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Positive, negative, zero and changing rates

Change and motion · level 5, lesson 2 of 6

A question to start

A hot-air balloon is coming down to land. Its height above the field drops by 2 metres every second. Its rate of change of height is \(-2\) metres per second: minus two.

Can a speed be negative? Can you go slower than standing still? Something is odd here. A minus sign is telling us something, but what? This lesson is about reading the sign of a rate (plus, minus or zero) as part of a story, and about what happens when the rate itself keeps changing.

The idea

The sign of the rise

In the last lesson, the slope of a line was rise over run. When time (or any other input) goes across from left to right, the run is always positive: we move forward in time. So the sign of the slope is decided by the rise:

  • If the up quantity gets bigger, the rise is positive, and the slope is positive. The line goes up from left to right.
  • If the up quantity gets smaller, the rise is negative (an end value minus a bigger start value), and the slope is negative. The line goes down.
  • If the up quantity stays the same, the rise is 0, and the slope is \(0 \div \text{run} = 0\). The line is flat.
positivegoing up negativegoing down zerono change uprightimpossible here

The bigger the number, ignoring the sign, the steeper the line and the faster the change. A slope of \(-6\) is steeper than a slope of \(+2\); it is just steep going down.

What about an upright line? Its run is 0, and we can never divide by 0, so its slope has no value at all: we say it is undefined. On a distance-time graph an upright line would mean being at many places at the same instant, or moving with no time passing. No real object can do that, so a real distance-time graph never has an upright piece.

What the sign means in a story

The sign always answers the question "is the up quantity growing or shrinking?". What that means depends on the story.

Graph Positive slope Zero slope Negative slope
distance from home against time walking away from home standing still walking back towards home
water in a tank against time filling neither filling nor emptying draining
bank balance against weeks saving no change spending more than earning
temperature against hours warming steady cooling
height of a balloon against time rising hovering coming down

Now the balloon puzzle. The rate of change of height is \(-2\) m/s. That does not mean the balloon goes "slower than stopped". It means the height is getting smaller, by 2 metres each second. The balloon's speed (how fast, ignoring direction) is 2 m/s; the minus sign tells the direction: down.

Scientists have a word for a rate that keeps the sign: velocity. The velocity of the balloon is \(-2\) m/s (taking up as positive). Its speed is 2 m/s, which is the size of the velocity without the sign. Speed is never negative; velocity can be.

A journey with every kind of slope

Here is Maya's afternoon, as distance from home against time.

ABCD 010152535 time (minutes) 04008001200 m
Part Rise Run Slope Story
A \(800 - 0 = 800\) m 10 min \(+80\) m/min walks to the shop
B \(800 - 800 = 0\) m 5 min \(0\) m/min inside the shop
C \(1200 - 800 = 400\) m 10 min \(+40\) m/min strolls on to the park, slowly
D \(0 - 1200 = -1200\) m 10 min \(-120\) m/min runs home, fast

Part D has the steepest line of all, and the most negative slope: she runs home at 120 metres per minute. The sign says "towards home", the size says "fast".

Changing rates: when the graph curves

A straight line has one slope everywhere: the rate is constant. A curve has a slope that changes as you move along it: the rate is changing. There are four basic shapes, and each tells a different story.

up, fasterand faster up, butslowing down down, fasterand faster down, butslowing down

To read a curve, ask two questions:

  1. Is it going up or down? That is the sign of the rate.
  2. Is it getting steeper or flatter? That says whether the change is speeding up or slowing down.

Some examples:

  • A snowball rolling downhill, its mass against time: up, faster and faster (the bigger it is, the more snow it picks up).
  • A cup of hot cocoa's temperature: down, but slowing down. It cools fast at first, then more and more gently as it gets close to room temperature.
  • A sprinter's distance in the first two seconds of a race: up, faster and faster (she is speeding up).
  • A child's height against age, from 18 to 25: up, but slowing down until it is almost flat.
  • Water in a cone-shaped funnel draining out of the bottom: the level goes down, faster and faster near the end, because the narrow part holds little water.

Average and changing: a careful word

On a curve, "the rate" has no single value. If someone says "the cocoa cooled at 3 degrees per minute", they must mean either an average over some time, or the rate at one particular moment. Lesson 3 is about the first; level 6 is all about the second.

Worked examples

1. A bank balance. Leo has 240 dollars on 1 March and 168 dollars on 1 April (31 days later). Find the rate of change, and say what it means.

Rate \(= \dfrac{168 - 240}{31} = \dfrac{-72}{31} \approx -2.32\) dollars per day. Negative: the balance went down, by about 2.32 dollars a day on average.

