Sara cycles from her home to a shop, stays at the shop, and then cycles home. The distance-time graph below shows information about her journey.
(a)Write down the distance from Sara's home when t = 15 minutes.(1)
(b)Work out how long Sara stayed at the shop.(1)
(c)Describe what is happening to Sara's distance from home between t = 25 and t = 45 minutes.(1)
(Total for Question 1 is 3 marks)
2
Use the distance-time graph from Question 1 to answer this question.
(a)Calculate Sara's average speed, in km/h, while cycling to the shop.(2)
(b)Calculate Sara's average speed, in km/h, while cycling home.(2)
(Total for Question 2 is 4 marks)
3
The graph below can be used to change between pounds (GBP) and euros. It is a straight line through the origin, passing through the point (100, 115).
(a)Use the graph to change 40 pounds to euros.(1)
(b)Use the graph to change 69 euros to pounds.(1)
(c)Work out the number of euros equal to 1 pound.(1)
(Total for Question 3 is 3 marks)
4
Containers P, Q and R are shown below. Each is filled with water at a constant rate. P is a cylinder (the same width all the way up). Q is a cone with its point at the bottom, getting wider towards the top. R is a bottle with a wide base that narrows to a thin neck at the top. Graphs A, B and C show depth of water against time as each container is filled. Graph A is a straight line. Graph B is a curve that starts steep and becomes less steep over time. Graph C is a curve that starts shallow and becomes steeper over time.
(a)Which graph, A, B or C, matches container P?(1)
A
B
C
(b)Which graph, A, B or C, matches container Q?(1)
A
B
C
(c)Which graph, A, B or C, matches container R?(1)
A
B
C
(Total for Question 4 is 3 marks)
5
A vase is made from a wide cylindrical base joined to a narrower cylindrical top section. Water is poured into the vase at a constant rate. Sketch a graph to show how the depth of water in the vase changes with time. Label your axes Time and Depth.
(Total for Question 5 is 2 marks)
6
The speed-time graph below shows a car's journey. Speed (m/s) is on the vertical axis, 0 to 20, and Time (seconds) is on the horizontal axis, 0 to 40. The graph rises in a straight line from (0,0) to (10,20), is horizontal from (10,20) to (25,20), then falls in a straight line from (25,20) to (40,0). Which section of the graph shows the car travelling at a constant speed?
A) 0 to 10 seconds
B) 10 to 25 seconds
C) 25 to 40 seconds
(Total for Question 6 is 1 mark)
7
Use the speed-time graph described in Question 6 to answer this question.
(a)Describe how the speed of the car changes during the first 10 seconds.(1)
(b)Describe how the speed of the car changes between t = 25 and t = 40 seconds.(1)
(c)State the maximum speed reached by the car.(1)
(Total for Question 7 is 3 marks)
8
A train leaves Uppington station at 09:00. It travels 120 km to Lowbridge station, arriving at 10:30. It waits at Lowbridge for 20 minutes, then travels back to Uppington, non-stop, arriving at 12:25.
(a)Draw a distance-time graph to show the train's journey.(3)
(b)Calculate the average speed of the train on the outward journey, in km/h.(2)
(c)Calculate the average speed of the train on the return journey, in km/h. Give your answer correct to 1 decimal place.(2)
(d)Calculate the average speed of the train for the whole journey (Uppington to Lowbridge and back to Uppington), including the time spent waiting at Lowbridge. Give your answer correct to 3 significant figures.(3)
(Total for Question 8 is 10 marks)
9
Two mobile phone tariffs are shown on the graph below as straight lines of cost (£) against minutes used, for up to 200 minutes. Tariff A has no fixed charge and costs 15p per minute (a straight line through the origin passing through (200, 30)). Tariff B has a fixed monthly charge of £10 plus 5p per minute (a straight line starting at (0, 10) and passing through (200, 20)).
(a)Write down the fixed monthly charge for Tariff B.(1)
(b)Work out the cost per minute for Tariff A.(2)
(c)A customer uses 120 minutes in a month. Use the graphs to determine which tariff is cheaper, and by how much.(2)
(Total for Question 9 is 5 marks)
10
Priya walks from home to the bus stop, waits for the bus, then travels quickly by bus to college. Sketch a distance-time graph to show Priya's journey. Label your axes Time and Distance from home.
(Total for Question 10 is 3 marks)
11
Tom walks 3 km from home to the park in 40 minutes. He then jogs 5 km from the park to the leisure centre in 25 minutes.
