GCSE Maths · Higher · Chapter 23: Graphs · Lesson 1

Distance-time and velocity-time graphs

Grades 4–7About 40 minutesCalculator allowed31 marks

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NameClassDate

Do Now

Grades 4–6

The first question is what today's lesson needs. The others bring back earlier topics.

A1
Needed today
A train travels at 90 km/h for 40 minutes. How far does it go?
[1 mark]
Answer
A2
From chapter 17
Write $x^{2} + 6 x + 5$ in the form $(x + p)^{2} + q$ .
[1 mark]
Answer
A3
From chapter 17
Solve $y = x^{2}$ and $y = 9$ together.
[1 mark]
Answer
A4
From chapter 20
A tangent at $A$ and the chord $A B$ make an angle of $6 1^{\circ}$ . $C$ is on the circle on the other side of $A B$ . Find angle $A C B$ .
[1 mark]
Answer

Example, then your turn

Grades 4–6

Your teacher works through each example with you. Then try the one beside it.

Example 1

A van drives 48 km in 0.8 h, waits 0.45 h, then drives 36 km in 0.75 h. Work out its average speed for the whole journey.

Hint Total distance

\div

total time, wait included.

B1
A bus drives 39 km in 0.6 h, stops for 0.25 h, then drives 30 km in 0.65 h. Work out its average speed for the whole journey.
[3 marks]
Answer

Example 2

A tram speeds up steadily from rest to 15 m/s in 6 s, holds 15 m/s for 20 s, then slows steadily to rest in 9 s. Find its first acceleration and its total distance.

Not drawn accurately

Hint Sketch the graph. Gradient gives acceleration; area gives distance.

B2
A lift speeds up steadily from rest to 4 m/s in 2.5 s, moves at 4 m/s for 7 s, then slows steadily to rest in 2 s. Find its first acceleration and its total distance.
[4 marks]
Answer $\dots\dots$ m/s² and $\dots\dots$ m

Practice

Grades 5–6

C1
The graph shows a car's journey. Work out the car's speed between 0.5 hours and 2 hours.
[2 marks]
Answer
C2
The graph shows the velocity of a cart. Work out its deceleration as it comes to rest.
[2 marks]
Answer

C3
A car speeds up steadily from 8 m/s to 20 m/s in 12 s. Find the distance it travels.
[2 marks]
Answer
C4
A cyclist accelerates from rest to 9 m/s at 1.5 m/s². How long does it take?
[2 marks]
Answer
C5
One section of a graph is a horizontal line. What does that section tell you (a) on a distance-time graph, (b) on a velocity-time graph?
[2 marks]

Exam-style questions

Grades 6–7

D1
Jo drives 45 km at 30 km/h, then another 45 km at 90 km/h. She says, "My average speed was 60 km/h."
Explain why Jo is wrong, and work out her average speed.
[3 marks]

D2
Nadia walks from home to a café, stops there, and then walks home. The graph shows her walk.
(a)
Work out her speed on the way home.
[2 marks]
Answer
(b)
Work out her average speed for the whole walk, including her stop.
[2 marks]
Answer
Total for D2: 4 marks

Extension

Grade 8 and beyond

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
A car is moving at 6 m/s. It speeds up steadily for $T$ seconds until it reaches 18 m/s, then at once slows steadily to rest in 12 seconds. Its average speed for the whole journey is 10 m/s. Find the value of $T$ .
[3 marks]
Hint 1 · What to try
Write the total distance and the total time in terms of $T$ .
Hint 2 · The first line
Distance $= \frac{1}{2} (6 + 18) T + \frac{1}{2} \times 18 \times 12 = 12 T + 108$ , and time $= T + 12$ .
Hint 3 · The full method
$\frac{12 T + 108}{T + 12} = 10$ , so $12 T + 108 = 10 T + 120$ , $2 T = 12$ and $T = 6$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can find a speed from a distance-time graph and an average speed for a whole journey.
I can find an acceleration and a distance from a velocity-time graph.

Answers and mark scheme

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