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Speeding up and slowing down

Change and motion · level 2, lesson 5 of 6

A question to start

Throw a ball straight up into the air. It goes up, up, up... and then comes down.

Here is the puzzle: at the very top, for a tiny moment, how fast is the ball going?

Some people say "very fast, it was just thrown". Some say "it is falling, so it must be moving". But think about it: just before the top the ball is going up, and just after the top it is going down. To change from going up to going down, it must, for one instant, be going neither way. At the very top, the ball is stopped.

In this lesson you will learn how to see speeding up, slowing down and stopping, using a simple trick: mark where something is every second.

The idea

Marking positions every second

Imagine taking a photo of a moving thing once every second and marking where it was on the ground. You get a row of marks. The gap between two marks tells you how far it went in that one second.

  • A big gap means it went far in one second: it was going fast.
  • A small gap means it went only a little way: it was going slowly.
  • No gap (two marks on the same spot) means it did not move: it was stopped.

The marks are like footprints. From footprints in the sand you can tell whether someone was walking, running or standing still.

A bike starting off

Ana gets on her bike and starts to pedal. A friend puts a chalk mark on the path where her front wheel is at each second.

Second 0 1 2 3 4 5
Distance from start (metres) 0 1 3 6 10 15
Gap since last second (metres) 1 2 3 4 5
0 s1 s2 s 3 s4 s5 s 123 45 gaps in metres: bigger each second

The gaps get bigger every second: 1, 2, 3, 4, 5 metres. Ana is speeding up. In each second she goes further than in the second before.

Notice that this is the "not steady, speeding up" pattern from the last lesson. The change row (the gaps) grows.

A car braking

A car is driving along and sees a red light. The driver brakes gently. Marks every second:

Second 0 1 2 3 4 5
Distance from where braking began (metres) 0 8 14 18 20 20
Gap since last second (metres) 8 6 4 2 0
0 s1 s2 s 3 s4 s, 5 s 8642 gaps in metres: smaller each second, then none

The gaps get smaller: 8, 6, 4, 2, and then 0. The car is slowing down, and between second 4 and second 5 it does not move at all: it has stopped. The last two marks are on the same spot.

Look how evenly it slows: each gap is 2 metres less than the one before. The brakes take away the same amount of speed every second.

Steady speed

If all the gaps are the same, the mover is going at a steady speed: not speeding up, not slowing down. A train on a long straight track might leave marks 20 metres apart every second, again and again.

The gaps... The mover is...
get bigger speeding up
stay the same going at a steady speed
get smaller slowing down
are zero stopped

The ball thrown up

Now the ball from the start. Here are its heights every quarter of a second, for a ball thrown straight up quite gently. The left column shows it going up, the right column shows it coming down.

going up coming down top: stopped
  • Going up, the gaps get smaller and smaller. The ball is slowing down.
  • At the top, the last two dots on the way up are very close, and the first two on the way down are very close. The ball is barely moving. At the very top, for an instant, it is stopped.
  • Coming down, the gaps get bigger and bigger. The ball is speeding up.

Why? Gravity pulls the ball down all the time, whether it is going up or down. While the ball goes up, the pull is against it, so it takes away a little speed every moment, like brakes. When all the upward speed is used up, the ball stops. Then gravity keeps pulling, and now the ball goes down faster and faster.

Notice the picture is the same on both sides, just flipped. A ball spends as long going up as coming down (if the air does not slow it much), and it is going just as fast when it lands as when it was thrown.

On Mars, gravity pulls less hard, only a bit more than a third as strongly as on Earth. A ball thrown up at the same speed would slow down more gently, so it would go about two and a half times as high and stay up about two and a half times as long.

Worked examples

Example 1. Ria's bike ride. Ria writes down how far she has ridden at the end of each minute.

Minute 0 1 2 3 4 5
Total distance (metres) 0 200 450 750 900 900

What happened in each minute?

