Speed, velocity and acceleration

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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 infl uence of gravity.

2.1 Speed

In everyday life we think of speed as how fast something is travelling. However, this is too vague for scientifi c purposes.

Speed is defi ned as the distance travelled in unit time. It can be calculated from the formula:

speed distance ________ 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.

Speed, velocity and acceleration

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Speed, velocity and acceleration

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-fl ash photograph. A light is set up to fl ash 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.

Successive images of the car are equal distances apart, showing that the car is travelling at a constant speed. To fi nd the speed, we measure the distance between two images and divide by the time between each fl ash.

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.

An athlete runs at a steady speed and covers 60 m in 8.0 s. Calculate her speed.

speed distance _______ time 60 ___ 8.0 m/s 7.5 m/s

WORKED EXAMPLES

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.

QUESTIONS

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Figure 2.2

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Using graphs

Distance–time graphs

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.

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 fl ash, 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.

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.

Figure 2.3

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Figure 2.4

Distance changing at a steady state.

distance

time

Figure 2.5

Increasing distances with time travelled.

distance

time

Figure 2.6 Story of a car journey.

P distance

time

2.3 Describe the journeys shown in the diagrams below.

QUESTIONS

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Speed–time graphs

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.

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

speed distance _______

time Rearrange the equation:

distance speed time

Look at Figure 2.9. The object is travelling at a constant speed, v, for time t. 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 _ 1₂ v t

1

_

2 v is the average speed of the object and distance travelled is given by

average speed time, so once again the distance travelled is equal to the area under the graph.

The general rule is that the distance travelled is equal to the area under the speed–time graph.

Figure 2.7

Speed changing at steady rate.

speed

time

Figure 2.8

Story of speed on a journey.

speed

time

Figure 2.9

Area under graph of constant speed. t

v

speed

time

Figure 2.10

Area under graph of constant acceleration. t

v

speed

time

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.

Time passed (4.5 0.5) s 4.0 s Initial speed 0 m/s

Final speed 120 m/s

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

_{_}1 2 4.0 120 m 240 m

WORKED EXAMPLES

0 40 80 120 160 0 1 2 3 4 5 time (s) speed (m/s) Figure 2.11

Distance travelled by a car.

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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).

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:

acceleration ⴝ change in velocity________________ 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.

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.

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 fi nishes at the same point as it started, so there has been no net displacement in any direction.

1 A racing car on a straight, level test track accelerates from rest to 34 m/s in 6.8 s. Calculate its acceleration.

Acceleration change of velocity_______________ time

__________________________ (fi nal velocity initial velocity)

time (34 _______ 0) 6.8 m/s2 5.0 m/s2

WORKED EXAMPLES

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Notice that the acceleration is negative, which shows that it is a deceleration.

Calculation of acceleration from a

velocity–time graph

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.

Acceleration (12.5 5)_________ (4 1) m/s2 7.5 ___ ₃ m/s2

2.5 m/s2

Mathematically this is known as the gradient of the graph. Gradient ⴝ ___________ increase in y

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 fi nal answer.

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.

Acceleration change of velocity_______________ time

__________________________ (fi nal velocity initial velocity)

time (0 _______ 16) 2.5 m/s2 ⴚ6.4 m/s2 Figure 2.13 Velocity–time graph. 0 5 10 15 20 0 1 2 3 4 5 time (s) speed (m/s)

2.4 Describe the motion of the object shown in the graph in Figure 2.14.

QUESTIONS

Figure 2.14

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. c) Calculate the acceleration of the object. Figure 2.15 S distance time 0 1.0 2.0 3.0 4.0 0 0.1 0.2 0.3 0.4 0.5 time (s) speed (m/s) 02_phys_012_020.indd 17 02_phys_012_020.indd 17 6/11/08 10:19:016/11/08 10:19:01

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that it is accelerating. Figure 2.17 shows the speed–time graph of the ball.

The graph is a straight line, which tells us that the acceleration is constant. We can calculate the value of the acceleration by measuring the gradient. Use the points (0.10, 0.50) and (0.45, 3.9).

Gradient ___________ (3.9 0.50) (0.45 0.10) m/s ____ _s 3.4 ____ _0.35 m/s2 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.

We will also look, in Chapter 3, at what happens if there is signifi cant air resistance. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 Figure 2.16 Falling steel ball.

0 1.0 2.0 3.0 4.0 0 0.1 0.2 0.3 0.4 0.5 time (s) speed (m/s) Figure 2.17

Speed–time graph of falling steel ball.

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.

QUESTIONS

S

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2.7 Describe what happens to speed in the two journeys described in the graphs

a) b)

2.8 Describe how the speed changes in the two journeys described in the graphs.

a) b)

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 speed-time 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.

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 fi nd

(i) the fi nal 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.

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. time distance speed time speed time speed (m/s) time (s) A B C 0 10 20 30 40 0 10 20 30 40 50 S distance time 02_phys_012_020.indd 19 02_phys_012_020.indd 19 6/11/08 10:19:026/11/08 10:19:02

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• defi ne speed

• recall and use the equation speed distance _______

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

• understand that velocity and acceleration are vectors

• recall and use the equation acceleration change in velocity_______________

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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