Cambridge International
AS & A Level Mathematics:
Mechanics
Practice Book
,Contents
How to use this book
1 Velocity and acceleration
2 Force and motion in one dimension
3 Forces in two dimensions
4 Friction
5 Connected particles
6 General motion in a straight line
7 Momentum
8 Work and energy
9 The work-energy principle and power
Answers
, How to use this book
Throughout this book you will notice particular features that are designed to help your learning.
This section provides a brief overview of these features.
■ Relate force to acceleration.
■ Use combinations of forces to calculate their effect on an object.
■ Include the force on an object due to gravity in a force diagram and calculations.
■ Include the contact force on a force diagram and in calculations.
Learning objectives indicate the important concepts within each chapter and help you to navigate
through the practice book.
TIP
The velocity jumps suddenly when t = 5. This can happen, for
example, in a collision. The displacement still needs to be continuous.
Tip boxes contain helpful guidance about calculating or checking your answers.
WORKED EXAMPLE 2.1
A block of mass 16 kg is pushed across a smooth horizontal surface using a constant
horizontal force of T newtons. The block starts from rest and takes 5 seconds to travel 20
metres. Find the value of T.
Answer
s = 20, u = 0, t = 5 Constant force so constant
acceleration.
a = 1.6 m s−2
Newton’s second law: Resultant force = mass × acceleration.
T = 16 × 1.6 = 25.6
Worked examples provide step-by-step approaches to answering questions. The left side shows a fully
worked solution, while the right side contains a commentary explaining each step in the working.
END-OF-CHAPTER REVIEW EXERCISE 7
, 1 A raindrop of mass 0.005 kg has momentum of magnitude between 0.01 Ns and 0.02 Ns.
Calculate the range of speeds of the raindrop.
2 A train of mass 300 000 kg is travelling along a straight track. The train speeds up from 4 m
s−1 to 6.5 m s−1. Calculate the increase in the momentum of the tr sured in the direction of
travel.
The End-of-chapter review exercise contains exam-style questions covering all topics in the chapter.
You can use this to check your understanding of the topics you have covered.
E
Extension material goes beyond the syllabus. It is highlighted by a red line to the left of the text.
Throughout each chapter there are multiple exercises containing practice questions. The questions are
coded:
PS These questions focus on problem solving.
P These questions focus on proofs.
M These questions focus on modelling.
You should not use a calculator for these questions.
You can use a calculator for these questions.
,Chapter 1
Velocity and acceleration
■ Work with scalar and vector quantities for distance and speed.
■ Use equations of constant acceleration.
■ Sketch and read displacement–time graphs and velocity–time graphs.
■ Solve problems with multiple stages of motion.
,1.1 Displacement and velocity
WORKED EXAMPLE 1.1
A plane flies from Warsaw to Athens, a distance of 1600 km, at an average speed of 640 km
h−1.
How long does the flight take?
Answer
State the equation to use.
Rearrange the equation to make time the subject.
Use consistent units, substitute values into the
equation.
Flight takes 2 hours 30 minutes. Convert the decimal answer into hours and
minutes.
EXERCISE 1A
1 How long will an athlete take to run 1500 metres at 7.5 m s−1?
2 A train maintains a constant velocity of 60 m s−1 due south for 20 minutes. What is its
displacement in that time? Give the distance in kilometres.
3 Some Antarctic explorers walking towards the South Pole expect to average 1.8
kilometres per hour. What is their expected displacement in a day in which they walk for
14 hours?
Questions 4 and 5 refer to the four points, A, B, C and D, which lie in a straight line with
distances between them shown in the diagram. The displacement is measured from left
to right.
4 Find:
a i the displacement from D to A
ii the displacement from D to B
b i the distance from D to B
ii the distance from C to A
c i the total displacement when a particle travels from B to C and then to A
ii the total displacement when a particle travels from C to D and then to A.
TIP
Remember displacement is a vector quantity, and distance is a
scalar quantity.
PS 5 a i A particle travels from A to C in 23 seconds and then from C to B in 18 seconds.
Find its average speed and average velocity.
ii A particle travels from B to D in 38 seconds and then from D to A in 43 seconds.
, Find its average speed and average velocity.
b i A particle travels from B to D in 16 seconds and then back to B in 22 seconds.
Find its average speed and average velocity.
ii A particle travels from A to C in 26 seconds and then back to A in 18 seconds.
Find its average speed and average velocity.
TIP
Remember speed is a scalar quantity, and velocity is a vector
quantity.
6 Here is an extract from the diary of Samuel Pepys for 4 June 1666, written in London.
‘We find the Duke at St James’s, whither he is lately gone to lodge. So walking through
the Parke we saw hundreds of people listening to hear the guns.’
These guns were at the battle of the English fleet against the Dutch off the Kent coast, a
distance of between 110 and 120 km away. The speed of sound in air is 344 m s−1. How
long did it take the sound of the gunfire to reach London?
7 Light travels at a speed of 3.00 × 108 m s−1. Light from the star Sirius takes 8.65 years
to reach the Earth. What is the distance of Sirius from the Earth in kilometres?