  1. Minute 1: from 0 to 200. She rode 200 metres.
  2. Minute 2: from 200 to 450. She rode 250 metres, more than before, so she sped up.
  3. Minute 3: from 450 to 750. She rode 300 metres. Faster again. This was her fastest minute.
  4. Minute 4: from 750 to 900. Only 150 metres. She slowed down, maybe going up a hill.
  5. Minute 5: from 900 to 900. No distance at all. She stopped.

The total distance only ever goes up (or stays the same). But the speed went up and then down. To see the speed, look at the gaps, not the totals.

Example 2. Which is which? Three toy cars leave marks every second, with these gaps (in cm):

  • Car A: 5, 5, 5, 5. Steady speed.
  • Car B: 12, 9, 6, 3, 0. Slowing down, then stopped.
  • Car C: 2, 4, 6, 8. Speeding up.

Example 3. Finding the marks from the gaps. A sled starts at the top of a hill. In each second it goes 2 metres more than in the second before, starting with 2 metres in the first second. How far from the top is it after 4 seconds?

  1. Gaps: 2, 4, 6, 8 metres.
  2. Add them up (small changes add up to the big change, from lesson 1): 2 + 4 + 6 + 8 = 20 metres.
  3. Marks: 0, 2, 6, 12, 20. It is speeding up.

Example 4. Stopping distance. A bike is braking. Its gaps are 6, 4, 2, 0 metres. How far does it travel while braking?

  1. Add the gaps: 6 + 4 + 2 + 0 = 12 metres.
  2. This is why cyclists and drivers must brake early: even while slowing down, they keep moving for a while.

Try it

  1. A runner's gaps each second are 7, 7, 7, 7 metres. Is she speeding up, slowing down, or steady?
  2. A scooter's gaps each second are 1, 3, 5, 7 metres. What is happening? How far did it go in 4 seconds?
  3. A ball rolling on grass has gaps 10, 7, 4, 1, 0 (in cm). What is happening? When did it stop?
  4. Marks for a toy train every second are at 0, 3, 6, 9, 12 cm. Write the gaps. Is it steady?
  5. Marks for a skateboard are at 0, 1, 4, 9, 16 metres. Write the gaps. What is it doing?
  6. A car's total distances at the end of each minute are 0, 500, 1000, 1000, 1400 metres. Describe the trip.
  7. A ball is thrown up. Where is it moving slowest? Why?
  8. Grown-up question: in the braking-car table, the gaps go down by 2 each second. If the car had started braking when it was going twice as fast (first gap 16 metres, then going down by 2 each second), how far would it travel before stopping?
Answers
  1. Steady: the gaps are all the same.
  2. The gaps grow: it is speeding up. Distance: 1 + 3 + 5 + 7 = 16 metres.
  3. The gaps shrink: it is slowing down. The last gap is 0, so it had stopped by the end of the fourth second: in the fifth second it did not move at all.
  4. Gaps: 3, 3, 3, 3. Steady.
  5. Gaps: 1, 3, 5, 7. The gaps grow, so it is speeding up (perhaps rolling down a hill).
  6. First minute: 500 m. Second minute: 500 m, the same, steady. Third minute: 0 m, stopped (maybe at a red light or in traffic). Fourth minute: 400 m, moving again, a bit slower than at first.
  7. At the very top. It has used up all its upward speed and is about to start falling. For an instant, it is stopped.
  8. Gaps: 16, 14, 12, 10, 8, 6, 4, 2, 0. Add them: 16 + 14 + 12 + 10 + 8 + 6 + 4 + 2 = 72 metres. That is more than three and a half times the 20 metres before, although the speed was only doubled. Going twice as fast takes about four times as far to stop, which is why speed limits matter.

Common mistake

Looking at the totals instead of the gaps. In Ria's ride, the total distance keeps going up even when she slows down. Speeding up or slowing down shows in the gaps between marks, not in the marks themselves.

Thinking a ball at the top of its throw is going fast. It is at its highest, but its speed is at its lowest: zero, for an instant. Highest and fastest are different questions, just like "warmest" and "warming fastest" in lesson 3.

Where it leads