Non Uniform Acceleration

6 Key excerpts on "Non Uniform Acceleration"

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  eBook - PDF

  Dynamics of Particles and the Electromagnetic Field

  (With CD-ROM)

  • Slobodan Danko Bosanac(Author)
  • 2005(Publication Date)
  • WSPC

    (Publisher)

  Chapter 9 Non-Uniform Motion Everything that has been discussed so far involved reference frames that move in the relative uniform motion. In particular it was shown how the space-time and other dynamics parameters for the particle and fields trans- form between them if Axiom 4 is implemented. Although this analysis is important uniform motion is of a very special kind because one important aspect of it was neglected: in transforming from one coordinate system into the other it is necessary to apply acceleration (or deceleration) in order to change the relative velocity of the reference frames. The non-uniform mo- tion was neglected because of the assumption that once the uniform was reinstated the field was analyzed after sufficiently long time when its steady state was achieved. However, it is precisely that during the interval of the non-uniform motion the most important effect occurs: exchange of energy between the particle and the field, and for its proper understanding the non-uniform motion should be understood. Furthermore study of the non- uniform motion is essential for understanding the nature of a very special kind of force: gravity. For a particle in a uniform motion the following symmetry applies (the essence of Axiom 4): (a) particle moves uniformly with respect to a ref- erence frame and (b) particle is at rest but the reference frame moves uniformly. It is tempting to generalize this symmetry statement for the non-uniform motion, which in essence should state that there is no way to distinguish between the following two situations: (a) a force acts on the particle and the reference frame is not moving, and (b) the particle is not moving but the reference frame accelerates. At the center of attention is now the force, in one case it acts on the particle and in the other on the reference frame, and the question, however, is whether this symmetry is justified or not. The immediate answer is that there is no reason why 167

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  eBook - PDF

  Applied Mathematics

  Made Simple

  • Patrick Murphy(Author)
  • 2014(Publication Date)
  • Butterworth-Heinemann

    (Publisher)

  50 Applied Mathematics Made Simple C H A P T E R F O U R MOTION WITH UNIFORM ACCELERATION IN A STRAIGHT LINE Restricting the motion to a straight line has the advantage that any changes in displacement and velocity must take place along the line. For example, if we are told that a body moves along a straight line with a speed of 27 m s 1 , all we need to do in order to speak of the velocity of the body is to relate the direction of motion to a positive or negative sign. _ r -r : -+ Fig. 46 Consider a body moving along the line X~OX + shown in Fig. 46. The position of the body at any time is given by the point P. Taking 0 as the base point or origin, we adopt the usual convention of measuring displacements to one side (in this case to the right) as positive and displacements to the other side (in this case to the left) as negative. Thus when a point Q is said to be —10 m from 0 , we know that it lies to the left of 0 as illustrated. The position of Ρ would be described as + 2 5 m from 0 ; more usually we would just say Ρ is 25 m from 0 , i.e. all numbers are taken as positive unless otherwise stated. A similar relation applies when we speak of the velocity of a point moving along the line X~OX + . For example, a velocity of — 13 m s 1 means that the point is moving from right to left. (1) Acceleration We have already met in Chapter Three motions in which the velocity does not remain uniform: such changes in the velocity of a body are related to what we call acceleration. DEFINITION: Acceleration is the rate of change of velocity with respect to time. If the increase or decrease in the velocity is the same for every second of the motion, then the acceleration is said to be uniform. It should be noted that acceleration has magnitude, direction, and sense. When the velocity of a body is increasing, the motion of the body is said to be 'accelerating'; when the velocity is decreasing, the motion is said to be 'retarding'.

