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Fast at first, then slower

Change and motion · level 3, lesson 5 of 6

A question to start

You make a mug of hot cocoa at 84 °C. The kitchen is at 20 °C. Ten minutes later the cocoa is 52 °C: it has cooled by 32 degrees. If it keeps cooling at 32 degrees every ten minutes, after another ten minutes it would be 20 °C, and after another ten it would be −12 °C, colder than a freezer.

That cannot be right. A mug of cocoa on a kitchen table does not freeze. So what really happens? How does a hot drink cool down?

The idea

Watching the cocoa

Here is how a mug of cocoa might cool, measured every ten minutes. The last row shows how much hotter than the room it is.

Minutes 0 10 20 30 40 50 60
Temperature (°C) 84 52 36 28 24 22 21
Change −32 −16 −8 −4 −2 −1
Hotter than room by 64 32 16 8 4 2 1

Look at the "change" row. Every change is a minus number, so the cocoa is always getting cooler. But the changes get smaller: 32 degrees in the first ten minutes, then 16, then 8, then 4. The cocoa cools fast at first, then slower and slower.

(These are tidy numbers chosen to show the idea clearly. A real mug cools in very much this way, though its exact numbers depend on the mug, the drink and the room.)

What the curve looks like

0102030405060 time (minutes) 020406080 °C room temperature, 20 °C

The graph is not a straight line. It is a curve. It drops steeply at the start (fast cooling), then gets gentler and gentler (slower cooling). Near the end it is almost flat, lying just above the dashed line for the room's temperature. We say the curve levels off: it gets closer and closer to a level line without crossing it.

Why does it slow down?

Heat moves from hot things to cooler things. And here is the key fact: the bigger the difference in temperature, the faster the heat moves.

  • At the start, the cocoa is 64 degrees hotter than the room. Heat pours out of it quickly.
  • Later, it is only 8 degrees hotter. Heat leaks out slowly.
  • When it is just 1 degree hotter, heat barely moves at all.

So as the cocoa cools, it gets closer to the room's temperature, and so it cools more slowly. The cooling slows itself down. Look at the last row of the table: in these tidy numbers, every ten minutes the gap above the room is cut in half: 64, 32, 16, 8, 4, 2, 1.

And it can never go below the room temperature. Once the cocoa is as cool as the room, there is no difference left, so no more heat flows. The room is the cocoa's floor. That is why the puzzle at the start was wrong: the cooling rate does not stay at 32 degrees every ten minutes. It keeps shrinking.

Other things that slow down as they go

The same shape appears in many places, whenever change uses itself up:

  • A growing plant. A young plant shoots up quickly. As it gets near its full size, it grows more and more slowly, and then stops.
  • Learning something new. When you start learning to juggle or type, you improve fast. Later, each small improvement takes much more practice.
  • A bath draining. When the bath is full, the water pushes hard on the plughole and rushes out. When only a little is left, it trickles.
  • An ice lolly warming up in a warm room: it warms quickly at first, then more slowly as it gets close to room temperature. This is the cocoa story upside down: the curve rises steeply and then levels off.

Three shapes of change

Here are the three shapes you now know, side by side. In each, time goes across.

steady: same changesspeeding up: bigger changesslowing down: smaller changes

You can tell them apart from a table, without drawing anything: look at the differences (from the last lesson). Same differences: steady, a straight line. Growing differences: a curve that gets steeper. Shrinking differences: a curve that gets flatter.

Worked examples

1. When can I drink it? Cocoa is nice to drink below 40 °C. Using the table, when does it get there?

At 10 minutes it is 52 °C (too hot). At 20 minutes it is 36 °C (fine). So it reaches 40 °C somewhere between 10 and 20 minutes, closer to 20, because 40 is much nearer to 36 than to 52. (A careful calculation says about 17 minutes.)

2. Fast and slow rates. How fast did the cocoa cool in the first ten minutes, and in the last ten minutes, in degrees per minute?

