Physics
Velocity and Position by Integration
Velocity and position by integration refer to the process of determining an object's velocity and position by integrating its acceleration and velocity, respectively, over a given time interval. This involves using calculus to find the area under the curve of the acceleration function to obtain the velocity function, and similarly for determining the position function from the velocity function.
Written by Perlego with AI-assistance
Related key terms
1 of 5
6 Key excerpts on "Velocity and Position by Integration"
eBook - PDFWorkshop Physics Activity Guide Module 1
Mechanics I
- Priscilla W. Laws, David P. Jackson, Brett J. Pearson(Authors)
- 2023(Publication Date)
- Wiley(Publisher)
Unless the position graph consists of a single, straight line, the average velocity between two points will, in general, depend on the points you choose. In addition to the average velocity, we may also want to know the instanta- neous velocity, defined as the slope of the line tangent to the graph at a particular time. Mathematically, the instantaneous velocity is given by the derivative of UNIT 3: INTRODUCTION TO ONE-DIMENSIONAL MOTION 71 the position-time graph (the slope between two points that are infinitely close together): x = lim Δt→0 Δx Δt = dx dt (3.1) The Velocity Vector When dealing with more than one dimension, an object’s position should be considered a vector quantity, r = x x + y y + z z, so it becomes difficult to plot “the position” of an object in two (or three) dimensions. In these cases the concept of the “slope” of the position graph begins to lose its meaning. However, we can still define the average velocity to be the change in the position vector divided by the change in time: avg ≡ ⟨ ⟩ = Δ r Δt = Δx Δt x + Δy Δt y + Δz Δt z (3.2) where the bracket around denotes the average. When written this way, we see that the vector nature of velocity is a direct result of the vector nature of position. As you might guess, the instantaneous velocity vector is defined as the derivative of the position vector with respect to time: = lim Δt→0 Δ r Δt = d r dt = dx dt x + dy dt y + dz dt z = x x + y y + z z (3.3) Stated another way, velocity is the rate of change of position with respect to time. The instantaneous velocity vector points in the direction the object is moving at that instant and, as with any vector, the magnitude of the velocity vector is given by its length, = | | = √ x 2 + y 2 + z 2 . As we saw in Unit 1, a vector can be represented by an arrow (whether in one or more dimensions).
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
11 C H A P T E R 2 Motion Along a Straight Line 2-1 POSITION, DISPLACEMENT, AND AVERAGE VELOCITY What Is Physics? One purpose of physics is to study the motion of objects—how fast they move, for example, and how far they move in a given amount of time. NASCAR engineers are fanatical about this aspect of physics as they determine the performance of their cars before and during a race. Geologists use this physics to measure tectonic-plate motion as they attempt to predict earthquakes. Medical researchers need this physics to map the blood flow through a patient when diagnosing a par- tially closed artery, and motorists use it to determine how they might slow suf- ficiently when their radar detector sounds a warning. There are countless other 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. Key Ideas ● The position x of a particle on an x axis locates the par- ticle with respect to the origin, or zero point, of the axis. ● The position is either positive or negative, according to which side of the origin the particle is on, or zero if the particle is at the origin. The positive direction on an axis is the direction of increasing positive numbers; the opposite direction is the negative direction on the axis. ● The displacement Δx of a particle is the change in its position: Δx = x 2 − x 1 . ● Displacement is a vector quantity. It is positive if the particle has moved in the positive direction of the x axis and negative if the particle has moved in the negative direction. ● When a particle has moved from position x 1 to position x 2 during a time interval Δt = t 2 − t 1 , its average velocity during that interval is v avg = Δx Δt = x 2 − x 1 t 2 − t 1 . ● The algebraic sign of v avg indicates the direction of motion (v avg is a vector quantity).
