Velocity and Acceleration

10 Key excerpts on "Velocity and Acceleration"

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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)

  Section 2 Velocity and Acceleration 2.1. Motion When the position of one body relative to another is continuously changing the bodies are said to be in relative motion. Motion, in fact, is always relative. In very many cases we are concerned with the motion of a body relative to the earth, and in such cases the word relative is generally omitted. We are accustomed to refer-ring simply to the motion of a motor vehicle or aircraft. We may say, for example, that a caj* is travelling at a speed of 30 km/h. When we do so, it must be understood that the speed is relative to the earth. 2.2. Velocity The velocity of a body is the rate at which the body is changing its position. Because direction is involved, velocity is a vector quantity. The magnitude, or numerical value, of a velocity is called the speed. The average speed of a body during a given interval of time IS measured by the ratio total distance^raven.d in given time W h e n a body travels equal distances during equal intervals of time, what-ever the magnitude of the time interval, the body is said to travel with constant speed. A velocity may change because of change in speed, or in the direction of motion or because of a change v in both of these. 32 Velocity and Acceleration When a body moves in such a way that its velocity does not change, it is said to move with constant, or uniform, velocity. Hence to move with uniform velocity, a body must travel at constant speed in a straight line. EXAMPLE. A vehicle travels a distance of 840 m in 30 sec. Express its average speed in km/h. Distance travelled = 840 m = 0-84 km. 30 s = ai 0-84 km Time interval = 30s = g§öö h = iiö-h .*. average speed — —j = 100-8 km/h. EXAMPLE. The straight-line distance between two towns, A and B, is 49 km. Town A is due north-west from B. The road distance between the towns is 56 km. A motorist leaves A at 13.20 h and arrives at B at 14.10 h. Calculate (a) his average speed; (b) his average velocity.

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  General Engineering Science in SI Units

  In Two Volumes

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

    (Publisher)

  Section 2 Velocity and Acceleration 2.1. Motion When the position of one body relative to another is continuously changing the bodies are said to be in relative motion. Motion, in fact, is always relative. In very many cases we are concerned with the motion of a body relative to the earth, and in such cases the word relative is generally omitted. We are accustomed to refer-ring simply to the motion of a motor vehicle or aircraft. We may say, for example, that a car is travelling at a speed of 30 km/h. When we do so, it must be understood that the speed is relative to the earth. 2.2. Velocity The velocity of a body is the rate at which the body is changing its position. Because direction is involved, velocity is a vector quantity. The magnitude, or numerical value, of a velocity is called the speed. The average speed of a body during a given interval of time is measured by the ratio total distance um^d m given time w h e n a body travels equal distances during equal intervals of time, what-ever the magnitude of the time interval, the body is said to travel with constant speed. A velocity may change because of change in speed, or in the direction of motion or because of a change in both of these. 32 Velocity and Acceleration When a body moves in such a way that its velocity does not change, it is said to move with constant, or uniform, velocity. Hence to move with uniform velocity, a body must travel at constant speed in a straight line. EXAMPLE. A vehicle travels a distance of 840 m in 30 sec. Express its average speed in km/h. Distance travelled = 840 m = 0-84 km. Time interval = 3 0 s = a|ööh = 1 ^-h 0-84 km .·. average speed = —j I2Ö h = 100-8 km/h. EXAMPLE. The straight-line distance between two towns, A and B, is 49 km. Town A is due north-west from B. The road distance between the towns is 56 km. A motorist leaves A at 13.20 h and arrives at B at 14.10 h. Calculate (a) his average speed; (b) his average velocity.

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  Inquiry into Physics

  • Vern Ostdiek, Donald Bord(Authors)
  • 2017(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  (d) can be perpendicular to one of the original velocities. ANSWERS: 1. (c) 2. (a) 3. vector 4. False 5. (b) 1.3 Acceleration The physical world around us is filled with motion. But think about this for a moment: cars, bicycles, pedestrians, airplanes, trains, and other vehicles all change their speed or direction often. They start, stop, speed up, slow down, and make turns. The velocity of the wind usually changes from moment to mo- ment. Even Earth as it moves around the Sun is constantly changing its direc- tion of motion and its speed, though not by much as reckoned on a daily basis. The main thrust of Chapter 2 is to show how the change in velocity of an object is related to the force acting on it. For these reasons, a very important concept in physics is acceleration. Physical Quantity Metric Units English Units Acceleration ( a ) meter per second 2 (m/s 2 ) foot per second 2 (ft/s 2 ) mph per second (mph/s) Whenever something is speeding up or slowing down, it is undergoing accel- eration. As you travel in a car, anytime the speedometer’s reading is changing, the car is accelerating. Acceleration is a vector quantity, which means it has both magnitude and direction. Note that the relationship between acceleration and velocity is the same as the relationship between velocity and displacement. Acceleration indicates how rapidly velocity is changing, and velocity indicates how rapidly displacement is changing. EXAMPLE 1.3 A car accelerates from 20 to 25 m/s in 4 seconds as it passes a truck (Figure 1.15). What is its acceleration? Acceleration Rate of change of velocity. The change in velocity divided by the time elapsed. a 5 D v D t DEFINITION Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-300 26 Chapter 1 The Study of Motion SOLUTION Because the direction of motion is constant, the change in velocity is just the change in speed—the later speed minus the earlier speed.

