stops going up and starts coming down.

If someone says a speedometer or a pace watch is “just a guess”, you can tell them how it works: it compares two measurements taken so close together that they’re practically the same instant.