15.04 Speed, distance and time (Extended)
Speed, distance and time
The average speed of a vehicle is related to the total distance travelled and the total time taken by the formula:
$\text{average speed}=\frac{\text{total distance travelled}}{\text{total time taken}}$average speed=total distance travelledtotal time taken
$S=\frac{D}{T}$S=DT
We can rearrange the formula for speed to make either $D$D or $T$T the subject. For example, rearranging to make $T$T the subject:
| $S$S | $=$= | $\frac{D}{T}$DT |
|
|
| $S\times T$S×T | $=$= | $\frac{D}{T}\times T$DT×T |
multiply both sides by $T$T |
|
| $ST$ST | $=$= | $D$D |
|
|
| $T$T | $=$= | $\frac{D}{S}$DS |
divide both sides by $S$S |
Similarly we can rearrange to make $D$D the subject and the formulas are below.
| Unknown time | Unknown distance |
|---|---|
| $T=\frac{D}{S}$T=DS | $D=S\times T$D=S×T |
Worked example
Example 1
Jim drives his car from Sydney to Canberra, a distance of $279$279 km. The journey takes him $3$3 hours and $45$45 minutes, including lunch and fuel stops along the way.
(a) What is Jim's average speed for the journey, correct to the nearest km/h?
Think: Whenever we use a formula, we must keep units consistent. In this case, speed is expressed in kilometres per hour and distance is in kilometres, so time must be expressed in hours.
Our first step is to convert the time of $3$3 hours $45$45 minutes into hours. Because there are $60$60 minutes in an hour, $45$45 minutes is equivalent to $\frac{45}{60}$4560 or $0.75$0.75 hours. Therefore $3$3 hours $45$45 minutes is the same as $3.75$3.75 hours.
Do: Substituting into the speed-distance-time formula:
| $S$S | $=$= | $\frac{D}{T}$DT |
|
| $=$= | $\frac{279}{3.75}$2793.75 |
substitute $D=279$D=279 and $T=3.75$T=3.75 |
|
| $=$= | $74.4$74.4 |
|
|
| $=$= | $74$74 km/h (nearest km/h) |
|
(b) How far could Jim travel at this same average speed in $40$40 minutes. Round your answer to the nearest kilometre.
Think: A time of $40$40 minutes is equivalent to $\frac{40}{60}$4060 hours, which simplifies to $\frac{2}{3}$23 of an hour. In this case it is easier to leave the time as a fraction because the decimal is recurring. It is also easier to substitute our values into the formula when $D$D is the subject.
Do:
| $D$D | $=$= | $ST$ST |
|
| $=$= | $74\times\frac{2}{3}$74×23 |
substitute $S=74$S=74 and $T=\frac{2}{3}$T=23 |
|
| $=$= | $\frac{148}{3}$1483 |
|
|
| $=$= | $49.\overline{3}$49.3 |
|
|
| $=$= | $49$49 km (nearest km) |
|
(c) If Jim travelled at this same average speed, how long would it take him to drive from Sydney to Melbourne, a distance of $875$875 km. Give your answer in hours and minutes.
Think: This time, it will be easier to substitute our values into the formula when $T$T is the subject.
Do:
| $T$T | $=$= | $\frac{D}{S}$DS |
|
| $T$T | $=$= | $\frac{875}{74}$87574 |
substitute $D=875$D=875 and $S=74$S=74 |
| $T$T | $=$= | $11.824\ldots$11.824… |
|
| $T$T | $=$= | $11$11 hours $49$49 minutes (nearest minute) |
|
Reflect: In the final steps of the calculation, notice that our value for time was equal to $11.82432\ldots$11.82432… hours. To convert this time into hours and minutes, we simply multiplied the decimal portion by $60$60. To isolate the decimal portion, subtract $11$11 first, then multiply by $60$60. This gives $49.4594\ldots$49.4594… minutes, or $49$49 minutes, rounded to the nearest minute.
Practice questions
Question 1
A man drives a truck $144$144 km in $2$2 hours, then stops to refuel and eat lunch at a petrol station for an hour. He then drives the truck for another $5$5 hours, covering $96$96 km.
What was the truck’s average speed throughout the whole journey (in kilometres per hour)?
Question 2
If a galapagos turtle travels $0.306$0.306 km at a speed of $0.09$0.09 km/hr, how long will it take the animal to cross the whole distance?
Question 3
Two animals were tracked and their movements were measured over different time periods. The cat travelled $10$10 km in $0.1$0.1 hours, while the rabbit travelled $34$34 km in $0.3$0.3 hours.
Continuing at the rate measured, how far would the cat travel in $0.3$0.3 hours?
Question 4
Sally states that her trip between Brisbane and Townsville had an average speed of $91$91 km/h. The table shown displays the distances between two towns (in kilometres). For example, the distance between Adelaide and Brisbane is $2063$2063 km.
How long did it take Sally to travel from Brisbane to Townsville? Give your answer correct to two decimal places.
| Adelaide | ||||||||
| $1542$1542 | Alice Springs | |||||||
| $2063$2063 | $3012$3012 | Brisbane | ||||||
| $3143$3143 | $2324$2324 | $1717$1717 | Cairns | |||||
| $3053$3053 | $1511$1511 | $3415$3415 | $2727$2727 | Darwin | ||||
| $728$728 | $2270$2270 | $1674$1674 | $3054$3054 | $3781$3781 | Melbourne | |||
| $2724$2724 | $3630$3630 | $4384$4384 | $5954$5954 | $4045$4045 | $3452$3452 | Perth | ||
| $1420$1420 | $2644$2644 | $996$996 | $2546$2546 | $4000$4000 | $868$868 | $4144$4144 | Sydney | |
| $2525$2525 | $2096$2096 | $1467$1467 | $374$374 | $2556$2556 | $2857$2857 | $5728$5728 | $2494$2494 | Townsville |
Question 5
Carl wanted to fly from Perth to Adelaide with an overnight stop in Sydney.
Carl's first plane took $4$4 hours of flight time. If the airplane was travelling at an average speed of $825$825 km/h, what is the distance between Perth and Sydney as it appears on the map, measured in scaled units on the map?
Give your answer to one decimal place.
Carl's second trip took $2.2$2.2 hours. If the distance between Sydney and Adelaide appears to be $2.2$2.2 units on the map, what was the average speed of the airplane? Give your answer to the nearest integer.
Carl's overnight stay at a hotel in Sydney costs $$120$120. If the flight company charges $11$11 cents/km for the flight, what was the total cost of the trip?
Give your answer to the nearest cent.