Chapter
2
Speed, velocity and acceleration
Figure 2.1
What determines the maximum
height that a pole-vaulter can
reach?
In this chapter we look at moving bodies, how their speeds can be
measured and how accelerations can be calculated. We also look at what
happens when a body falls under the influence of gravity.
2.1 Speed
In everyday life we think of speed as how fast something is travelling.
However, this is too vague for scientific purposes.
Speed is defined as the distance travelled in unit time.
It can be calculated from the formula:
distance
speed ________
time
Units
The basic unit of distance is the metre and the basic unit of time is the
second. The unit of speed is formed by dividing metres by seconds,
giving m/s.
An alternative unit is the kilometre per hour (km/h) often used when
considering long distances.
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Speed, velocity and acceleration
WORKED EXAMPLES
An athlete runs at a steady speed and covers 60 m in 8.0 s.
Calculate her speed.
distance
speed _______
time
60
___
m/s
8.0
7.5 m/s
Measurement of speed
We can measure the speed of an object by measuring the time it takes
to travel a set distance. If the speed varies during the journey, the
calculation gives the average speed of the object. To get a better idea of the
instantaneous speed we need to measure the distance travelled in a very
short time.
One way of doing this is to take a multi-flash photograph. A light is set
up to flash at a steady rate. A camera shutter is held open while the object
passes in front of it. Figure 2.2 shows a toy car moving down a slope.
QUESTIONS
2.1 A car travels 200 m in 8.0 s.
Calculate its speed.
2.2 A cricketer bowls a ball at
45 m/s at a batsman 18.0 m
away from him. Calculate
the time taken for the ball to
reach the batsman.
Figure 2.2
<ph_0202>
NOW ARTWORK
PLEASE SUPPLY BRIEF
Successive images of the car are equal distances apart, showing that the
car is travelling at a constant speed. To find the speed, we measure the
distance between two images and divide by the time between each flash.
Acceleration
So far we have looked at objects travelling at constant speed. However,
in real life this is quite unusual. When an object changes its speed it is
said to accelerate. If the object slows down this is often described as a
deceleration.
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Figure 2.3 shows a multi-flash photograph of the toy car rolling down a
steeper slope. This time its speed increases as it goes down the slope – it is
accelerating.
Figure 2.3
<ph_0203>
NOW ARTWORK
PLEASE SUPPLY BRIEF
distance
Using graphs
Distance–time graphs
time
Figure 2.4
Distance changing at a steady state.
We can draw distance–time graphs for the two journeys of the car in
Figures 2.2 and 2.3.
In Figure 2.2 the car travels equal distances between each flash, so the total
distance travelled increases at a steady rate. This produces a straight line
as shown in Figure 2.4. The greater the speed, the steeper the slope (or
gradient) of the line.
distance
time
Figure 2.5
Increasing distances with time
travelled.
distance
Graphs are used a lot in science and in other mathematical situations.
They are like pictures in a storybook, giving a lot of information in a
compact manner.
In Figure 2.3 the car travels increasing distances in each time interval. This
leads to the graph shown in Figure 2.5, which gradually curves upwards.
The graph in Figure 2.6 shows the story of a journey. The car starts at
quite a high speed and gradually decelerates before coming to rest at
point P.
P
QUESTIONS
2.3 Describe the journeys shown in the diagrams below.
time
distance
distance
Figure 2.6
Story of a car journey.
time
time
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Speed, velocity and acceleration
Speed–time graphs
speed
Instead of using a graph to look at the distance travelled over a period of
time we can look at how the speed changes.
Figure 2.7 appears similar to Figure 2.4. However closer inspection shows
that it is the speed which is increasing at a constant rate, not the distance.
This graph is typical for one in which there is a constant acceleration.
In this case the gradient of the graph is equal to the acceleration. The
greater the acceleration the larger the gradient.
The graph in Figure 2.8 shows the story of the speed on a journey.
time
Figure 2.7
Speed changing at steady rate.
speed
This is a straight-line graph, with a negative gradient. This shows constant
deceleration, sometimes described as negative acceleration.
