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Filling and emptying

Change and motion · level 1, lesson 4 of 6

A question to start

You have two containers: a tall thin vase and a wide flat bowl. You fill each one from the same tap, turned on the same amount. In which one does the water climb up faster?

Now a harder one. A cup has a small hole in the bottom. You fill it with water and watch it drip out. Does the water go down by the same amount every minute? Or does it go down fast at first and then slower? Or slow at first and then faster?

Have a guess before you read on. Then, if you can, try it at the sink with a paper cup and a pencil hole.

The idea

Filling and emptying are changes

When you fill something, the amount inside it gets bigger over time. When you empty something, the amount inside gets smaller over time. So filling and emptying are changes, like growing and shrinking.

But there are two different questions we can ask:

  1. Is it getting fuller or emptier? (Which way is it changing?)
  2. Is it changing fast or slow? And does it stay the same speed, or change speed?

The second question is the new one in this lesson.

Same all the way: a straight-sided tub

Turn on a tap so that it pours steadily into a tub with straight sides. Every minute, the same amount of water comes out of the tap. Because the sides are straight, every bit of water spreads out over the same floor, so it lifts the water's top by the same amount each time.

123 45 123 45 straight tub V-shaped glass

In the straight tub, the marks after 1, 2, 3, 4 and 5 minutes are evenly spaced, like the steps of a ladder. The water rises the same all the way. Grown-ups call this a steady change.

Fast at first, then slower: a V-shaped glass

Now pour from the same tap into a glass shaped like a V: narrow at the bottom and wide at the top.

At the bottom the glass is narrow, so a little water makes a tall column. The water climbs fast. Higher up, the glass is wide, so the same amount of water spreads out thin and only lifts the top a little. The water climbs slowly.

Look at the marks in the picture: big gaps at the bottom, small gaps at the top. Same tap, same water every minute, but the shape changes how fast the level climbs.

That answers the first question at the start: the water climbs much faster in the tall thin vase than in the wide bowl. A narrow container needs only a little water to get a lot higher.

The dripping cup: fast at first, then slower

Now the cup with a hole. When the cup is full, there is a lot of water on top of the hole, pushing down. The water squirts out quickly. As the cup empties, there is less water pushing, so it comes out more gently. The drips get slower and slower, and the last bit takes the longest.

full: fast half: slower low: slow drips

You can see this in your bathtub too. When you pull the plug in a full bath, the water rushes out with a loud gurgle. The last little bit near the bottom drains slowly.

The sand timer: about the same all the way

A sand timer (also called an hourglass) has two glass bulbs joined by a tiny hole. Sand pours from the top bulb into the bottom bulb.

Here is a surprise: sand does not behave like water. The grains jam against each other near the hole, so the weight of the sand on top does not push it through faster. The sand trickles through at about the same amount each moment, whether the top bulb is full or almost empty. That is exactly why people used sand timers to measure time: a steady trickle is like a steady clock.

Three kinds of change

So far we have found three kinds:

Kind What it looks like Example
Same all the way (steady) The same change each minute A tap into a straight tub; sand in a sand timer
Fast at first, then slower Big changes first, then small A V-glass filling; a cup draining through a hole
Slow at first, then faster Small changes first, then big Filling a glass that is wide at the bottom and narrow at the top

Notice: the fast or slow is about how much changes in each minute, not about how much there is.

Worked examples

Example 1. Counting cups. You fill a straight-sided bucket using a cup. After 1 cup the water is 2 fingers deep. After 2 cups it is 4 fingers deep. How deep after 3 cups? After 5 cups?

  1. Each cup adds the same water, and the bucket has straight sides, so each cup lifts the water by the same amount: 2 fingers.
  2. After 3 cups: 4 + 2 = 6 fingers.
  3. After 4 cups: 8 fingers. After 5 cups: 10 fingers.

This is a steady change: 2, 4, 6, 8, 10, counting by twos.

Example 2. A funnel-shaped glass. A glass is wide at the top and narrow at the bottom. You pour in one spoon of juice at a time. Will the first spoon or the last spoon lift the juice more?

  1. The first spoon goes into the narrow bottom, so it makes a tall column.
  2. The last spoon goes into the wide top, so it spreads out thin.
  3. The first spoon lifts the juice more. This is "fast at first, then slower".

Example 3. A bottle upside down. You turn a full water bottle upside down over the sink. Glug, glug, glug. Does it empty in the same amount of time as one with a big hole? Think about it: the bottle's neck is narrow, so the water and the air have to take turns getting through. That is why it glugs. A wider opening would empty faster. The size of the hole changes how fast the emptying is.

Example 4. Which shape? A glass is filled steadily from a tap. The water climbs slowly at first, then faster and faster. What shape could the glass be?

  1. Climbing slowly means the water is spreading out wide.
  2. Climbing fast means the glass is narrow there.
  3. So the glass is wide at the bottom and narrow at the top, like a bottle or an upside-down cone.

Try it

  1. You put 1 spoon of rice into a straight tube and the rice is 1 cm deep. If you keep adding 1 spoon at a time, how deep will it be after 4 spoons?
  2. Which fills up to the top faster from the same tap: a little cup or a big bucket?
  3. A bath has its plug pulled. When does the water go down fastest: at the start or at the end?
  4. Why could people use a sand timer as a clock, but not a cup of water with a hole in it?
  5. A vase is narrow at the bottom and wide at the top. Water goes in steadily. Does the level climb fast at first, or slow at first?
  6. Draw a glass where the water would climb the same all the way from a steady tap.
  7. You are filling a straight tub. After 1 minute the water is 3 cm deep. After 2 minutes it is 6 cm deep. How deep after 3 minutes?
  8. Grown-up question: a round fishbowl is narrow at the bottom, widest in the middle, and narrow again at the top. From a steady tap, how does the level climb?
Answers
  1. Each spoon adds 1 cm in a straight tube: 1, 2, 3, 4. 4 cm.
  2. The little cup. It needs much less water, so the same tap fills it sooner.
  3. At the start, when there is the most water pushing down on the plug hole.
  4. Sand trickles at about the same amount each moment, so equal amounts of sand mean equal amounts of time. Water from a hole slows down as the cup empties, so the first half and the second half take different times.
  5. Fast at first (narrow part), then slower (wide part).
  6. Any glass with straight up-and-down sides, like a can or a tube, the same width all the way up.
  7. It adds 3 cm each minute: 9 cm.
  8. Fast at first (narrow bottom), slowest in the middle (widest part), then fast again near the top (narrow again). It is a change that slows down and then speeds up.

Common mistake

Thinking the same tap always makes the water climb at the same speed. The tap gives the same amount of water each minute, but the shape of the container decides how high that water climbs. Only straight-sided containers climb the same all the way.

Mixing up "a lot of water" and "filling fast". A big bucket may hold a lot, but it fills slowly. Fast and slow are about how much changes in each minute.

Where it leads