Graphs in Practical Situations (Copy)
Question 1
The graph of distance (km) against time (hours) shows a car travelling at a constant speed of 60 km/h.
(a) Draw and describe the graph for 3 hours.
(b) State the gradient and explain its meaning.
Question 2
A bus leaves a town at 9:00 a.m. and travels 120 km at 40 km/h.
(a) Sketch the distance–time graph.
(b) At what time does the bus reach its destination?
Question 3
A runner jogs for 10 minutes at 12 km/h, rests for 5 minutes, then jogs again at 6 km/h for another 10 minutes.
Draw a distance–time graph to represent the journey.
Question 4
The table shows the exchange rate between rupees and dollars:
- $1 = Rs. 300
Draw a conversion graph for 0 to $20.
Use it to convert Rs. 4500 into dollars.
Question 5
A train accelerates uniformly from rest to 20 m/s in 10 s.
(a) Sketch the speed–time graph.
(b) Calculate the acceleration.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
Question 6
A car moves at a constant speed of 25 m/s for 30 s.
(a) Draw the speed–time graph.
(b) Find the distance travelled.
Question 7
The velocity–time graph of a cyclist shows constant acceleration from 0 to 12 m/s in 8 seconds.
(a) Calculate the gradient.
(b) State its physical meaning.
Question 8
A lorry decelerates uniformly from 15 m/s to rest in 5 seconds.
(a) Sketch the speed–time graph.
(b) Find the deceleration.
Question 9
A bus travels 100 km in 2 hours and then 80 km in 3 hours.
(a) Plot a distance–time graph.
(b) Use the graph to find the average speed of the whole journey.
Question 10
The speed–time graph of a car is a straight line from 0 to 30 m/s in 12 s, then constant for 8 s, then decelerates to 0 in 6 s.
(a) Sketch the graph.
(b) Calculate the total distance travelled.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
Question 11
The gradient of a tangent to a distance–time graph at a point is 20 km/h.
Explain what this represents.
Question 12
A conversion graph shows litres of petrol against rupees.
If 1 litre = Rs. 260,
(a) Draw the graph for up to 10 litres.
(b) Find the cost of 7.5 litres.
Question 13
A boy cycles at 12 km/h for 45 minutes, rests for 15 minutes, then cycles back home at 15 km/h.
Draw and interpret the distance–time graph.
Question 14
The speed–time graph of a car is a horizontal line at 25 m/s from t = 0 to t = 20.
(a) Find the distance travelled.
(b) Explain the meaning of the flat line.
Question 15
The acceleration of a motorcycle is represented by a straight-line graph rising from 0 to 18 m/s in 6 s.
(a) Find the acceleration.
(b) Calculate the area under the graph.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
Question 16
A conversion graph shows temperature in Celsius against Fahrenheit.
Use the graph to find the Fahrenheit value of 40°C.
Question 17
The distance–time graph of a lorry is a straight line through (0,0) and (4,200).
(a) Find the speed.
(b) How far has it travelled in 2.5 hours?
Question 18
A car accelerates from 0 to 25 m/s in 5 seconds.
(a) Draw the speed–time graph.
(b) Calculate the acceleration.
(c) Find the distance travelled.
Question 19
A train moves at 72 km/h for 1.5 hours.
(a) Draw a distance–time graph.
(b) Find the distance travelled.
Question 20
A car decelerates from 30 m/s to 10 m/s in 5 seconds, then continues at constant speed.
(a) Sketch the speed–time graph.
(b) Find the deceleration.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
Question 21
The gradient of a distance–time graph for a runner is 5 m/s.
What does this tell you about his motion?
Question 22
A car accelerates from rest at 4 m/s² for 10 s.
(a) Draw the speed–time graph.
(b) Find the speed at t = 10 s.
(c) Find the distance travelled.
Question 23
The table shows the speed of a bus at different times:
t (s): 0, 5, 10, 15, 20
v (m/s): 0, 5, 10, 10, 0
(a) Plot the speed–time graph.
(b) Estimate the distance travelled.
Question 24
A petrol tank fills at 5 litres per minute.
(a) Draw a conversion graph of time against volume.
(b) Use it to find the volume after 18 minutes.
Question 25
A motorbike travels at a constant speed of 72 km/h.
(a) Draw the distance–time graph for 3 hours.
(b) Use the gradient to calculate the speed.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
Question 26
The speed–time graph of a bus rises from 0 to 20 m/s in 10 s, stays constant for 20 s, then returns to 0 in 5 s.
(a) Draw the graph.
(b) Find the total distance travelled.
