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18 Speed, Distance and Time
18.1 Speed
In this section we introduce the idea of speed, considering both instantaneous speed and average speed.
Instantaneous speed = speed at any instant in time
Average speed = distance travelled time taken
If a car travels 100 miles in 2 hours, average speed = (^1002) = 50 mph The car does not travel at a constant speed of 50 mph; its speed varies during the journey between 0 mph and, perhaps, 70 mph. The speed at any time is called the instantaneous speed. The following table lists units in common use for speed and their abbreviations:
Example 1
Judith drives from Plymouth to Southampton, a distance of 160 miles, in 4 hours. She then drives from Southampton to London, a distance of 90 miles, in 1 hour and 30 minutes. Determine her average speed for each journey.
Distance Time Speed Abbreviation mile hours miles per hour mph kilometres hours kilometres per hour km/h metres hours metres per hour m/h metres seconds metres per second m/s feet seconds feet per second f.p.s. or ft. per sec. centimetres seconds centimetres per cm/sec or cm/s second
Solution
Plymouth to Southampton Average speed = (^1604) = 40 mph Southampton to London Time taken = 1 hour and 30 minutes = 1 12 hours or^32 hours
Average speed = 90 ÷^32
= 90 ×^23 = 60 mph
Example 2
John can type 960 words in 20 minutes. Calculate his typing speed in: (a) words per minute, (b) words per hour.
Solution
(a) Typing speed = (^96020) = 48 words per minute (b) Typing speed = 48 × 60 = 2880 words per hour
- Eva drives from Edinburgh to Dover in 3 stages: Start Time Finish Time Distance Edinburgh to Leeds 0620 0920 210 miles Leeds to London 1035 1305 200 miles London to Dover 1503 1703 78 miles
Calculate her average speed for each stage of her journey.
- Delia drives 220 km in 3 12 hours. Calculate her average speed correct to the nearest km/h.
18.2 Calculating Speed, Distance and Time
In this section we extend the ideas of speed to calculating distances and times , using the following formulae:
Speed = DistanceTime
Distance = Speed × Time
Time = DistanceSpeed
Example 1
Jane drives at an average speed of 45 mph on a journey of 135 miles. How long does the journey take?
Solution
Time = distancespeed
= 3 hours
Example 2
Chris cycles at an average speed of 8 mph. If he cycles for 6 12 hours, how far does he travel?
Solution
Distance = speed ×time
= 8 × (^6 )
= 52 miles
Example 3
Nikki has to travel a total of 351 miles. She travels the first 216 miles in 4 hours. (a) Calculate her average speed for the first part of the journey. (b) If her average speed remains the same, calculate the total time for the complete journey.
Solution
(a) Average speed = distancetime
= (^2164) = 54 mph
(b) Time = distance speed
= (^35154) = 6.5 hours
(b) How far can Andrew row in: (i) 12 seconds, (ii) 3 12 minutes, (iii) 4 hours?
- A snail moves 5 m in 2 hours, If the snail moves at the same speed, calculate: (a) the time it takes to move 20 m, (b) the distance it would move in 3 1 2 hours, (c) the time it takes to moves 1 m, (d) the distance that it moves in 15 minutes.
- Laura drives for 3 hours at 44 mph. Clare drives 144 miles in 4 hours. (a) Who travels the greater distance? (b) Whose speed is the slower? (c) How far would Laura travel if she drove for 3 hours at the same speed as Clare?
- A lorry travels for 3 hours at 48 mph and then for 2 hours at 53 mph. (a) What is the total distance travelled by the lorry? (b) What is the average speed for the whole journey?
- Sally drives for 2 1 2 hours at 50 mph, then drives 80 miles at 40 mph, and finally drives for 30 minutes at 60 mph. (a) Calculate the total distance that Sally drives. (b) Calculate the time that Sally takes for the journey. (c) Calculate her average speed for the whole journey.
18.3 Problems with Mixed Units
In this section we consider working with mixed units, and with changing units used for speeds.
Example 1
(a) Convert 1 hour 24 minutes to hours (decimal). (b) Write 2.32 hours in hours and minutes.
