Speed, velocity and acceleration


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1 Chapter Speed, velocity and acceleration Figure.1 What determines the maximum height that a polevaulter can reach? 1 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..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 travelled in unit. It can be calculated from the formula: speed Units The basic unit of is the metre and the basic unit of 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 s. _phys_1_.indd 1 6/11/8 1:18:55
2 Speed, velocity and acceleration WORKED EXAMPLES An athlete runs at a steady speed and covers 6 m in 8. s. Calculate her speed. speed 6 8. m/s 7.5 m/s Measurement of speed We can measure the speed of an object by measuring the it takes to travel a set. 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 travelled in a very short. One way of doing this is to take a multiflash 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. shows a toy car moving down a slope. QUESTIONS.1 A car travels m in 8. s. Calculate its speed.. A cricketer bowls a ball at 45 m/s at a batsman 18. m away from him. Calculate the taken for the ball to reach the batsman. Figure. <ph_> NOW ARTWORK PLEASE SUPPLY BRIEF Successive images of the car are equal s apart, showing that the car is travelling at a constant speed. To find the speed, we measure the between two images and divide by the 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. _phys_1_.indd 13 6/11/8 1:18:58 13
3 Figure.3 shows a multiflash photograph of the toy car rolling down a steeper slope. This its speed increases as it goes down the slope it is accelerating. Figure.3 <ph_3> NOW ARTWORK PLEASE SUPPLY BRIEF Figure.4 Distance changing at a steady state. Figure.5 Increasing s with travelled. Using graphs Distance 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 graphs for the two journeys of the car in Figures. and.3. In Figure. the car travels equal s between each flash, so the total travelled increases at a steady rate. This produces a straight line as shown in Figure.4. The greater the speed, the steeper the slope (or gradient) of the line. In Figure.3 the car travels increasing s in each interval. This leads to the graph shown in Figure.5, which gradually curves upwards. The graph in Figure.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.3 Describe the journeys shown in the diagrams below. Figure.6 Story of a car journey. 14 _phys_1_.indd 14 6/11/8 1:18:58
4 Speed, velocity and acceleration Speed graphs Instead of using a graph to look at the travelled over a period of we can look at how the speed changes. Figure.7 appears similar to Figure.4. However closer inspection shows that it is the speed which is increasing at a constant rate, not the. 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.8 shows the story of the speed on a journey. This is a straightline graph, with a negative gradient. This shows constant deceleration, somes described as negative acceleration. Using a speed graph to calculate travelled speed Rearrange the equation: speed Look at Figure.9. The object is travelling at a constant speed, v, for t. The travelled v t We can see that it is the area of the rectangle formed. Now look at Fig..1, 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 travelled 1_ v t 1_ v is the average speed of the object and travelled is given by average speed, so once again the travelled is equal to the area under the graph. The general rule is that the travelled is equal to the area under the speed graph. WORKED EXAMPLES Use the graph in Figure.11 to calculate the travelled by the car in the interval from.5 s to 4.5 s. Time passed (4.5.5) s 4. s Initial speed m/s Final speed 1 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 1_ base height area under the graph the travelled 1_ 4. 1 m 4 m speed Figure.7 Speed changing at steady rate. speed Figure.8 Story of speed on a journey. speed v t Figure.9 Area under graph of constant speed. speed v t Figure.1 Area under graph of constant acceleration (s) Figure.11 Distance travelled by a car. _phys_1_.indd 15 6/11/8 1:18:59 15
5 Figure.1 The lap of the track is 3. m, and the car completes a full lap in 6. s. The average speed of the car is 5. 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. S. 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). 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..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 Units change in velocity taken The basic unit of speed is metres per second (m/s) and the basic unit of is the second. The unit of acceleration is formed by dividing m/s by seconds. This gives the unit m/s. 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 (final velocity initial velocity) (34 ) m/s m/s 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. 16 _phys_1_.indd 16 6/11/8 1:19:
6 Speed, velocity and acceleration A boy on a bicycle is travelling at a speed of 16 m/s. He applies his brakes and comes to rest in.5 s. Calculate his acceleration. You may assume the acceleration is constant. change of velocity Acceleration (final velocity initial velocity) ( 16) m/s m/s Notice that the acceleration is negative, which shows that it is a deceleration. Calculation of acceleration from a velocity graph Look at the graph in Figure.13. We can see that between 1. s and 4. s the speed has increased from 5. m/s to 1.5 m/s. (1.5 5) Acceleration m/s (4 1) m/s.5 m/s Mathematically this is known as the gradient of the graph. Gradient increase in y increase in x (s) Figure.13 Velocity graph. We see that acceleration is equal to the gradient of the speed 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.4 Describe the motion of the object shown in the graph in Figure a) Describe the motion of the object in shown in the graph in Figure.15. b) Calculate the travelled by the object. S c) Calculate the acceleration of the object Figure (s) Figure.15 _phys_1_.indd 17 6/11/8 1:19:1 17
7 Figure.16 shows a multiflash photograph of a steel ball falling. The light flashes every.1 s. We can see that the ball travels further in each interval, so we know that it is accelerating. Figure.17 shows the speed graph of the ball. S (s) Figure.17 Speed graph of falling steel 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 (.1,.5) and (.45, 3.9). (3.9.5) Gradient m/s (.45.1) s m/s 9.7 m/s The acceleration measured in this experiment is 9.7 m/s. All objects in free fall near the Earth s surface have the same acceleration. The recognised value is 9.8 m/s, although it is quite common for this to be rounded to 1 m/s. The result in the above experiment lies well within the uncertainties in the experimental procedure. This is somes 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 significant air resistance.. Figure.16 Falling steel ball. QUESTIONS.6 An aeroplane travels at a constant speed of 96 km/h. Calculate the it will take to travel from London to Johannesburg, a of 9 km. 18 _phys_1_.indd 18 6/11/8 1:19:
8 Speed, velocity and acceleration.7 Describe what happens to speed in the two journeys described in the graphs a) b).8 Describe how the speed changes in the two journeys described in the graphs. a) speed b) speed.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.75 s. Draw a speed graph to show the deceleration, and use your graph to calculate a) the travelled once the brakes are applied b) the deceleration of the car. S.1 A car accelerates from rest at m/s for 8 seconds. a) Draw a speed graph to show this motion. b) Use your graph to find (i) the final speed of the car (ii) the travelled by the car..11 The graph shows how the speed of an aeroplane changes with. 4 B C 3 1 A (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. _phys_1_.indd 19 6/11/8 1:19: 19
9 Summary Now that you have completed this chapter, you should be able to: define speed recall and use the equation speed understand that acceleration is a change of speed draw and interpret  graphs draw and interpret speed graphs calculate travelled from a speed graph recognise that the steeper the gradient of a speed 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 calculate acceleration from the gradient of a speed graph describe an experiment to measure the acceleration of free fall. _phys_1_.indd 6/11/8 1:19:3
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