A-Level Maths (Edexcel 9MA0) · Mechanics Year 1 · Chapter 9: Constant acceleration · Lesson 1

Displacement–time and velocity–time graphs

127 marks

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Date

Do Now

Quick questions to start.

A1
Needed today
Convert $54 km h^{- 1}$ to metres per second.
[2 marks]
Answer
A2
Needed today
A trapezium has parallel sides $12$ cm and $20$ cm, which are $5$ cm apart. Find its area.
[2 marks]
Answer
A3
Common slip
A particle starts from rest and its velocity increases uniformly to $18 m s^{- 1}$ in $6$ seconds. Find its acceleration, giving the correct units.
[2 marks]
Answer
A4
Common slip
A runner goes $300$ m due east, then $100$ m due west, in a total of $80$ s. Find her average speed and her average velocity.
[3 marks]
Answer

Example, then your turn

A-Level

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

Example 1

The diagram is the displacement–time graph of a cyclist who rides in a straight line from $O$ , stops, and then rides back to $O$ . Find her velocity in each stage of the journey in km h $^{- 1}$ , her average velocity and her average speed for the whole hour.

Hint Velocity is the gradient of a displacement–time graph. A negative gradient means she is moving back towards

O

. Average velocity uses the displacement; average speed uses the total distance.

B1
The diagram is the displacement–time graph of a walker who moves along a straight path from $O$ . Find her velocity in each stage in km h $^{- 1}$ , and her average velocity and her average speed for the whole $90$ minutes.
[5 marks]
Answer

Example 2

The diagram is the velocity–time graph of a car moving along a straight road. Find the acceleration, the deceleration, and the total distance the car travels.

Hint Acceleration is the gradient of a velocity–time graph. Distance is the area under it: a trapezium, or a triangle, a rectangle and a triangle. Keep the

\frac{1}{2}

in every triangle and trapezium.

B2
The diagram is the velocity–time graph of a cyclist. Find her acceleration in the first stage, her deceleration in the last stage, and the total distance she travels.
[5 marks]
Answer

Example 3

A train starts from rest at a station. It accelerates uniformly at $0.5 m s^{- 2}$ for $40$ s, then moves at constant speed for $T$ s, then decelerates uniformly to rest in $50$ s at the next station. The stations are $3.6$ km apart. Sketch a velocity–time graph of the journey and find $T$ .

Hint Find the top speed first. On the sketch put the top speed on the velocity axis and

40

40 + T

and

90 + T

on the time axis. The area under the graph is the distance, in metres.

B3
A tram starts from rest. It accelerates uniformly at $0.4 m s^{- 2}$ for $30$ s, then moves at constant speed for $T$ s, then decelerates uniformly to rest in $20$ s. It travels $2.1$ km in all. Sketch a velocity–time graph of the motion and find $T$ .
[6 marks]
Answer

Example 4

A particle moves along a straight line. The diagram is its velocity–time graph for $0 \leq t \leq 14$ . Find the time at which it is at rest, the distance it travels in the $14$ s, its displacement from its start at $t = 14$ , and the time at which it passes its start again.

Hint The velocity falls by

15 m s^{- 1}

10

s. Area above the axis is distance moved forwards; area below the axis is distance moved backwards. Distance adds the two; displacement subtracts.

B4
The diagram is the velocity–time graph of a particle moving along a straight line, for $0 \leq t \leq 11$ . Find the time at which it is at rest, the distance it travels in the $11$ s, its displacement from its start at $t = 11$ , and the time at which it passes its start again.
[6 marks]
Answer

Example 5

At $t = 0$ car $A$ passes a point $P$ moving at a constant $15 m s^{- 1}$ . At the same moment car $B$ starts from rest at $P$ and moves in the same direction, accelerating uniformly at $2.5 m s^{- 2}$ until it reaches $25 m s^{- 1}$ , then keeping that speed. The diagram shows both velocity–time graphs. Find when the cars have the same speed, the greatest distance between them before $B$ catches $A$ , and the time and distance from $P$ at which $B$ catches $A$ .

Hint Until the speeds are equal,

A

is going faster, so the gap grows. Each car's distance from

P

is the area under its own graph.

B

catches

A

when the two areas are equal.

B5
At $t = 0$ a motorbike passes a junction $J$ moving at a constant $18 m s^{- 1}$ . At the same moment a car starts from rest at $J$ and moves the same way. It accelerates uniformly at $2 m s^{- 2}$ for $15$ s and then keeps a constant speed. Find when they have the same speed, the greatest distance between them before the car catches the motorbike, and the time and distance from $J$ at which the car catches it.
[6 marks]
Answer

Practice

A-Level

C1
In the last $8$ s of a race a runner slows down uniformly from $10 m s^{- 1}$ to $U m s^{- 1}$ and covers $64$ m. Priya writes $\frac{1}{2} \times 8 \times U = 64$ , so $U = 16$ . Explain why her answer cannot be right, and find $U$ .
[4 marks]
Answer

C2
The diagram shows the velocity–time graph of a particle for the first $8$ s of its motion. Leo says the acceleration in the first $5$ s is $\frac{1}{2} (2 + 17) \times 5 = 47.5 m s^{- 1}$ . Explain his mistake, then find the acceleration and the distance travelled in the first $5$ s.
[4 marks]
Answer
C3
A car starts from rest and accelerates uniformly to a speed of $V m s^{- 1}$ in $12$ s. It keeps this speed for $30$ s, then decelerates uniformly to rest. The deceleration is twice the acceleration. The car travels $780$ m in all. Find $V$ and the acceleration.
[5 marks]
Answer
C4
A lift rises from rest. It accelerates uniformly at $0.8 m s^{- 2}$ for $2.5$ s, then rises at a constant speed, then decelerates uniformly at $0.5 m s^{- 2}$ until it stops. It rises $30$ m in all. Find the time for which it moves at constant speed and the total time of the journey.
[5 marks]
Answer
C5
A cyclist starts from rest, accelerates uniformly at $2 m s^{- 2}$ for $10$ s, then rides at a constant $20 m s^{- 1}$ . Find the time at which she has travelled $64$ m. Jay says she has travelled $300$ m after $\frac{300}{20} = 15$ s. Explain why Jay is wrong and find the correct time.
[5 marks]
Answer