Using a speed–time graph to calculate distance travelled
distance
speed _______
time
Rearrange the equation:
time
Figure 2.8
Story of speed on a journey.
distance speed time
Look at Figure 2.9. The object is travelling at a constant speed, v, for time t.
speed
v
The distance travelled v t
We can see that it is the area of the rectangle formed.
Now look at Fig. 2.10, which shows a journey with constant acceleration
from rest. The area under this graph is equal to the area under the triangle
that is formed.
The distance travelled _1v t
t time
Figure 2.9
Area under graph of constant speed.
2
_1 v is the average speed of the object and distance travelled is given by
2
average speed time, so once again the distance travelled is equal to the
area under the graph.
speed
v
The general rule is that the distance travelled is equal to the area under
the speed–time graph.
t time
WORKED EXAMPLES
Figure 2.10
Area under graph of constant acceleration.
160
Time passed (4.5 0.5) s 4.0 s
Initial speed 0 m/s
Final speed 120 m/s
120
In this case, the area under the line forms a triangle and the area of a
triangle is found from the formula:
area _12 base height
area under the graph the distance travelled
_12 4.0 120 m 240 m
speed (m/s)
Use the graph in Figure 2.11 to calculate the distance travelled by the car
in the time interval from 0.5 s to 4.5 s.
80
40
0
0
1
2
3
time (s)
4
5
Figure 2.11
Distance travelled by a car.
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S
2.2 Velocity
Velocity is very similar to speed. When we talk about speed we do
not concern ourselves with direction. However, velocity does include
direction. So an object travelling at 5 m/s due south has a different velocity
from an object travelling at 5 m/s northwest.
It is worth observing that the velocity changes if the speed increases, or
decreases, or if the direction of motion changes (even if the speed remains
constant).
Figure 2.12
The lap of the track is 3.0 m, and the
car completes a full lap in 6.0 s.
The average speed of the car is 5.0 m/s.
However its average velocity is zero!
Velocity is a vector and the car
finishes at the same point as it started,
so there has been no net displacement
in any direction.
There are many quantities in physics which have direction as well as size.
Such quantities are called vectors. Quantities, such as mass, which have
only size but no direction are called scalars.
2.3 Acceleration
We have already introduced acceleration as occurring when an object
changes speed. We now explore this idea in more detail.
If a body changes its speed rapidly then it is said to have a large
acceleration, so clearly it has magnitude (or size). Acceleration can be
found from the formula:
change in velocity
acceleration ⴝ ________________
time taken
Units
The basic unit of speed is metres per second (m/s) and the basic unit of time
is the second. The unit of acceleration is formed by dividing m/s by seconds.
This gives the unit m/s2. This can be thought of as the change in velocity
(in m/s) every second.
You will also notice that the formula uses change of velocity, rather than
change of speed. It follows that acceleration can be not only an increase
in speed, but also a decrease in speed or even a change in direction of the
velocity. Like velocity, acceleration has direction, so it is a vector.
WORKED EXAMPLES
1
A racing car on a straight, level test track accelerates from rest to
34 m/s in 6.8 s. Calculate its acceleration.
change of velocity
Acceleration _______________
time
(final velocity initial velocity)
__________________________
time
(34
0)
_______ m/s2
6.8
5.0 m/s2
It is important that the track is straight and level or it could be argued that
there is a change of direction, and therefore an extra acceleration.
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Speed, velocity and acceleration
2
A boy on a bicycle is travelling at a speed of 16 m/s. He applies his
brakes and comes to rest in 2.5 s. Calculate his acceleration. You may
assume the acceleration is constant.
change of velocity
Acceleration _______________
time
(fi
nal
velocity
initial velocity)
__________________________
time
(0 16)
_______ m/s2
2.5
ⴚ6.4 m/s2
Notice that the acceleration is negative, which shows that it is a
deceleration.
Calculation of acceleration from a
velocity–time graph
speed (m/s)
20
Look at the graph in Figure 2.13. We can see that between 1.0 s and 4.0 s
the speed has increased from 5.0 m/s to 12.5 m/s.
(12.5 5)
Acceleration _________ m/s2
(4 1)
7.5 m/s2
___
3
2.5 m/s2
15
10
5
0
Mathematically this is known as the gradient of the graph.