2. Night temperatures. At 8 p.m. it is 14 °C; at 2 a.m. it is 5 °C. Find the rate of change. If it keeps cooling at that rate, what is the temperature at 6 a.m.?

  1. From 8 p.m. to 2 a.m. is 6 hours. Rate \(= (5 - 14) \div 6 = -9 \div 6 = -1.5\) °C per hour.
  2. From 2 a.m. to 6 a.m. is 4 hours: change \(= -1.5 \times 4 = -6\) °C. Temperature \(= 5 - 6 = -1\) °C.

3. An elevator. An elevator is at floor height 36 m. It moves for 12 seconds and ends at 6 m. Find its velocity and its speed.

Velocity \(= (6 - 36) \div 12 = -30 \div 12 = -2.5\) m/s (going down). Speed \(= 2.5\) m/s.

4. Maya's whole trip. Using the graph above, what was Maya's average velocity for the whole 35 minutes, and her average speed?

  • She ends where she started: change in distance from home \(= 0 - 0 = 0\). Average velocity \(= 0 \div 35 = 0\) m/min.
  • But she really walked \(800 + 400 + 1200 = 2400\) m. Average speed \(= 2400 \div 35 \approx 68.6\) m/min.

The two averages answer different questions: "how far from home did I end up?" against "how much ground did I cover?". That is why the sign matters.

5. Which shape? A bath is being filled, but the tap is slowly turned down. Sketch water against time.

The water keeps going up (positive rate), but each minute adds less than the minute before. So the curve is up, but slowing down: the second shape in the picture.

Try it

  1. Give the sign (positive, negative or zero) of the rate in each story: (a) a candle's height while it burns; (b) a parked car's distance from home; (c) the number of people in a stadium as fans arrive.
  2. A diver's depth below the surface goes from 3 m to 18 m in 10 s. Using "depth below the surface" as the up quantity, what is the rate? Using "height above the surface" (so the diver is at \(-3\) m, then \(-18\) m), what is it?
  3. A tank holds 500 L and drains at a steady 25 L/min. Write the rate with the right sign. When is it empty?
  4. In Maya's graph, which part had the gentlest non-zero slope? What was her speed in it, in km/h?
  5. Which of these is the steepest: slope \(+3\), slope \(-5\), slope \(0.5\), slope \(-1\)?
  6. A ball is thrown straight up and caught at the same height. Describe its height-against-time graph using the four curve shapes. Where is the rate zero?
  7. Draw (or describe) a graph of the water level in a bathtub for this story: the tap runs steadily, then someone gets in, then the tap is turned off, then the plug is pulled and the water drains, slowly at the end.
  8. A town's population went from 12 400 to 11 650 in 5 years. Find the average rate of change. What would the population be after 3 more years at the same rate?
Answers
  1. (a) negative (it gets shorter); (b) zero (no change); (c) positive.
  2. Depth: \((18 - 3) \div 10 = +1.5\) m/s (depth growing). Height: \((-18 - (-3)) \div 10 = -15 \div 10 = -1.5\) m/s. Same motion, opposite sign: the sign depends on which way you call "up". Always say what you measure.
  3. Rate \(= -25\) litres per minute. Empty after \(500 \div 25 = 20\) minutes.
  4. Part C, \(+40\) m/min. In an hour that is \(40 \times 60 = 2400\) m, so 2.4 km/h.
  5. Slope \(-5\): its size, 5, is the biggest. It is steep going down.
  6. Going up the curve is "up, but slowing down" (the ball slows as it rises); at the top it is flat for an instant, where the rate is zero; coming down it is "down, faster and faster" (the ball speeds up as it falls).
  7. Steady rise (straight line up); a sudden jump up when the person gets in (the level rises at once though no extra water came in); a flat piece when the tap is off; then a falling curve while draining that is "down, but slowing down", flattening near the end as the water gets shallow and drains slowly.
  8. \((11\,650 - 12\,400) \div 5 = -750 \div 5 = -150\) people per year. After 3 more years: \(11\,650 - 3 \times 150 = 11\,200\).

Common mistake

Thinking negative means slow. A slope of \(-120\) is fast: it is the size that tells how fast, and the sign that tells which way. A slope near 0, like \(+0.5\) or \(-0.5\), is the slow one.

Mixing up "going down" with "slowing down". A graph going down has a negative rate. A graph getting flatter has a rate getting closer to zero, which means slowing down. A curve can go up while slowing down (a plant near its full height), or go down while speeding up (a stone falling, measured as height).

Where it leads