12. Time and Motion

Lesson

When we talk about speed, we need to put it in context by using appropriate units. If someone said that they ran at a speed of $5$5 yesterday, you wouldn't know whether to be impressed or not.

If they had run $5$5 kilometres an hour, that's a decent walking speed but it's not anywhere near a running speed. If they ran $5$5 metres a second that would be more impressive. Road signs often don't have a unit given, but that is because they are standardised across a country. In many countries the standard is kilometres per hour (km/h), but in some countries like the United States and Great Britain the standard is miles per hour instead.

Measurements of speed are a rate. A rate is a ratio between two measurements with different units. Speed is the ratio between distance and time.

To calculate speed, we can quite simply use the fact that $\text{Speed}=\frac{\text{Distance}}{\text{Time}}$Speed=DistanceTime

Some units that are commonly used for speed are:

- metres per second (m/sec)
- kilometres per hour (km/h)
- miles per hour (mph)

When considering which measurement of speed is appropriate to use, it is important to think about the likely distance that would be travelled and how much time that would probably take.

Speed can be measured in any combination of distance and time, but for numbers to really be useful to us we want them to be easy to compare.

For example, if we were told that a fighter jet travels $680556$680556 mm/s this sounds fast, but it is hard to interpret what that really means due to the unit and the size of the number. On the other hand, we know that travelling at $110$110 km/h is quite fast for a car, so if we were told that a fighter jet can travel at $2450$2450 km/h, then we get a much better idea of how fast that really is. Think of useful comparisons to help choose appropriate units of speed.

When choosing an appropriate unit of speed, it is also important to consider the size of the numbers.

In July 2015, Matt Stonie ate $62$62 hot dogs in $10$10 minutes. This sounds like a lot, but what is $62$62 hot dogs/$10$10 minutes as a unitary rate? We could think of it as $372$372 hot dogs/hour, but Matt didn't keep eating for an hour, and this number is quite a large amount to think about. We could also think of this as $0.10\overline{3}$0.103 hot dogs/second, but that is a very small number that doesn't mean much in the context. When did you last think about how much of a hot dog you can eat in a second? It's difficult to know if this is a fast or slow speed. Instead, we can think of this as $6.2$6.2 hot dogs/minute, and by choosing these more appropriate units we get a better grasp of just how fast that is (quite fast!)

What unit is most appropriate for measuring the speed of a person while running?

m/h

Akm/h

Bcm/min

Ccm/s

D

Sophia captured footage of a hawk diving $200$200 metres in $10$10 seconds. Using this as a comparison, what units would be most appropriate for measuring the speed of a feather falling to the ground?

cm/h

Amm/h

Bcm/s

Ckm/s

D

We express speed per a single unit of the described amount of time, for example, $50$50 km/hour means $50$50 km per $1$1** **hour, $30$30 m/min means $30$30 metres for every $1$1** **minute. Understanding this can help us calculate speed and convert units of speed.

Calculate the speed in kilometres per minute, of a vehicle which travels $120$120 km in $2$2 hours.

**Think: ** First convert to km/h, that is kilometres in $1$1** **hour.

**Do: **

$\frac{120\text{ km}}{2\text{ h}}$120 km2 h | $=$= | $\frac{120\text{ km}\div2}{2\text{ h}\div2}$120 km÷22 h÷2 |

$=$= | $\frac{60\text{ km}}{1\text{ h}}$60 km1 h | |

$=$= | $60$60 km/h |

Now, convert this speed to km/min.

**Think:** $\frac{60\text{ km}}{1\text{ h}}=\frac{60\text{ km}}{60\text{ min}}$60 km1 h=60 km60 min, and we want to find the speed per $1$1 min.

**Do:**

$\frac{60\text{ km}}{60\text{ min}}$60 km60 min | $=$= | $\frac{60\text{ km}\div60}{60\text{ min}\div60}$60 km÷6060 min÷60 |

$=$= | $\frac{1\text{ km}}{1\text{ min}}$1 km1 min | |

$=$= | $1$1 km/min |

Remember!

Length unit conversion chart:

Calculate the speed in kilometres per hour (km/h) for each of the following situations.

A car which travels $400$400 km in $5$5 hours.

A cyclist who cycles $50$50 km in $3$3 hours.

Round your answer to one decimal place.

A train which travels $650$650 km in $6$6 hours.

Round your answer to one decimal place.

Convert each of the following speeds from kilometres per hour to metres per minute.

$60$60 km/hr

$5$5 km/hr

$24$24 km/hr