TIP
Consider how many seconds there are in 8.65 years.
,1.2 Acceleration
WORKED EXAMPLE 1.2
A skateboarder travels down a hill in a straight line with constant acceleration. She starts
with speed 1.5 m s−1 and finishes with speed 9.5 m s−1. The length of the hill is 22 m.
a Find the time taken.
b Find the acceleration of the skateboarder.
Answer
a s = 22, u =1.5, v = 9.5 Begin by listing the information given.
so State the equation to be used and substitute in the
known values.
t=4 Rearrange the equation to find the time.
Time taken = 4 seconds Include the units in the final answer.
b Choose the equation to be used.
Substitute in the known values.
=2
Acceleration = 2 m s−2 Include the units in the final answer.
EXERCISE 1B
1 Write the following quantities in the specified units, giving your answers to 3 significant
figures.
a i 3.6 km h−1 in m s−1
ii 62 km h−1 in m s−1
b i 5.2 m s−1 in km h−1
ii 0.26 m s−1 in km h−1
c i 120 km h−2 in m s−2
ii 450 km h−2 in m s−2
d i 0.82 m s−2 in km h−2
ii 2.7 m s−2 in km h−2
TIP
Use velocities not speeds.
2 A police car accelerates from 15 m s−1 to 35 m s−1 in 5 seconds. The acceleration is
constant. Illustrate this with a velocity–time graph. Use the equation v = u + at to
calculate the acceleration. Find also the distance travelled by the car in that time.
3 A marathon competitor running at 5 m s−1 puts on a sprint when she is 100 metres from
the finish, and covers this distance in 16 seconds. Assuming that her acceleration is
constant, use the equation to find how fast she is running as she crosses
the finishing line.
, 4 Starting from rest, an aircraft accelerates to its take-off speed of 60 m s−1 in a distance
of 900 metres. Assuming constant acceleration, find how long the take-off run lasts.
Hence calculate the acceleration.
TIP
‘Rest’ means not moving, so the velocity is zero.
5 A train is travelling at 80 m s−1 when the driver applies the brakes, producing a
deceleration of 2 m s−2 for 30 seconds. How fast is the train then travelling, and how far
does it travel while the brakes are on?
PS 6 A balloon at a height of 300 m is descending at 10 m s−1 and decelerating at a rate of
0.4 m s−2. How long will it take for the balloon to stop descending, and what will its
height be then?
7 A train goes into a tunnel at 20 m s−1 and emerges from it at 55 m s−1. The tunnel is
1500 m long. Assuming constant acceleration, find how long the train is in the tunnel for,
and the acceleration of the train.
PS 8 A cyclist riding at 5 m s−1 starts to accelerate, and 200 metres later she is riding at 7 m
s−1. Find her acceleration, assumed constant.
, 1.3 Equations of constant acceleration
WORKED EXAMPLE 1.3
A train is travelling at 55 m s−1. The driver needs to reduce the speed to 35 m s−1 to pass
through a junction. The deceleration must not exceed 0.6 m s−2. How far ahead of the
junction should the driver begin to slow down the train?
Answer
Using the maximum deceleration:
u = 55, v = 35, a = −0.6 Begin by listing the information given. As we have
deceleration, the acceleration is a negative value.
v2= u2+ 2as so 1225 = 3025 − State the equation to be used and substitute in the
1.2s known values.
s = 1500 Rearrange to find the distance.
The driver should start to slow Include the units in the final answer and clarify the
down at least 1500 m ahead of answer in the contextof the question.
the junction.
EXERCISE 1C
P 1 Use the formulae v = u + at and to prove that .
TIP
Decide which variable to eliminate.
P 2 a Use the formulae and v = u + at to derive the formula
.
b A particle moves with constant acceleration 3.1 m s−2. It travels 300 m in the first 8
seconds. Find its speed at the end of the 8 seconds.
P 3 Use the formulae v = u + at and to derive the formula v2 = u2 + 2as.
4 An ocean liner leaves the harbour entrance travelling at 3 m s−1, and accelerates at 0.04
m s−2 until it reaches its cruising speed of 15 m s−1.
a How far does it travel in accelerating to its cruising speed?
b How long does it take to travel 2 km from the harbour entrance?
5 A boy kicks a football up a slope with a speed of 6 m s−1. The ball decelerates at 0.3 m s
−2. How far up the slope does it roll?
6 A cyclist comes to the top of a hill 165 metres long travelling at 5 m s−1, and free-wheels
down it with an acceleration of 0.8 m s−2. Write expressions for his speed and the
distance he has travelled after t seconds. Hence find how long he takes to reach the
bottom of the hill, and how fast he is then travelling.
PS 7 A particle reduces its speed from 20 m s−1 to 8.2 m s−1 while travelling 100 m.
Assuming it continues to move with the same constant acceleration, how long will it take
to travel another 20 m?
PS 8 A particle moves with constant deceleration of 3.6 m s−2. It travels 350 m while its
speed halves. Find the time it takes to do this.
