Physics
Graphs of Motion
Graphs of motion are visual representations of an object's position, velocity, and acceleration over time. They typically include distance-time graphs, velocity-time graphs, and acceleration-time graphs. These graphs provide a clear way to analyze and understand an object's motion, including its speed, direction, and changes in velocity.
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9 Key excerpts on "Graphs of Motion"
eBook - PDF- Stephen Lee(Author)
- 2014(Publication Date)
- CRC Press(Publisher)
In addition, a picture in the form of a diagram or graph can often be used to show the information more clearly. Figure 1.4 is a diagram showing the direction of motion of the marble and relevant distances. The direction of motion is indicated by an arrow. Figure 1.5 is a graph showing the position above the level of your hand against the time. Notice that it is not the path of the marble. 1.25 m top position 1.25 m zero position positive direction hand 4 AN INTRODUCTION TO MATHEMATICS FOR ENGINEERS : MECHANICS Q UESTION 1.3 What are the positions of the particles A, B and C in the diagram below? Figure 1.3 What is the displacement of B i) relative to A ii) relative to C? –4 A B C –3 –2 –1 0 1 2 3 4 5 x Figure 1.4 Figure 1.5 Note When drawing a graph it is very important to specify your axes carefully. Graphs showing motion usually have time along the horizontal axis. Then you have to decide where the origin is and which direction is positive on the vertical axis. In this graph the origin is at hand level and upwards is positive. The time is measured from the instant the marble leaves your hand. Notation and units As with most mathematics, you will see in this book that certain letters are commonly used to denote certain quantities. This makes things easier to follow. Here the letters used are: ● s , h , x , y and z for position ● t for time measured from a starting instant ● u and v for velocity ● a for acceleration. The S.I. (Système International d’Unités) unit for distance is the metre (m), that for time is the second (s) and that for mass the kilogram (kg). Other units follow from these so speed is measured in metres per second, written ms 1 . S.I. units are used almost entirely in this book but occasional references are made to imperial and other units. –1 position (metres) time (s) 0.5 A B C 1.5 1 0 1 2 1.25 m A H B C 1 m MOTION ALONG A STRAIGHT LINE 5 Q UESTION 1.4 The graph in figure 1.5 shows that the position is negative after one second (point B).
eBook - PDF- Michael Tammaro(Author)
- 2019(Publication Date)
- Wiley(Publisher)
From Equations 2.5.3 and 2.5.1, the position and velocity are x at v at and 1 2 2 = = Thus, the position versus time graph has the shape of a parabola, whereas the velocity versus time graph is a straight line. Run the animation in Animated Figure 2.6.3 and watch I N T E R A C T I V E F E A T U R E 60 | Chapter 2 carefully. The car leaves a mark along its path every second and, because its speed is increasing, the distance between the dots increases. This is reflected in the position versus Animated Figure 2.6.3 A car accelerates forward, from rest, at 2.00 m/s 2 . The position versus time graph is a parabola, whereas the velocity versus time graph is a straight line. I N T E R A C T I V E F E A T U R E time graph. The velocity, on the other hand, increases by the same amount each second because the acceleration is constant. The slope of the velocity versus time graph, v t / ∆ ∆ , is the acceleration. The slope of the position versus time graph is changing. The slope of the tangent line at a particular point is the instantaneous velocity. For example, the slope of the tangent line (the blue line) at 2.00 s is equal to 4.00 m/s, which is confirmed by the velocity versus time graph. 2.7 Use the equations of one-dimensional kinematics to solve free-fall problems. Any two objects attract each other with a force called gravity. For example, your left shoe pulls on your right, and your right shoe pulls back on your left. As we will see in Chapter 10, the force between terrestrial objects, such as shoes, is so weak that it is negligible. The force between a shoe and the Earth, on the other hand, is not negligible, because a dropped shoe will fall. The Earth pulls down on objects near its surface, and this is what causes things to fall, to be planted firmly on the Earth’s surface, or to arc through the air when thrown. For an object near the surface of the Earth, we say that it is in a state of free fall if the pull of Earth’s gravity is the only force acting on it.