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  eBook - ePub

  Describing Motion

  The Physical World

  • Robert Lambourne(Author)
  • 2019(Publication Date)
  • CRC Press

    (Publisher)

  Most objects do not move at a precisely constant velocity. If you drop an apple it will fall downwards, but it will pick up speed as it does so (Figure 1.20), and if you drive along a straight road you are likely to encounter some traffic that will force you to vary your speed from time to time. For the most part, real motions are non-uniform motions. Figure 1.20 A falling apple provides an example of nonuniform motion. A sequence of pictures taken at equal intervals of time reveals the increasing speed of the apple as it falls. Figure 1.21 The Position–time graph for a car accelerating from rest. Figure 1.21 shows the Position–time graph of an object that has an increasing velocity over the period r = 0tor = 20s;a car accelerating from rest. As you can see, the Position–time graph is curved. There is relatively little change in position during the first few seconds of the motion but as the velocity increases the car is able to change its position by increasingly large amounts over a given interval of time. This is shown by the increasing steepness of the graph. In everyday language we would say that the graph has an increasing gradient, but you saw in the last section that the term gradient has a precise technical meaning in the context of straight-line graphs. Is it legitimate to extend this terminology to cover curved graphs, and if so, how exactly should it be done? Extending the concept of gradient to the case of curved graphs is actually quite straightforward. The crucial point to recognize is that if you look closely enough at a small part of a smooth curve, then it generally becomes indistinguishable from a straight line. (In a similar way, the surface of the Earth is clearly curved when viewed from space, but each region is approximately flat when seen close-up.) So, if we choose a point on a curve we can usually draw a straight line passing through that point which has the same slope as the curve at the point of contact

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  eBook - PDF

  General Engineering Science in SI Units

  In Two Volumes

  • G. W. Marr, N. Hiller(Authors)
  • 2016(Publication Date)
  • Pergamon

    (Publisher)

  The definition of acceleration indicates that a unit of acceler-ation is obtained from the ratio u ^uoft^me y · T h u s i n t h e SI s y s t e m the basic unit is the metre s ^ n s d econd or metre per second per second, usually abbreviated to m/s 2 . EXAMPLE. A cyclist, starting from rest, attains a speed of 18 km/h in 55 s, while travelling in a straight line. Calculate his aver-age acceleration. Change in velocity = 18 — 0 = 18 km/h 18X10 3 , = -T6ÖÖ-m/s = 5 m/s Time required for change = 55 s Λ average acceleration = ^ m/s 2 ^ 0091 m/s 2 EXAMPLE. A train pulls out of a station with a uniform accelera-tion of 15 cm/s 2 . How long will it take to reach a speed of 54 km/h? Change in speed = 54 — 0 km/h = 54 km/h 54X10 3 3 600 = 15 m/s m/s A Λ . change in speed Acceleration = %— Λ change in speed • · * im e required = acceleration 15 m/s 15XlO-2 m/s 2 100 s 36 VELOCITY AND ACCELERATION 2.4. Distance-time Graphs A distance-time graph is a graph in which the distance travelled in a given time is plotted as ordinate, the corresponding time being plotted as abscissa. For non-uniform motion the graph will be a curve, such as that shown in Fig. 2.2. A general idea of the motion can be deduced from such a graph. Over the portion OA of the graph the curve is concave upward; that is, the graph is W QJ tt> E. w m to u c n V) Q 80 70 bO 50 40 30 .20 10 u 10 20 30 40 50 Time, t seconds F I G . 2.2. rising more and more steeply. The distance travelled during any particular time interval within this total period is thus greater than the distance travelled during the preceding equal interval of time. Hence the average velocity is increasing and the motion is therefore accelerated motion. Over the portion AB, the concavity of the curve is downward; the velocity is progressively decreasing and therefore the moving body is decelerating. During the time interval (i 2 — h), the distance travelled is O2—$i), as indicated in Fig. 2.3. The average velocity during this period is 37

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  eBook - PDF

  General Engineering Science in SI Units

  The Commonwealth and International Library: Mechanical Engineering Division

  • G. W. Marr, N. Hiller(Authors)
  • 2013(Publication Date)
  • Pergamon

    (Publisher)