Solution
(a) 2460 = 0.
Therefore, 1 hr 24 mins = 1.4 hours (b) 0.32 × 60 = 19. Therefore, 2.32 hours = 2 hrs 19.2 mins
Example 2
A car travels 200 miles in 3 hours and 20 minutes. Calculate the average speed of the car in mph.
Solution
3 hours 20 minutes = (^3 )
= 3 13 hours Speed = distance ÷ time = 200 ÷ (^3 )
= 200 ÷^103
= 200 × 103 = 60 mph
Example 3
An athlete runs 1500 m in 3 minutes and 12 seconds. Calculate the average speed of the athlete in m/s.
Solution
3 minutes 12 seconds = 3 × 60 + 12 = 192 seconds
Exercises
- Convert the following times from hours and minutes to hours, giving your answers as mixed numbers and decimals, correct to 2 decimal places. (a) 1 hour 40 minutes (b) 3 hours 10 minutes (c) 1 hour 6 minutes (d) 2 hours 18 minutes (e) 3 hours 5 minutes (f) 6 hours 2 minutes (g) 1 hour 7 minutes (h) 2 hours 23 minutes
- Change the following times to hours and minutes: (a) 1 1 4 hours (b) 1.2 hours (c) 3.7 hours (d) 4.4 hours (e) 1.45 hours (f) 3.65 hours
- A car travels 60 miles in 50 minutes. Calculate the average speed of the car in mph.
- Jane drives 80 miles in 1 hour and 40 minutes. Calculate her average speed.
- Convert the following speeds to km/h: (a) 60 mph (b) 43 m/s (c) 66 m/s (d) 84 mph
- Convert the following speeds to mph: (a) 16 m/s (b) 82 km/h (c) 48 km/h (d) 7 m/s
- Alec drives 162 km in 2 hours and 12 minutes. Calculate his average speed in: (a) km/h (b) m/s (c) mph Give your answers to 2 decimal places.
- Jai drives 297 miles in 5 hours and 24 minutes. (a) Calculate his average speed in mph. (b) He then drives for a further 1 hour and 28 minutes at the same average speed. How far has he travelled altogether? Give your answers to 2 decimal places.
- A train travels at 40 m/s. Calculate the time it takes to travel: (a) 30 000 m, (b) 50 km, (c) 200 miles.
- A long distance runner runs at an average speed of 7 mph. How long will it take the runner to run: (a) 20 miles, (b) 15 km, (c) 10 000 m?
18.4 Distance-Time Graphs
Graphs that show distance against time can be used to describe journeys. The vertical scale shows the distance from the starting point or reference point.
Distance From Starting Point
Time
Travelling away from starting point at a constant speed Returning to starting point at a constant speed Not moving
The graph above illustrates 3 parts of a journey. The gradient of a straight line gives the speed of the moving object. Gradient is a measure of the speed. Note that a negative gradient indicates that the object is moving towards the starting point rather than away from it.
Distance
Time
Step
Rise
Gradient = RiseStep
During the third stage as the child returns, he travels 1000 m in 100 seconds. Speed = distancetime
= (^1000100) = 10 m/s
Example 2
On a journey, Rebecca drives at 50 mph for 2 hours, rests for 1 hour and then
drives another 70 miles in 1 1 2 hours.
Draw a distance-time graph to illustrate this journey.
Solution
(^00 12 1 1 12 2 2 12 3 3 12 44 )
20
40
60
80
100
120
140
160
180
200
Distance (miles)
Time (hours)
First stage Second stage Travels 100 miles in 2 hours. Rests, so distance does not change.
Third stage
Travels 70 miles in 1 12 hours.
Example 3
The graph shows how Tom's distance from home varies with time, when he visits Ian.
(a) How long does Tom spend at Ian's? (b) How far is it from Tom's home to Ian's? (c) For how long does Tom stop on the way to Ian's? (d) On which part of the journey does Tom travel the fastest? (e) How fast does Tom walk on the way back from Ian's?