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  College Physics

  • Michael Tammaro(Author)
  • 2019(Publication Date)
  • Wiley

    (Publisher)

  Hence- forth, when we refer to an object’s velocity, we mean its instantaneous velocity, with the magnitude of the instantaneous velocity being the object’s (instantaneous) speed. 2.4 Solve problems involving average acceleration, velocity, and time. 2.4.1 Interpret or determine the direction of the acceleration. If your friend tells you that her velocity has changed, what she probably means is that the magnitude of her velocity has increased or decreased; that is, the speed with which she is moving has changed. Your friend may use the word acceleration or deceleration when speaking about her changing speed. These words have very specific meanings in physics and, like velocity, the way they are used in everyday conversation is sometimes inconsist- ent with their scientific definitions. When an object’s velocity changes, we say that there is acceleration. Suppose that the velocity changes from v 0 to v in a time t t t 0 ∆ = − . We define the object’s average acceler- ation in the following way: a v t v v t t avg 0 0 = ∆ ∆ = − − (2.4.1) The SI unit of acceleration is / m s 2 . Acceleration is a vector quantity, so it has a magni- tude and a direction, with the direction of the acceleration being the same as the direction of the change in velocity. There is acceleration if there is a change in the direction of the velocity, in the magnitude of the velocity (i.e., the speed), or in both. To illustrate the use of Equation 2.4.1, consider the situation depicted in Animated Figure 2.4.1. A rocket, moving in the positive x direction, is coasting in space at a speed 2.4 ACCELERATION Learning Objectives Animated Figure 2.4.1 A rocket fires its forward thrusters and its speed increases. I N T E R A C T I V E F E A T U R E I N T E R A C T I V E F E A T U R E Acceleration | 51 A car is traveling at a speed of 21.8 m/s. The driver taps the brakes for 3.2 s during which time the magnitude of the average acceleration is 1.5 m/s 2 .

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  Applied Mechanics for Engineers

  The Commonwealth and International Library: Mechanical Engineering Division

  • C. B. Smith, N. Hiller, G. E. Walker(Authors)
  • 2013(Publication Date)
  • Pergamon

    (Publisher)

  CHAPTER 4 Velocity and Acceleration Velocity Velocity is rate of change of position with respect to time, so it is necessary to specify its magnitude, direction and the datum with respect to which it is measured. EXAMPLE 4.1. A motor-car travelling along a straight road has a velocity of 60 mile/h or 88 ft/sec relative to the road. A second car approaching from the opposite direction has a velocity of 30 mile/h or 44 ft/s relative to the road. Then the velocity of the second car relative to the first car is 90 m.p.h. or 132 ft/s, which it will be seen is the difference between the velocities of the two cars relative to the road allowing for the fact that one car is going in the opposite direction to the other. Like force, a velocity is a vector quantity and may be represented by a line; the length of the line being the magnitude of the velocity and its position showing the direction. Problems in relative velocity may be dealt with either by cal-culation or by graphical construction. EXAMPLE 4.2. A cyclist is travelling due north at 15 mile/h in a wind blowing from the south-east at 20 mile/h. Find the direction in which a flag mounted on the front of his cycle will point. Calculation. Velocity of cycle relative to the ground in a northerly direction = 15 mile/h. Velocity of wind relative to ground in a northerly direction = 20 sin 45° = 20x0-7071 = 14-142 mile/h. 71 72 APPLIED MECHANICS FOR ENGINEERS Λ velocity of wind relative to cycle in a northerly direction = vel. of wind—vel. of cycle = 14-142-15 = -0-858 mile/h. Velocity of cycle relative to ground in a westerly direction = 0 mile/h. Velocity of wind relative to ground in a westerly direction = 20 cos 45° = 20x0-7071 = 14-142 mile/h. .*. velocity of wind relative to cycle in a westerly direction = 14-142-0 = 14-142 mile/h. .'. actual velocity of wind relative to cycle = y/( — 0-858) 2 + (14-142) 2 = 14-17 m.p.h. and this will be in a direction 14-142 t a n 1 — — — = t a n 1 -16-48^93° 31' W.