Exam-style questions

Edexcel exam style

D1
A train starts from rest at station $A$ and accelerates uniformly for $50$ s until its speed is $72 km h^{- 1}$ . It keeps this speed for a time, then decelerates uniformly at $0.5 m s^{- 2}$ , coming to rest at station $B$ . The distance $A B$ is $5$ km.
(a)
Sketch a velocity–time graph for the journey from $A$ to $B$ .
[3 marks]
Answer
(b)
Find the time for which the train moves at constant speed.
[4 marks]
Answer
(c)
Find the average speed of the train between $A$ and $B$ .
[2 marks]
Answer
(d)
State one way in which the velocity–time graph model may not be realistic.
[2 marks]
Total for D1: 11 marks

D2
A sprinter runs $100$ m in a straight line in $12$ s. She is modelled as accelerating uniformly from rest for $4$ s up to a speed of $V m s^{- 1}$ , then running at $V m s^{- 1}$ to the finish. Kai says her top speed is $\frac{100}{12} = 8.33 m s^{- 1}$ .
Kai is wrong.
(a)
Find $V$ .
[3 marks]
Answer
(b)
Explain Kai's mistake.
[2 marks]
(c)
Find the time at which she has run $50$ m.
[3 marks]
Answer
(d)
State one way in which the model is not realistic.
[2 marks]
Total for D2: 10 marks
D3
A van passes a point $P$ at time $t = 0$ , moving along a straight road at a constant $20 m s^{- 1}$ . A car is at rest at $P$ . At $t = 5$ s the car starts to move in the same direction as the van. It accelerates uniformly at $2.5 m s^{- 2}$ until its speed is $30 m s^{- 1}$ , then keeps this speed.
(a)
Sketch, on the same axes, the velocity–time graphs of the van and the car for $t \geq 0$ .
[3 marks]
Answer
(b)
Find the time at which the car reaches $30 m s^{- 1}$ .
[2 marks]
Answer
(c)
Find the time at which the car overtakes the van, and the distance from $P$ at which this happens.
[5 marks]
Answer
Total for D3: 10 marks
D4
A train travels between two stations $6.3$ km apart, starting and finishing at rest. Its greatest speed is $30 m s^{- 1}$ , its acceleration is $0.5 m s^{- 2}$ and its deceleration is $0.75 m s^{- 2}$ . Find the shortest possible time for the journey.
[7 marks]
Answer
D5
A particle moves along a straight line. The diagram, which is not drawn to scale, is its velocity–time graph. The velocity falls uniformly from $8 m s^{- 1}$ to $- 4 m s^{- 1}$ in the first $6$ s, stays at $- 4 m s^{- 1}$ for $T$ s, then rises uniformly to $0$ in $2$ s. At the end of the motion the particle is back at its starting point.
(a)
Find the deceleration in the first $6$ s.
[2 marks]
Answer
(b)
Find the time at which the particle is instantaneously at rest.
[2 marks]
Answer
(c)
Find $T$ .
[3 marks]
Answer
(d)
Find the total distance travelled by the particle.
[2 marks]
Answer
Total for D5: 9 marks
D6
A car moves along a straight road from rest at $A$ to rest at $B$ . It accelerates uniformly for $T$ seconds up to a speed of $V m s^{- 1}$ , moves at $V m s^{- 1}$ for $4 T$ seconds, then decelerates uniformly to rest in $2 T$ seconds. The journey takes $84$ s and $A B = 1320$ m.
(a)
Sketch a velocity–time graph for the journey.
[3 marks]
Answer
(b)
Show that $T = 12$ .
[2 marks]
Answer
(c)
Find $V$ .
[3 marks]
Answer
(d)
Find the acceleration and the deceleration of the car.
[2 marks]
Answer
Total for D6: 10 marks

Extension

Stretch

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

E1
A car travels along a straight road from rest at $A$ to rest at $B$ . Its acceleration and its deceleration each have the same constant size $a m s^{- 2}$ . On one day it accelerates and then at once decelerates, with no constant-speed stage, and the journey takes $40$ s. On another day it keeps to a speed limit of $15 m s^{- 1}$ : it accelerates at $a m s^{- 2}$ to $15 m s^{- 1}$ , keeps this speed, then decelerates at $a m s^{- 2}$ to rest at $B$ . This journey takes $50$ s. Find $a$ and the distance $A B$ .
[10 marks]
Hint 1 · What to try
Sketch both velocity–time graphs. The first is a triangle; by symmetry its peak is at $t = 20$ .
Hint 2 · The first line
From the first journey, $A B = 400 a$ . In the second, speeding up and slowing down each take $\frac{15}{a}$ s.
Hint 3 · The full method
You get a quadratic in $a$ . For each root, check whether the first journey's top speed is more than $15 m s^{- 1}$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can find velocities, average velocity and average speed from a displacement–time graph.
I can find accelerations as gradients and distances as areas on a velocity–time graph.
I can sketch a velocity–time graph from a description and use the area to find an unknown time or speed.
I can tell distance from displacement when the velocity changes sign, and compare two vehicles on one graph.

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