Solution
(a) The longer horizontal part of the graph represents the time that Tom is at Ian's. Time = 90 − 40 = 50 minutes (b) 3000 m (c) Tom stops for 10 minutes, represented by the smaller horizontal part on the graph. (d) He travels fastest on the second part of the journey to Ian's. This is where the graph is steepest. He travels 2000 m in 10 minutes. Speed = (^200010) = 200 m/minute = 200 60 1000
×
= 12 km/h
Distance (m)
Time (mins)
500
1000
0
1500
2000
2500
3000
0 10 20 30 40 50 60 70 80 90 100 110 120
- Describe the 5 parts of the journey (labelled (a), (b), (c), (d) and (e)) represented by the following distance-time graph:
Distance (km)
Time (hours)
0 1 2 3 4 5 6 7 8
100
200
0
300
400
500
600
(a)
(c) (b)
(d)
(e)
- Ray walks 420 m from his house to a shop in 7 minutes. He spends 5 minutes at the shop and then walks home in 6 minutes. (a) Draw a distance-time graph for Ray's shopping trip. (b) Calculate the speed at which Ray walks on each part of the journey.
- Mary sprints 200 m in 30 seconds, rests for 45 seconds and then walks back in 1 12 minutes to where she started the race. (a) Draw a distance-time graph for Mary. (b) Calculate the speed at which Mary runs. (c) Calculate the speed at which Mary walks.
- After morning school, Mike walks home from school to have his lunch. The distance-time graph on the next page describes his journey on one day, showing his distance from home. (a) How far is Mike's home from school? (b) How long does it take Mike to walk home? (c) At what speed does he walk on the way home? Give your answer in m/s. (d) How long does Mike spend at home?
(e) At what speed does he walk back to school? Give your answer in m/s.
Distance (m)
Time (mins)
0 10 20 30 40 50 60 70
300
600
0
900
1200
1500
Home
School
- Helen cycles for 20 minutes at 5 m/s and then for a further 10 minutes at 4 m/s. (a) How far does she cycle altogether? (b) Draw a distance-time graph for her ride.
- The distance-time graph shown is for a 3000 m cross-country race, run by Rachel and James.
Distance (m)
Time (mins)
0 5 10 15 20 25 30 35
500
1000
0
1500
2000
2500
3000
Rachel James
18.5 Other Compound Measures
In the section so far we have considered speed in several different contexts. We will now look at other things such as goals per game, and postage rates.
Example 1
In a football season, Ivor Boot scores 27 goals in 40 matches. Calculate his average scoring rate in goals per match, goals per minute and goals per hour.
Solution
Scoring rate = (^2740)
= 0.675 goals per match
Scoring rate = 27 40 × 90 (there are 90 minutes per match)
= 0.0075 goals per minute
Scoring rate = (^40 271 ) × 2
= 0.45 goals per hour
Example 2
A package has a mass of 200 grams. It can be posted first class for 60p, or second class for 47p. Calculate the cost per gram for first and second class post.
Solution
First Class Cost per gram = 60 200 = 0.3p
Second Class Cost per gram = 20047
= 0.235p
Exercises
- Three boys play football for a school team. The numbers of goals scored and matches played are listed below:
(a) Who scores the most goals per match? (b) Who scores the least goals per match?
- Alison plays 20 games for her school hockey team and scores 18 goals. Each match lasts 90 minutes. Calculate her scoring rate in: (a) goals per hour, (b) goals per minute, (c) goals per match.
- When playing football, Jai claims to be able, on average, to score a goal every 40 minutes. How many goals would you expect him to score in: (a) 90 minutes, (b) 1 hour, (c) 5 matches, (d) 40 matches?
- It costs 96p to send an air mail letter of mass 40 grams to Africa, and 107p to send it to China. (a) Calculate the cost per gram for each destination. (b) If the same rates apply to a 50 gram letter, calculate the cost for each destination.
- A package of mass 80 grams costs 39p to post first class and 31p to post second class. Calculate the cost per gram for first and second class post.
- A taxi driver charges £3.20 for a 4 km journey. How much does he charge: (a) per km, (b) per metre?
Number of Goals Scored Number of Matches Played Ian 16 32 Ben 22 40 Sergio 9 20