  • First ten minutes: 32 degrees in 10 minutes, \(32 \div 10 = 3.2\) degrees per minute.
  • Last ten minutes (50 to 60): 1 degree in 10 minutes, \(1 \div 10 = 0.1\) degrees per minute.

At the start it was cooling 32 times as fast as at the end.

3. A plant that slows down. A plant is measured every week:

Week 0 1 2 3 4 5 6 7
Height (cm) 0 30 55 75 90 100 106 109

Differences: 30, 25, 20, 15, 10, 6, 3. They shrink, so the plant grows fast at first, then slower. In which week did it grow most? The first: 30 cm. A sensible guess for week 8: the differences are shrinking towards 0, so maybe 1 or 2 cm more, about 110 or 111 cm. It is reaching its full height.

4. Telling it from a story. "A ball rolled across the grass. It went 5 metres in the first second, 3 metres in the next, then 1.5 metres, then 0.5 metres, and then stopped." The differences in distance shrink, so the ball is slowing down. A distance-time graph would rise steeply, then flatten, then go flat at \(5 + 3 + 1.5 + 0.5 = 10\) metres.

Try it

  1. Which of these lists of changes shows "fast at first, then slower"? (a) 2, 4, 6, 8; (b) 10, 8, 5, 2; (c) 5, 5, 5, 5.
  2. In the cocoa table the gap above room temperature halves every 10 minutes. What would the temperature be at 70 minutes?
  3. An ice cube at −10 °C is put into a room at 20 °C. Ignore the melting. Will its temperature change fast at first or slowly at first? Why?
  4. A runner's distance from the start, minute by minute, is 0, 8, 15, 21, 26, 30 hundred metres. Find the differences. Is the runner speeding up, slowing down or steady?
  5. Can the cocoa cool to 15 °C if the kitchen stays at 20 °C? Explain.
  6. A bath empties fast at first and slowly at the end. Sketch, in words, the graph of the depth of water against time.
  7. Mo learns to type. His speeds in words per minute, week by week: 10, 25, 35, 42, 46, 48. Describe how his typing speed is changing, using the differences.
  8. Challenge: in the tidy cocoa table the gap above the room halves every 10 minutes. What is the gap after 100 minutes? Does it ever become exactly 0?
Answers
  1. (b): the changes are getting smaller. (a) is speeding up; (c) is steady.
  2. At 60 minutes the gap is 1 degree. Half of that is 0.5, so at 70 minutes it is \(20 + 0.5 = 20.5\) °C.
  3. Fast at first. At the start it is 30 degrees colder than the room, a big difference, so heat flows into it quickly. As it warms towards 20 °C the difference shrinks and it warms more slowly.
  4. Differences: 8, 7, 6, 5, 4 (hundreds of metres each minute). They get smaller, so the runner is slowing down, though still moving.
  5. No. It cools only because it is hotter than the room. Once it reaches 20 °C there is no difference, and so no more cooling. It can never go below the room temperature.
  6. The depth starts high. The line falls steeply at first, then less and less steeply, flattening out as it reaches 0.
  7. Differences: +15, +10, +7, +4, +2. His speed is always getting better, but by less and less each week. He improves fast at first and then more slowly.
  8. Ten halvings: 64 becomes 32, 16, 8, 4, 2, 1, 0.5, 0.25, 0.125, 0.0625. After 100 minutes the gap is 0.0625 degrees, so the cocoa is 20.0625 °C. Halving never reaches exactly 0, so in this model the gap never becomes exactly 0, just too small to measure.

Common mistake

Assuming the first change keeps going. The puzzle at the start made this mistake: "32 degrees in the first ten minutes, so 32 degrees every ten minutes". Only steady change keeps the same rate. Always look at several differences before predicting.

Thinking "slower" means "going down". A curve that rises steeply and then flattens is still going up the whole time. It is growing more slowly, not shrinking. Slowing down is about the size of the changes, not their direction.

Where it leads