Speed
Speed is a rate that compares distance with time, so a car that covers \(150\) km in \(2\) hours travels at \(150 \div 2 = 75\) km/h. In NSW Year 8 maths, students use the speed formula in all three forms to find speed, distance or time (NESA outcome MA4‑RAT‑C‑01).
You will change minutes into fractions or decimals of an hour and back, convert between km/h and m/s, and find the average speed for a trip with several parts and rests, rather than averaging the speeds. You will also compare runners, work out how long a bus stops each hour, find when a fast water taxi catches a ferry, and find the speed a bus needs to get back on time. It is part of Year 8 Time and Travel.
Every question with a fully worked solution.
- Speed - Video - Finding the average speed of a car in km per hour Watch
Theory
Speed is a rate: the distance travelled in each unit of time. \(S = \dfrac{D}{T}\), \(D = S \times T\) and \(T = \dfrac{D}{S}\). A car that goes \(270\) km in \(3\) h and \(160\) km in \(2\) h averages \(430 \div 5 = 86\) km/h. This Year 8 maths page shows how.
Speed is the distance travelled per unit of time. Its units name both, such as km/h or m/s.
One formula, three ways: \(S = \dfrac{D}{T}\), \(D = S \times T\), \(T = \dfrac{D}{S}\). Write the formula, substitute, then write the units.
The units must match. km/h needs kilometres and hours; m/s needs metres and seconds. Change units before substituting. m/s \(\times 3.6 =\) km/h and km/h \(\div 3.6 =\) m/s.
Minutes to hours: divide by \(60\): \(15\) min \(= 0.25\) h; \(20\) min \(= \dfrac{1}{3}\) h (keep the fraction). Decimal hours to minutes: multiply the decimal part by \(60\): \(4.2\) h \(= 4\) h \(12\) min.
Average speed \(=\) total distance \(\div\) total time, counting every stop. It is not the mean of the separate speeds.
Harder problems. Time to catch up \(=\) the gap \(\div\) the distance gained each hour (travellers moving towards each other close the gap at the sum of their speeds). Speed needed \(=\) distance left \(\div\) time left.
Minutes as part of an hour:
| Minutes | Of an hour | Decimal |
|---|---|---|
| 6 | \(\dfrac{1}{10}\) | 0.1 |
| 12 | \(\dfrac{1}{5}\) | 0.2 |
| 15 | \(\dfrac{1}{4}\) | 0.25 |
| 20 | \(\dfrac{1}{3}\) | \(0.333\ldots\) |
| 30 | \(\dfrac{1}{2}\) | 0.5 |
| 45 | \(\dfrac{3}{4}\) | 0.75 |
Solving a speed problem
- Write the formula you need.
- Match the units (change minutes to hours, or km to m).
- Substitute and calculate.
- Answer with units, changing decimal hours to hours and minutes if needed.
| \(3 \times 90\) | \(=\) | \(270 \text{ km}\) |
| \(270 + 160\) | \(=\) | \(430 \text{ km}\) |
| \(3 + 2\) | \(=\) | \(5 \text{ h}\) |
| \(430 \div 5\) | \(=\) | \(86\) |
\(86\) km/h
| \(\dfrac{30 \times 1000}{60 \times 60}\) | \(=\) | \(\dfrac{300}{36} = 8\dfrac{1}{3}\) |
About \(8\dfrac{1}{3}\) m/s (or \(30 \div 3.6\)).
| \(36 \text{ min}\) | \(=\) | \(\dfrac{3}{5} \text{ h}\) |
| \(45 \times \dfrac{3}{5}\) | \(=\) | \(27\) |
\(27\) km
(i) Find its speed in km/h.
(ii) Write it in m/s.
| \(10.8 \times 4\) | \(=\) | \(43.2\) |
| \(43.2 \div 3.6\) | \(=\) | \(12\) |
(i) \(43.2\) km/h (\(15\) min \(= \dfrac{1}{4}\) h)
(ii) \(12\) m/s (check: \(10\,800\) m \(\div\) \(900\) s \(= 12\))
| \(148 \div 40\) | \(=\) | \(3.7 \text{ h}\) |
| \(0.7 \times 60\) | \(=\) | \(42\) |
\(3\) h \(42\) min after 06:15: it arrives at 09:57.
(i) Find its speed on the first part.
(ii) Find its average speed for the whole trip.
| \(30 \div \dfrac{25}{60}\) | \(=\) | \(72\) |
| \(30 + 21\) | \(=\) | \(51\) |
| \(25 + 5 + 30\) | \(=\) | \(60 \text{ min}\) |
(i) \(72\) km/h
(ii) \(51\) km in \(1\) h: \(51\) km/h
(ii) A coach must cover \(240\) km in \(3\) h. The first \(100\) km take \(1.25\) h. What speed does it need for the rest?
(i) At 8:20 the cyclist is \(18 \times \dfrac{20}{60} = 6\) km ahead. The car gains \(54 - 18 = 36\) km each hour.
| \(6 \div 36 \times 60\) | \(=\) | \(10 \text{ min}\) |
| \(140 \div 1.75\) | \(=\) | \(80\) |
(i) At 8:30 am
(ii) \(140\) km in \(1.75\) h: \(80\) km/h
Common pitfalls
Frequently asked questions
What is the formula for speed?
Speed = distance ÷ time. Rearranged, distance = speed × time and time = distance ÷ speed.
How do I change km/h to m/s?
Divide by 3.6. To change m/s to km/h, multiply by 3.6.
How do I change minutes to hours?
Divide by 60. For example, 45 minutes is 0.75 hours.
How do I find average speed?
Divide the total distance by the total time, including any stops.
How do I change 3.6 hours to hours and minutes?
Multiply the decimal part by 60: 0.6 × 60 = 36, so 3 h 36 min.