(a)Calculate Tom's average speed while walking, in km/h.(2)
(b)Calculate Tom's average speed while jogging, in km/h.(2)
(c)Calculate Tom's average speed for the whole journey from home to the leisure centre, in km/h. Give your answer correct to 1 decimal place.(3)
(Total for Question 11 is 7 marks)
12
A motorbike accelerates uniformly from rest, reaching a speed of 18 m/s after 12 seconds. Calculate the acceleration of the motorbike during these 12 seconds. State the units of your answer.
(Total for Question 12 is 3 marks)
13
The motorbike from Question 12 then travels at a constant 18 m/s for a further 20 seconds, before decelerating uniformly to rest in the next 6 seconds. Calculate the total distance travelled by the motorbike over the whole 38 seconds, using the area under the speed-time graph.
(Total for Question 13 is 4 marks)
14
Chidi sketches this distance-time graph for a car journey. The graph includes a vertical section: it rises from (10, 20) straight up to (10, 35) before continuing. Explain why this graph cannot show a real journey.
(Total for Question 14 is 2 marks)
15
A cyclist's speed-time graph for a 29 second journey is described below. She accelerates uniformly from rest to 6 m/s in 4 seconds, maintains 6 m/s for 15 seconds, then decelerates uniformly to rest over the next 3 seconds. She remains stationary for 5 seconds, then accelerates uniformly to 4 m/s over the final 2 seconds.
(a)Calculate the total distance travelled by the cyclist during the 29 seconds shown on the graph.(4)
(b)Calculate the cyclist's average speed for the whole 29 seconds. Give your answer correct to 2 significant figures.(2)
(Total for Question 15 is 6 marks)
16
Freya leaves her house at 14:00 and cycles at a constant speed of 15 km/h towards her friend Meera's house, which is 20 km away, along a straight route. At the same time, 14:00, Meera leaves her own house and cycles towards Freya's house, along the same route, at a constant speed of 10 km/h.
(a)By setting up an equation, or otherwise, find the time at which Freya and Meera meet.(3)
(b)Find the distance from Freya's house at which they meet.(2)
(Total for Question 16 is 5 marks)
17
A cheetah's distance-time graph during a short sprint is a straight line passing through the origin and the point (2.5, 0.075), where time is measured in seconds and distance in kilometres.
(a)Calculate the cheetah's speed in km/s.(2)
(b)Convert this speed to km/h.(2)
(Total for Question 17 is 4 marks)
18
A garden pond is filled using a hose that delivers water at a constant rate of 15 litres per minute. The pond has a capacity of 900 litres. After 20 minutes of filling, the hose is turned off for 8 minutes to check the water is level, before being turned back on at the same rate until the pond is full.
(a)Draw a graph to show the volume of water in the pond, in litres, against time, in minutes, from when filling starts until the pond is full.(3)
(b)Work out the total time taken to fill the pond.(2)
(Total for Question 18 is 5 marks)
Mark scheme · 4.4 Real Life and Distance Time Graphs
Question 1
(a) B1 6 (km) cao
(a) Answer: 6 km
(b) B1 10 (minutes) cao
(b) Answer: 10 minutes
(c) B1 Sara cycles back home at a constant speed, arriving home at t = 45 minutes (oe)
(c) Answer: She cycles back home at a constant speed and arrives home at t = 45 minutes.
Question 2
(a) M1 6 divided by (15/60) oe, or 6 divided by 0.25
(a) A1 24 (km/h) cao
(a) Answer: 24 km/h
(b) M1 6 divided by (20/60) oe
(b) A1 18 (km/h) cao
(b) Answer: 18 km/h
Question 3
(a) B1 46 (euros) ft from a correct reading of the graph
(a) Answer: 46 euros
(b) B1 60 (pounds) ft from a correct reading of the graph
(b) Answer: 60 pounds
(c) B1 1.15 cao
(c) Answer: 1.15 euros
Question 4
(a) B1 A cao
(a) Answer: A
(b) B1 B cao
(b) Answer: B
(c) B1 C cao
(c) Answer: C
Question 5
B1 two straight line sections (not curves), with a clear change of gradient where the container changes width
B1 second section (narrow top) steeper than the first section (wide base)
Answer: A graph made of two straight line sections: a less steep line for the wide base, followed by a steeper straight line for the narrow top, meeting at the point where the container changes width.
Question 6
B1 B cao
Answer: B) 10 to 25 seconds
Question 7
(a) B1 the car accelerates steadily / speed increases at a constant rate from 0 to 20 m/s (oe)
(a) Answer: The car accelerates steadily from 0 m/s to 20 m/s.
(b) B1 the car decelerates steadily / slows down at a constant rate from 20 m/s to 0 m/s (oe)
(b) Answer: The car decelerates steadily from 20 m/s to 0 m/s (comes to rest).