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
13 C H A P T E R 2 Motion Along a Straight Line 2-1 POSITION, DISPLACEMENT, AND AVERAGE VELOCITY What Is Physics? One purpose of physics is to study the motion of objects—how fast they move, for example, and how far they move in a given amount of time. NASCAR engineers are fanatical about this aspect of physics as they determine the performance of their cars before and during a race. Geologists use this physics to measure tectonic-plate motion as they attempt to predict earthquakes. Medical researchers need this physics to map the blood flow through a patient when diagnosing a par- tially closed artery, and motorists use it to determine how they might slow suf- ficiently when their radar detector sounds a warning. There are countless other 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. Key Ideas ● The position x of a particle on an x axis locates the par- ticle with respect to the origin, or zero point, of the axis. ● The position is either positive or negative, according to which side of the origin the particle is on, or zero if the particle is at the origin. The positive direction on an axis is the direction of increasing positive numbers; the opposite direction is the negative direction on the axis. ● The displacement Δx of a particle is the change in its position: Δx = x 2 − x 1 . ● Displacement is a vector quantity. It is positive if the particle has moved in the positive direction of the x axis and negative if the particle has moved in the negative direction. ● When a particle has moved from position x 1 to position x 2 during a time interval Δt = t 2 − t 1 , its average velocity during that interval is v avg = Δx Δt = x 2 − x 1 t 2 − t 1 . ● The algebraic sign of v avg indicates the direction of motion (v avg is a vector quantity).
- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2023(Publication Date)
- Wiley(Publisher)
2.1 POSITION, DISPLACEMENT, AND AVERAGE VELOCITY KEY IDEAS 1. The position x of a particle on an x axis locates the particle with respect to the origin, or zero point, of the axis. 2. The position is either positive or negative, according to which side of the origin the particle is on, or zero if the particle is at the origin. The posi- tive direction on an axis is the direction of increasing positive numbers; the opposite direction is the negative direction on the axis. 3. The displacement Δ x of a particle is the change in its position: Δ x = x 2 − x 1 . 4. Displacement is a vector quantity. It is positive if the particle has moved in the positive direction of the x axis and negative if the particle has moved in the negative direction. 5. When a particle has moved from position x 1 to position x 2 during a time inter- val Δt = t 2 − t 1 , its average velocity during that interval is v avg = Δ x ___ Δt = x 2 − x 1 ______ t 2 − t 1 . 6. The algebraic sign of v avg indicates the direction of motion (v avg is a vector quantity). Average velocity does not depend on the actual distance a particle moves, but instead depends on its original and final positions. 7. On a graph of x versus t, the average velocity for a time interval Δt is the slope of the straight line connecting the points on the curve that represent the two ends of the interval. 8. The average speed s avg of a particle during a time interval Δt depends on the total distance the particle moves in that time interval: s avg = total distance ____________ Δt . 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.
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 - PDF- Pierluigi Zotto, Sergio Lo Russo, Paolo Sartori(Authors)
- 2022(Publication Date)
- Società Editrice Esculapio(Publisher)
which is called instantaneous scalar velocity (or speed). Analytically the ratio, being obtained from the ratio Δs Δt with Δt approaching zero, represents the derivative of space with respect to time, that is, if the function s(t) is known, i.e. the position time law has been found for instance by graphic interpolation, the instanta- Kinematics of Point-like Particles Chapter 3 34 neous velocity can be obtained in any point of the trajectory at any time by computing the time derivative of the law itself v t ( ) = d dt s t ( ) ⎡ ⎣ ⎤ ⎦ = ′ s t ( ) . The two definitions (ratio between infinitesimal quantities and first derivative) are equivalent, but, from the Physics point of view, notation v = ds dt makes more sense, since it reminds about the way the operation of passage to the limit occurs; thus it is the one usually preferred when describing Physics topics. Obviously, when the explicit calculation of a derivative is required, the well-known derivation rules of Mathematical Analysis are used. The derivative of function s(t) provides its slope: if v > 0, then s(t) increases, i.e the body moves away from the origin; if v < 0, then s(t) decreases and the body moves closer to the origin; if v = 0 the body is at rest in a certain position of the trajectory. A motion is called uniform motion, if speed is constant, that is if, once divided the tra- jectory travelled by a point-like particle into all equal portions of space Δs, each portion is covered in the same time interval Δt . 3.4 Determination of the Position Time Law from Velocity We could now introduce the methods of Mathematical Analysis without any further justification, but it is more convenient to proceed step by step in order to persuade our- selves that Mathematical Analysis actually provides the adequate instruments needed to deal with the problem. Consider therefore a portion Δs of the trajectory covered in time Δt and divide it into N intervals Δs i each one covered in its time Δt i .
Index pages curate the most relevant extracts from our library of academic textbooks. They’ve been created using an in-house natural language model (NLM), each adding context and meaning to key research topics.