eBook - PDFGeneral Engineering Science in SI Units
The Commonwealth and International Library: Mechanical Engineering Division
- G. W. Marr, N. Hiller(Authors)
- 2013(Publication Date)
- Pergamon(Publisher)
Such graphs are called velocity-time graphs, or speed-time graphs. In general the graph will be some form of smooth curve. When the velocity is constant, however, the graph is a straight line 39 GENERAL ENGINEERING SCIENCE IN SI UNITS parallel to the time axis (Fig. 2.6). During a time interval (/ 2 -/i), the distance travelled at a constant velocity v is vX{h-h) = (ordinate of graph) X (/ 2 - * 1) = ADXAB = area ABCD. ΰ .O 2 Time FIG. 2.6. This shows that the distance travelled is numerically equal to the area under the velocity-time graph bounded by the ordinates at t x and t 2 . Where the velocity is not constant, the velocity-time graph is not a straight line parallel to the time axis. Even in such cases, the area under the curve bounded by two ordinates t and t 2 (Fig. 2.7) still represents, to scale, the total distance travelled during the period {t 2 -t). That this is the case may be demonstrated by the following argument. The total distance travelled in time (/ 2 —/i) is vX(t 2 — ti), where v is the average velocity during the period. But v is the average height of the curve above the axis between ty and t 2 . .*. vX(i 2 — ti) = (average height of curve)X(base length) = area under curve between t and t 2 . 40 VELOCITY AND ACCELERATION Another special case is when the moving body starts from rest and travels with constant acceleration. The velocity increases by equal amounts for equal intervals of time and consequently the velocity-time graph is a straight line passing through the origin. Time, s FIG. 2.8. For example, if the body *noves with a constant acceleration of 5 m/s 2 , this means that during each successive second the velocity will increase by 5 m/s. The appropriate velocity-time graph is shown in Fig. 2.8. The graph in Fig. 2.9 also represents uniformly accelerated 41 GENERAL ENGINEERING SCIENCE IN SI UNITS t, u Time FIG. 2.9. motion starting from rest. In this case, the constant acceleration, a = — = gradient of graph.
eBook - PDF- Fatih Gozuacik, Denise Pattison, Catherine Tabor(Authors)
- 2020(Publication Date)
- Openstax(Publisher)
• The distance an object moves is the length of the path along which it moves. • Displacement is the difference in the initial and final positions of an object. 2.2 Speed and Velocity • Average speed is a scalar quantity that describes distance traveled divided by the time during which the motion occurs. • Velocity is a vector quantity that describes the speed and direction of an object. • Average velocity is displacement over the time period during which the displacement occurs. If the velocity is constant, then average velocity and instantaneous velocity are the same. 2.3 Position vs. Time Graphs • Graphs can be used to analyze motion. • The slope of a position vs. time graph is the velocity. • For a straight line graph of position, the slope is the average velocity. • To obtain the instantaneous velocity at a given moment for a curved graph, find the tangent line at that point and take its slope. 2.4 Velocity vs. Time Graphs • The slope of a velocity vs. time graph is the acceleration. • The area under a velocity vs. time curve is the displacement. • Average velocity can be found in a velocity vs. time graph by taking the weighted average of all the velocities. KEY EQUATIONS 2.1 Relative Motion, Distance, and Displacement Displacement 2.2 Speed and Velocity Average speed Average velocity 2.3 Position vs. Time Graphs Displacement . 2.4 Velocity vs. Time Graphs Velocity Acceleration Chapter 2 • Key Terms 81 CHAPTER REVIEW Concept Items 2.1 Relative Motion, Distance, and Displacement 1. Can one-dimensional motion have zero distance but a nonzero displacement? What about zero displacement but a nonzero distance? a. One-dimensional motion can have zero distance with a nonzero displacement. Displacement has both magnitude and direction, and it can also have zero displacement with nonzero distance because distance has only magnitude. b. One-dimensional motion can have zero distance with a nonzero displacement.