(c) B1 20 (m/s) cao
(c) Answer: 20 m/s
Question 8
(a) B1 correct straight line from (09:00, 0) to (10:30, 120)
(a) B1 correct horizontal line from (10:30, 120) to (10:50, 120)
(a) B1 correct straight line from (10:50, 120) to (12:25, 0)
(a) Answer: Straight line rising from (09:00, 0 km) to (10:30, 120 km); horizontal line from (10:30, 120 km) to (10:50, 120 km); straight line falling from (10:50, 120 km) to (12:25, 0 km).
(b) M1 120 divided by 1.5 oe
(b) A1 80 (km/h) cao
(b) Answer: 80 km/h
(c) M1 120 divided by (95/60) oe, or 120 divided by 1.5833...
(c) A1 awrt 75.8 (km/h)
(c) Answer: 75.8 km/h
(d) M1 total distance = 240 (km)
(d) M1 total time = 205 minutes oe (3 h 25 min, or 205/60 hours, or 3.4167 hours)
(d) A1 awrt 70.2 (km/h)
(d) Answer: 70.2 km/h
Question 9
(a) B1 10 (pounds) cao
(a) Answer: £10
(b) M1 30 / 200 oe
(b) A1 0.15 (pounds), or 15p, cao
(b) Answer: £0.15 per minute (15p)
(c) M1 correct cost for A (£18) or B (£16) found from the graph or by calculation
(c) A1 Tariff B cheaper by £2, cao
(c) Answer: Tariff B is cheaper, by £2 (Tariff A = £18, Tariff B = £16).
Question 10
B1 straight line with positive gradient from the origin (walking)
B1 horizontal section following the first line (waiting for the bus)
B1 straight line with a steeper positive gradient than the first section, following the horizontal section (bus, faster than walking)
Answer: Line rises from the origin (walking to the bus stop), then a horizontal section (waiting), then a steeper rising line (travelling by bus, faster than walking).
Question 11
(a) M1 3 divided by (40/60) oe
(a) A1 4.5 (km/h) cao
(a) Answer: 4.5 km/h
(b) M1 5 divided by (25/60) oe
(b) A1 12 (km/h) cao
(b) Answer: 12 km/h
(c) M1 total distance = 8 (km)
(c) M1 total time = 65 minutes oe (13/12 hours, or 1.0833 hours)
(c) A1 awrt 7.4 (km/h)
(c) Answer: 7.4 km/h
Question 12
M1 18 divided by 12 oe
A1 1.5 cao
B1 m/s2 (correct units, independent)
Answer: 1.5 m/s2
Question 13
M1 area of first triangle = 0.5 x 12 x 18 (= 108)
M1 area of rectangle = 20 x 18 (= 360)
M1 area of second triangle = 0.5 x 6 x 18 (= 54)
A1 522 (m) cao
Answer: 522 metres
Question 14
B1 identifies the vertical section as the fault
B1 correct reason, e.g. the distance changes (by 15 km) while the time does not change, meaning the car would travel instantly (infinite speed), which is impossible (oe)
Answer: The vertical section shows the distance increasing by 15 km while the time stays the same (t = 10), which would mean travelling 15 km instantly (infinite speed) - impossible for a real car.
Question 15
(a) M1 area of acceleration triangle = 0.5 x 4 x 6 (= 12), and area of constant-speed rectangle = 15 x 6 (= 90)
(a) M1 area of deceleration triangle = 0.5 x 3 x 6 (= 9)
(a) M1 recognises the stationary 5 seconds adds 0 distance, and area of the final acceleration triangle = 0.5 x 2 x 4 (= 4)
(a) A1 14:48 cao (or 48 minutes after they set off)
(a) Answer: 14:48 (48 minutes after they set off)
(b) M1 15 x 0.8 oe (ft from part a)
(b) A1 12 (km) cao
(b) Answer: 12 km
Question 17
(a) M1 0.075 divided by 2.5 oe
(a) A1 0.03 (km/s) cao
(a) Answer: 0.03 km/s
(b) M1 0.03 x 3600 oe, ft from part (a)
(b) A1 108 (km/h) cao
(b) Answer: 108 km/h
Question 18
(a) B1 correct straight line from (0,0) to (20,300)
(a) B1 correct horizontal line from (20,300) to (28,300)
(a) B1 correct straight line, same gradient as the first, from (28,300) to (68,900)
(a) Answer: Straight line from (0,0) to (20,300); horizontal line from (20,300) to (28,300); straight line (same gradient as the first) from (28,300) to (68,900).
(b) M1 (900 - 300) / 15 (= 40), or 900 / 15 (= 60) plus the 8 minute pause