0
1
2
3
time (s)
4
5
Figure 2.13
Velocity–time graph.
increase in y
Gradient ⴝ ___________
increase in x
We see that acceleration is equal to the gradient of the speed-time graph.
It does not matter which two points on the graph line are chosen, the
answer will be the same. Nevertheless, it is good practice to choose points
that are well apart; this will improve the precision of your final answer.
QUESTIONS
2.5 a) Describe the motion of
the object in shown in
the graph in Figure 2.15.
b) Calculate the distance
travelled by the object.
S
c) Calculate the
acceleration of the
object.
distance
time
Figure 2.14
4.0
speed (m/s)
2.4 Describe the motion of
the object shown in the
graph in Figure 2.14.
3.0
2.0
1.0
0
0
Figure 2.15
0.1
0.2
0.3
time (s)
0.4
0.5
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0
0.1
Figure 2.16 shows a multi-flash photograph of a steel ball falling. The light
flashes every 0.1 s.
We can see that the ball travels further in each time interval, so we know
that it is accelerating. Figure 2.17 shows the speed–time graph of the ball.
0.2
4.0
speed (m/s)
0.3
0.4
0.5
3.0
2.0
1.0
0.6
0
0
0.7
0.2
0.3
time (s)
0.4
0.5
Figure 2.17
Speed–time graph of falling steel ball.
0.8
0.9
0.1
The graph is a straight line, which tells us that the acceleration is constant.
S
1.0
We can calculate the value of the acceleration by measuring the gradient.
Use the points (0.10, 0.50) and (0.45, 3.9).
1.1
(3.9 0.50) m/s
Gradient ___________ ____
(0.45 0.10) s
1.2
3.4 m/s2
____
0.35
1.3
1.4
1.5
1.6
1.7
9.7 m/s2
The acceleration measured in this experiment is 9.7 m/s2.
All objects in free fall near the Earth’s surface have the same acceleration.
The recognised value is 9.8 m/s2, although it is quite common for this to
be rounded to 10 m/s2. The result in the above experiment lies well within
the uncertainties in the experimental procedure.
This is sometimes called the acceleration of free fall, or acceleration due to
gravity, and is given the symbol g.
In Chapter 3 we will look at gravity in more detail.
1.8
We will also look, in Chapter 3, at what happens if there is significant air
resistance.
1.9
2.0
Figure 2.16
Falling steel ball.
QUESTIONS
2.6 An aeroplane travels at a constant speed of 960 km/h.
Calculate the time it will take to travel from London to
Johannesburg, a distance of 9000 km.
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Speed, velocity and acceleration
2.7 Describe what happens to speed in the two journeys described in the
graphs
a) distance
b)
distance
time
time
2.8 Describe how the speed changes in the two journeys described in the
graphs.
a) speed
b) speed
time
time
2.9 A motorist is travelling at 15 m/s when he sees a child run into the
road. He brakes and the car comes to rest in 0.75 s. Draw a speedtime graph to show the deceleration, and use your graph to calculate
a) the distance travelled once the brakes are applied
b) the deceleration of the car.
S
2.10 A car accelerates from rest at 2m/s2 for 8 seconds.
a) Draw a speed-time graph to show this motion.
b) Use your graph to find
(i) the final speed of the car
(ii) the distance travelled by the car.
2.11 The graph shows how the speed of an aeroplane changes with time.
speed (m/s)
40
B
C
30
20
10
0
A
0
10
20
30
40
50
time (s)
a) Describe the motion of the aeroplane.
b) Calculate the acceleration of the aeroplane during the period B
to C.
c) Suggest during which stage of the journey these readings were
taken.
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Summary
Now that you have completed this chapter, you should be able to:
• define speed
distance
• recall and use the equation speed _______
time
• understand that acceleration is a change of speed
• draw and interpret distance-time graphs
• draw and interpret speed-time graphs
• calculate distance travelled from a speed-time graph
• recognise that the steeper the gradient of a speed-time graph the
greater the acceleration
• recognise that acceleration of free fall is the same for all objects
S
• understand that velocity and acceleration are vectors
change in velocity
• recall and use the equation acceleration _______________
time
• calculate acceleration from the gradient of a speed-time graph
• describe an experiment to measure the acceleration of free fall.
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