  The definition of acceleration indicates that a unit of acceler-ation is obtained from the ratio u ^ ^ ^ f . Thus in the SI system the basic unit is the metre J^^ econd or metre per second per second, usually abbreviated to m/s 2 . EXAMPLE. A cyclist, starting from rest, attains a speed of 18 km/h in 55 s, while travelling in a straight line. Calculate his aver-age acceleration. Change in velocity = 18 — 0 = 18 km/h 18X10 3 , = -3 W m / s = 5 m/s Time required for change = 55 s Λ average acceleration = / 5 m/s 2 Δ 0091 m/s 2 EXAMPLE. A train pulls out of a station with a uniform accelera-tion of 15 cm/s 2 . How long will it take to reach a speed of 54 km/h? Change in speed = 54 — 0 km/h = 54 km/h 54X10 3 . = -3 6ÖÖ~ m / s = 15 m/s A , Λ . change in speed Acceleration = 7 7 — time , · · , change in speed • · t i m e required = accelcration 15 m/s 15X10-2 m/s 2 100 s 36 VELOCITY AND ACCELERATION 2.4. Distance-time Graphs A distance-time graph is a graph in which the distance travelled in a given time is plotted as ordinate, the corresponding time being plotted as abscissa. For non-uniform motion the graph will be a curve, such as that shown in Fig. 2.2. A general idea of the motion can be deduced from such a graph. Over the portion OA of the graph the curve is concave upward; that is, the graph is 20 30 40 50 Time, t seconds FIG. 2.2. rising more and more steeply. The distance travelled during any particular time interval within this total period is thus greater than the distance travelled during the preceding equal interval of time. Hence the average velocity is increasing and the motion is therefore accelerated motion. Over the portion AB, the concavity of the curve is downward; the velocity is progressively decreasing and therefore the moving body is decelerating. During the time interval (/ 2 —Ί), the distance travelled is (^ 2 — ·?ι), as indicated in Fig. 2.3. The average velocity during this period is 37

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  eBook - PDF

  An Introduction to Physical Science

  • James Shipman, Jerry Wilson, Charles Higgins, Bo Lou, James Shipman(Authors)
  • 2020(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 2.3 Acceleration 35 When any of these changes occur, an object is accelerating. Examples are 1. a car speeding up (or slowing down) while traveling in a straight line, 2. a car rounding a curve at a constant speed, and 3. a car speeding up (or slowing down) while rounding a curve. Acceleration is defined as the time rate of change of velocity. Taking the symbol Δ (delta) to mean “change in,” the equation for average acceleration (a) can be written as average acceleration 5 change in velocity time for change to occur a 5 Dv Dt 5 v f 2 v o t 2.2 The change in velocity (Δv) is the final velocity v f minus the original velocity v o . Also, the interval Δt is commonly written as t (Δt 5 t 2 t o 5 t), with t o taken to be zero (t is understood to be an interval). v o is not zero if the car is initially moving. The units of acceleration in the SI are meters per second per second, (m/s)/s, or meters per second squared, m/s 2 . These units may be confusing at first. Keep in mind that an acceleration is a measure of a change in velocity during a given time period. Consider a constant acceleration of 9.8 m/s 2 . This value means that the velocity changes by 9.8 m/s each second. Thus, for straight-line motion, as the number of seconds increases, the velocity goes from 0 to 9.8 m/s during the first second, to 19.6 m/s (that is, 9.8 m/s 1 9.8 m/s) during the second second, to 29.4 m/s (that is, 19.6 m/s 1 9.8 m/s) during the third second, and so forth, adding 9.8 m/s each second. This sequence is illustrated in ● Fig. 2.7 for an object that falls with a constant acceleration due to gravity of 9.8 m/s 2 . Because the velocity increases, the distance traveled by the falling object each second also increases, but not uniformly.

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