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  eBook - ePub

  Understanding Physics

  • Michael M. Mansfield, Colm O'Sullivan(Authors)
  • 2020(Publication Date)
  • Wiley

    (Publisher)

  Section A.13 .

  Study Appendix A, Section A.11.1 ‐ 6 and Worked Example 4.1

  For problems based on the material presented in this section visit up.ucc.ie/4/ and follow the link to the problems.

  4.3 Velocity and Acceleration vectors

  Velocity

  Consider a point P which is moving in two dimensions (the plane of the page). As indicated in Figure 4.15 , the displacements of P from O at two points A and B on the path of P and the corresponding times are (r, t) and (r + Δr, t + Δt), respectively.

  Figure 4.15

  A point moves from A to B in a time Δt; its change of displacement during that time interval is Δr.

  We can now define velocity in vector form (recall Equation (2.5) for the one‐dimensional version). The velocity of the point P at the instant it is at A is defined as follows

  The quantity is a vector (Δr) multiplied by a scalar ( ) and hence v is a vector in the direction of Δr in the limit Δr → 0, that is in the direction of the tangent to the path at A. Thus the direction of the velocity vector is always tangential to the path of the moving point.

  The magnitude of the velocity vector, denoted by |v|, is called the speed. This is the only case in physics in which the magnitude of a vector is given a special name. Note that if an object is moving at constant speed but changes direction, for example from 30 km per hour due North to 30 km per hour due East, its velocity has changed although its speed has not.

  Acceleration

  In a similar way can define acceleration in vector form. As illustrated in Figure 4.16 , if the velocities and corresponding times at two points A and B along a point's path of motion are (v, t) and (v + Δv, t + Δt), respectively, the acceleration of the moving point at A at the instant t is defined as

  Note that the direction of a is in the direction defined by Δv in the limit Δt → 0, which in general is not the same direction as that of v. The velocity vector triangle, representing the addition (v+Δv) = v + Δv (Figure 4.16

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  eBook - PDF

  Physics

  • John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
  • 2015(Publication Date)
  • Wiley

    (Publisher)

  Or it may decrease, as it does when the driver applies the brakes to stop at a red light. In either case, the change in velocity may occur over a short or a long time interval. To describe how the velocity of an object changes during a given time interval, we now introduce the new idea of accel- eration. This idea depends on two concepts that we have previously encountered, velocity and time. Specifically, the notion of acceleration emerges when the change in the velocity is combined with the time during which the change occurs. 30 Chapter 2 | Kinematics in One Dimension The meaning of average acceleration can be illustrated by considering a plane during takeoff. Figure 2.4 focuses attention on how the plane’s velocity changes along the runway. During an elapsed time interval Dt 5 t 2 t 0 , the velocity changes from an initial value of v B 0 to a final velocity of v B . The change D v B in the plane’s velocity is its final velocity minus its initial velocity, so that D v B 5 v B 2 v B 0 . The average acceleration a B is defined in the following manner, to provide a measure of how much the velocity changes per unit of elapsed time. Definition of Average Acceleration Average acceleration 5 Change in velocity Elapsed time a B 5 v B 2 v 0 B t 2 t 0 5 Dv B Dt (2.4) SI Unit of Average Acceleration: meter per second squared (m/s 2 ) The average acceleration a B is a vector that points in the same direction as D v B , the change in the velocity. Following the usual custom, plus and minus signs indicate the two possible directions for the acceleration vector when the motion is along a straight line. We are often interested in an object’s acceleration at a particular instant of time. The instantaneous acceleration a B can be defined by analogy with the procedure used in Sec- tion 2.2 for instantaneous velocity: a B 5 lim D t B0 Dv B D t (2.5) Equation 2.5 indicates that the instantaneous acceleration is a limiting case of the average acceleration.

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  eBook - PDF

  Cutnell & Johnson Physics, P-eBK

  • John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler, Heath Jones, Matthew Collins, John Daicopoulos, Boris Blankleider(Authors)
  • 2020(Publication Date)
  • Wiley

    (Publisher)