- Paul Anthony Russell(Author)
- 2021(Publication Date)
- Reeds(Publisher)
The study of kinematics concentrates on describing motion in words, numbers, diagrams, graphs, and equations. These help the engineer develop cognitive understanding about the way objects behave in the material world. The abstract realism will not be divorced from the object and forces involved; although these are Kinematics • 51 not part of the discipline, some reference to force and objects does help in shaping the engineer’s thought processes. Case A represents a body that was moving at 5 m/s due east, having its velocity changed to 12 m/s due east; the vector of each velocity is drawn from a common point; the difference between the free ends of the vectors is the change of velocity – in this case it is 7 m/s. Case B is a body with an initial velocity of 9 m/s due east, being changed to 2 m/s due west; the vector diagram shows the vector of each velocity drawn from a common point; the difference between their free ends is the change of velocity, which is 11 m/s. Case C is that of a body with an initial velocity of 6 m/s due east changed to 8 m/s due south. The vector diagram is constructed on the same principle of the two vectors drawn from a common point. The change of velocity is, as always, the difference between the free ends of the two vectors, this is, 8 6 10 2 2 + = m/s. The direction for change of velocity is S 36° 52’ W due to change in velocity taking place in the direction of the applied force, which in this case is east to south-west. In all cases, the vector diagrams are constructed by drawing the velocity vectors from a common point. This technique is called vector subtraction. Space diagrams Vector diagrams A 5 m/s 9 m/s 2 m/s 6 m/s 8 m/s 12 m/s B C N S 5 7 12 W E 9 6 8 Change of velocity 11 2 ▲ Figure 2.10 Space and vector diagrams for a change in velocity 52 • Applied Mechanics Acceleration is the rate of change of velocity; therefore, in all of these cases the value of acceleration can be obtained by dividing change of velocity by time.
- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2023(Publication Date)
- Wiley(Publisher)
LEARNING OBJECTIVES Motion Along a Straight Line 13 14 CHAPTER 2 Motion Along a Straight Line examples. In this chapter, we study the basic physics of motion where the object (race car, tectonic plate, blood cell, or any other object) moves along a single axis. Such motion is called one-dimensional motion. Motion The world, and everything in it, moves. Even seemingly stationary things, such as a roadway, move with Earth’s rotation, Earth’s orbit around the Sun, the Sun’s orbit around the center of the Milky Way galaxy, and that galaxy’s migration relative to other galaxies. The classification and comparison of motions (called kinematics) is often challenging. What exactly do you measure, and how do you compare? Before we attempt an answer, we shall examine some general properties of motion that is restricted in three ways. 1. The motion is along a straight line only. The line may be vertical, horizontal, or slanted, but it must be straight. 2. Forces (pushes and pulls) cause motion but will not be discussed until Chapter 5. In this chapter we discuss only the motion itself and changes in the motion. Does the moving object speed up, slow down, stop, or reverse direction? If the motion does change, how is time involved in the change? 3. The moving object is either a particle (by which we mean a point-like object such as an electron) or an object that moves like a particle (such that every portion moves in the same direction and at the same rate). A stiff pig slipping down a straight playground slide might be considered to be moving like a particle; however, a tumbling tumbleweed would not. Position and Displacement To locate an object means to find its position relative to some reference point, often the origin (or zero point) of an axis such as the x axis in Fig. 2.1.1. The positive direction of the axis is in the direction of increasing numbers (coordinates), which is to the right in Fig.
eBook - PDFFrom Atoms to Galaxies
A Conceptual Physics Approach to Scientific Awareness
- Sadri Hassani(Author)
- 2010(Publication Date)
- CRC Press(Publisher)
Chapter 6 Kinematics: Describing Motion Chapter 4 taught us some basic knowledge of motion along a straight line. This chapter generalizes the concept of motion and introduces some fundamental quantities necessary for its description. 6.1 Position, Displacement, and Distance What is motion? Vaguely speaking, it is the change in the state of an object. More precisely, it is the change in the position of an object with time. Still more precisely, we must speak of the motion of an object relative to an observer , although the “object” could be a person, and the “observer” a thing. To analyze the motion, draw an arrow from the “observer” O to the “object” A , and call the arrow the position vector . 1 The very definition of the position vector assumes Position vector. that both O and A are points. The position vector, denoted commonly by r , determines the instantaneous position of the object A [Figure 6.1(a)] relative to an observer O . The word “instantaneous” is important because the position of the point object A is, in general, constantly changing. If we were to take snapshots of A at various times, t 1 , t 2 , t 3 , etc., and label the corresponding points at which A is located by A 1 , A 2 , A 3 , etc, we would have a situation depicted in Figure 6.1(b), with position vectors r 1 , r 2 , r 3 , etc. In this figure, only three out of an infinitude of possible snapshots are shown. Bear in mind that every directed line segment from O to a point on the curve in Figure 6.1 is a possible position vector. What do you know? 6.1. Can you say that the observer—as defined in the descrip-tion of motion—does not move? If the point A does not change, i.e., if the position vector r does not vary with time, we say that the object is stationary relative to O . You have to make a clear distinction between the distance between O and A , which is the length of r , and the position vector, which is the directed line segment r .