Question 27
A lorry travels 150 km in 3 hours, rests for 1 hour, then continues 100 km in 2 hours.
(a) Sketch the distance–time graph.
(b) Find the average speed for the whole journey.
Question 28
A cyclist accelerates uniformly from 5 m/s to 15 m/s in 4 seconds.
(a) Sketch the speed–time graph.
(b) Find the acceleration.
(c) Find the distance travelled during this time.
Question 29
The conversion graph of pounds (£) to rupees (Rs.) passes through (0,0) and (10,3500).
(a) Write the exchange rate.
(b) Use the graph to convert Rs. 7000 into pounds.
Question 30
The graph of a car’s motion is given by a straight line on the distance–time graph with gradient 80 km/h.
(a) What does the gradient represent?
(b) How long will it take the car to travel 200 km?
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
Answer key and explanations
1.
(a) Straight line from (0,0) to (3,180).
(b) Gradient = 60 km/h → speed.
2.
(a) Straight line from (0,0) to (3,120).
(b) Time = 120 ÷ 40 = 3 h → arrives at 12:00 noon.
3.
Piecewise graph: slope for jogging, flat for rest, smaller slope for slow jog.
Explanation: Gradient = speed.
4.
(a) Straight line through origin, slope 300.
(b) Rs. 4500 ÷ 300 = $15.
5.
(a) Straight line (0,0) to (10,20).
(b) Acceleration = 20/10 = 2 m/s².
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
6.
(a) Horizontal line at v = 25 from t = 0 to t = 30.
(b) Distance = area under graph = 25 × 30 = 750 m.
7.
(a) Gradient = rise/run = 12/8 = 1.5.
(b) Meaning: acceleration = 1.5 m/s².
8.
(a) Line from (0,15) to (5,0).
(b) Deceleration = (0 − 15)/5 = −3 m/s².
9.
(a) First line slope = 50 km/h, second = 80/3 km/h.
(b) Total = 180 km in 5 h → avg speed = 36 km/h.
10.
(a) Piecewise trapezium.
(b) Distance = area = ½(12×30) + 8×30 + ½(6×30) = 180 + 240 + 90 = 510 m.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
11.
Gradient of tangent = instantaneous speed = 20 km/h.
12.
(a) Straight line through origin, slope 260.
(b) 7.5 litres = 260 × 7.5 = Rs. 1950.
13.
Graph slopes up, flat, then slopes down faster.
Interpretation: faster return trip.
14.
(a) Area = 25 × 20 = 500 m.
(b) Flat line means constant speed.
15.
(a) Acceleration = 18/6 = 3 m/s².
(b) Area = ½ × 6 × 18 = 54 m.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
16.
Use formula: F = 9/5C + 32.
At 40°C → F = 104.
17.
(a) Gradient = 200/4 = 50 km/h.
(b) At 2.5 h → 125 km.
18.
(a) Straight line.
(b) Acceleration = 25/5 = 5 m/s².
(c) Distance = ½ × 5 × 25 = 62.5 m.
19.
(a) Straight line slope = 72 km/h.
(b) Distance = 72 × 1.5 = 108 km.
20.
(a) Line falls from (0,30) to (5,10).
(b) Deceleration = (10 − 30)/5 = −4 m/s².
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
21.
Gradient = speed → runner moves at constant 5 m/s.
22.
(a) Straight line from (0,0) to (10,40).
(b) Speed at 10 s = 40 m/s.
(c) Distance = ½ × 10 × 40 = 200 m.
23.
(a) Plot points.
(b) Distance = area under graph = (½×5×5) + (½×5×10 + 5×10) = 12.5 + 75 = 87.5 m.
24.
(a) Straight line slope 5.
(b) Volume after 18 min = 90 L.
25.
(a) Straight line slope 72 km/h.
(b) Gradient = 72 km/h = speed.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
26.
(a) Trapezium + triangle.
(b) Distance = ½(10×20) + (20×20) + ½(5×20) = 100 + 400 + 50 = 550 m.
27.
(a) Graph piecewise.
(b) Avg speed = total distance ÷ total time = 250 ÷ 6 = 41.7 km/h.
28.
(a) Straight line up.
(b) Acceleration = (15 − 5)/4 = 2.5 m/s².
(c) Distance = area trapezium = ½(4)(5 + 15) = 40 m.
29.
(a) 350 per £ → rate.
(b) 7000 ÷ 350 = £20.
30.
(a) Gradient = speed = 80 km/h.
(b) Time = distance/speed = 200/80 = 2.5 h.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course