  In fact, in the limit that Δt becomes infinitesimally small, the instantaneous velocity and the average velocity become equal, so that  v = lim Δt→0 Δ x Δt (2.3) The notation lim Δt→0 Δ x Δt means that the ratio Δ x∕Δt is defined by a limiting process in which smaller and smaller values of Δt are used, so small that they approach zero. As smaller values of Δt are used, Δ x also becomes smaller. However, the ratio Δ x∕Δt does not become zero but, rather, approaches the value of the instantaneous velocity. For brevity, we will use the word velocity to mean ‘instantaneous velocity’ and speed to mean ‘instantaneous speed’. 2.3 Acceleration LEARNING OBJECTIVE 2.3 Define one-dimensional acceleration. PHOTO 2.1 As this sprinter explodes out of the starting block, her velocity is changing, which means that she is accelerating. In a wide range of motions, the velocity changes from moment to moment, such as in the case of the sprinter in photo 2.1. To describe the manner in which it changes, the concept of acceleration is needed. The veloc- ity of a moving object may change in a num- ber of ways. For example, it may increase, as it does when the driver of a car steps on the gas pedal to pass the car ahead. Or it may decrease, as it does when the driver applies the brakes to stop at a red light. In either case, the change in velocity may occur over a short or a long time interval. To describe how the velocity of an object changes during a given time interval, we now introduce the new idea of acceleration. This idea depends on two concepts that we have previously encountered, velocity and time. Specifically, the notion of acceleration emerges when the change in the velocity is combined with the time during which the change occurs. The meaning of average acceleration can be illustrated by considering a plane during take-off. Figure 2.4 focuses attention on how the plane’s velocity changes along the runway.

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  eBook - ePub

  Doing Physics with Scientific Notebook

  A Problem Solving Approach

  • Joseph Gallant(Author)
  • 2012(Publication Date)
  • Wiley

    (Publisher)

  The average speed is the total distance traveled divided by the elapsed time.

  There is an important difference between the average speed and average velocity. The average speed is always positive and conveys no information about direction. The average velocity contains information about direction and can be positive, negative or zero. It tells us how fast the object moved and the direction of the motion.

  The instantaneous velocity tells us how fast the object is moving right now, at this instant. As the time interval in Eq. (2.2) gets smaller, the average velocity gives a better approximation to the instantaneous velocity. The instantaneous velocity is the average velocity calculated over a vanishingly small time interval. Of course, “vanishingly small” brings us dangerously close to calculus. However, as you will soon see, we can derive equations for the instantaneous velocity without using calculus.

  Acceleration

  As an object moves, its velocity can change. The object’s average acceleration is the rate of change in its velocity

  (2.4)

  where Δv = v - v0 is the change in velocity and Δt is the finite time interval. For constant acceleration, the average and instantaneous acceleration are always the same. The SI unit for acceleration is the meter per second per second (m/ s2 ). The American unit for acceleration is the foot per second per second (ft/ s2 ), although the mile per hour per second (mi/ h/ s) is also used.

  Example 2.3

  A test drive

  What is the constant acceleration of a car that goes from zero to sixty in 4 seconds?

  Solution.

  The phrase “zero to sixty” refers to miles per hour, so you need to convert 60.0 miles per hour to meters per second. The following expression converts the final velocity with the Solve Method .

  You can use Eq. (2.4) and Evaluate

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  Fundamentals of Physics, Volume 1

  • David Halliday, Robert Resnick, Jearl Walker(Authors)
  • 2021(Publication Date)
  • Wiley

    (Publisher)

  Thus, the accel- eration relative to Alex is Observers on different frames of reference that move at constant velocity rela- tive to each other will measure the same acceleration for a moving particle. 86 CHAPTER 4 MOTION IN TWO AND THREE DIMENSIONS 4.7 RELATIVE MOTION IN TWO DIMENSIONS Learning Objective After reading this module, you should be able to . . . 4.7.1 Apply the relationship between a particle’s posi- tion, velocity, and acceleration as measured from two reference frames that move relative to each other at constant velocity and in two dimensions. Key Ideas ● When two frames of reference A and B are moving relative to each other at constant velocity, the velocity of a particle P as measured by an observer in frame A usually differs from that measured from frame B. The two measured velocities are related by v → PA = v → PB + v → BA , where v → BA is the velocity of B with respect to A. Both observers measure the same acceleration for the particle: a → PA = a → PB . Relative Motion in Two Dimensions Our two observers are again watching a moving particle P from the origins of refer- ence frames A and B, while B moves at a constant velocity v → BA relative to A. (The corresponding axes of these two frames remain parallel.) Figure 4.7.1 shows a cer- tain instant during the motion. At that instant, the position vector of the origin of B relative to the origin of A is r → BA . Also, the position vectors of particle P are r → PA relative to the origin of A and r → PB relative to the origin of B. From the arrangement of heads and tails of those three position vectors, we can relate the vectors with r → PA = r → PB + r → BA . (4.7.1) By taking the time derivative of this equation, we can relate the velocities v → PA and v → PB of particle P relative to our observers: v → PA = v → PB + v → BA . (4.7.2) By taking the time derivative of this relation, we can relate the accelera- tions a → PA and a → PB of the particle P relative to our observers.

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