eBook - PDF- John Matolyak, Ajawad Haija(Authors)
- 2013(Publication Date)
- CRC Press(Publisher)
25 © 2010 Taylor & Francis Group, LLC Motion in One Dimension This chapter addresses deriving and using the equations of motion that describe the time depen-dence of an object’s displacement, velocity, and acceleration. Relationships between displacement, velocity, and acceleration are also of importance and will be derived. This chapter starts with the basic definitions of displacement and average velocity of an object moving in one dimension. Such a simplified start will help to lead a complete set of equations of motion for an object moving along one dimension, east–west, north–south, or up and down. In the context of coordinate systems that were treated in the previous chapter, the one-dimensional motion will reduce the time and effort needed on the study of motion in two dimensions, which is the subject of the next chapter. 2.1 DISPLACEMENT Motion can be defined as a continuous change in position and that change could occur in one, two, or three dimensions. As the treatment here will be limited to motions only in one dimension, one axis of the coordinate system described in Chapter 1 will suffice. Choosing this axis as the x-axis, we depict on it a point, O, which will be considered as an origin of zero coordinate. Any point on the right of the origin O will have a positive coordinate and any point on the left of O will have a negative coordinate (Figure 2.1). The displacement, denoted by Δ x, of an object as it moves from an initial position x i to a final position x f along the x-axis can be defined as the change in the object’s position along the x-axis. That is Δ x = x f – x i . (2.1) 2.1.1 S PECIAL R EMARKS The quantities in Equation 2.1 are treated as vectors. x i is the initial position vector, x f the final posi-tion vector, and Δ x the displacement vector. Accordingly, all these quantities have a direction and magnitude.
eBook - PDFApplied Mechanics
Made Simple
- George E. Drabble(Author)
- 2013(Publication Date)
- Made Simple(Publisher)
But since the speed is increasing all the time, it follows that the distance covered (i.e. the displacement x) will increase at a pro-gressively-increasing rate, and the graph will be a curve of the form shown. This particular form of graph is called a parabola and will be familiar to many readers. It occurs here because it can be shown that the distance covered by the body is proportional, not directly to the time, as in column (a), but to the square of the time. Thus, if the moving object travels 4 metres t (a) t (b) / t (0 Fig. 4. (a) Motion with zero acceleration, (b) Motion with constant acceleration, (c) Motion with variable acceleration. 14 Applied Mechanics Made Simple in the first second of motion, it will travel 16 metres (4 X 2 2 ) in the first two seconds, 36 metres (4 x 3 2 ) in the first three seconds, and so on. Examples of this type of motion are rare in practice; an approximation is found in the case of a freely falling body, if the resistance to motion of the air is neglected. This is why the trajectory of a cannon shell, or similar projectile fired upwards at an angle, has the approximate form of a parabola. Column (c) of Fig. 4 shows the motion of a body under a steadily increasing acceleration. Now it is the turn of the velocity to increase parabolically, and the displacement increases at an even greater rate. There is no simple example of this special type of motion. These three examples are merely chosen for the simplicity of the motion they illustrate. The actual motion of most real objects is usually extremely complex. But it is surprising how much can be achieved by considering even only the first two examples. It is possible, frequently, to make simple approxi-mations, so that the motion may be assumed to be either constant acceleration or zero acceleration.
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