Velocity and Acceleration

8 Key excerpts on "Velocity and Acceleration"

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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 - ePub

  Foundations of Mechanical Engineering

  • A. D. Johnson(Author)
  • 2017(Publication Date)
  • Routledge

    (Publisher)

  Motion of bodies 2

  2.1 Aims

  • To introduce the concepts of linear and angular motion.
  • To explain the relationships between displacement, Velocity and Acceleration.
  • To explain the relationship between absolute and relative velocities.
  • To define the equations used to analyse linear and angular motion.
  • To introduce an approach by which linear and angular motion problems can be analysed.
  • To explain related topics such as ‘falling bodies’, ‘trajectories’ and vector methods.

  2.2 Introduction to Motion

  When traffic lights turn to green a car will move away with increasing velocity. The car will cover a distance in a particular direction and will possess a particular velocity at any instant. During this process the car possesses the three basic constituents of motion, namely: displacement, Velocity and Acceleration. It should be noted that since the car runs on wheels, these will also be in motion and therefore possess displacement, Velocity and Acceleration. However, the car moves in a linear direction, while the wheels move in an angular direction.

  2.2.1 Displacement

  If a man walks 10 km, there is an indication of the distance between the start position and the final position, but there is no indication of the direction. The 10 km is merely the distance covered and, as such, is a scalar quantity, i.e. possessing magnitude only. Displacement, however, implies a change in position or movement over a distance and gives the position and direction from the start point. Thus displacement is a vector quantity possessing both magnitude and direction.

  Fig. 2.1 Displacement diagram.

  Figure 2.1 gives an example of a man who walks 3 km east then 4 km north. He has actually walked a distance of 7 km but has been displaced from his start point by only 5 km.

  2.2.2 Velocity

  Velocity is the value of displacement measured over a period of time. It is the rate over which a distance/displacement is traversed. The magnitude of velocity is often expressed in convenient units such as kilometres per hour or miles per hour; however, these should be regarded as observation and comparison units. For analysis purposes velocity is better expressed in SI units of m/s.

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  eBook - ePub

  Vectors in Physics and Engineering

  • Alan Durrant(Author)
  • 2019(Publication Date)
  • Routledge

    (Publisher)

  2 were concerned with vectors that remained constant in time. We now consider vectors that vary in time, i.e. time-dependent vectors. An important application is the study of moving bodies. The position vector of a moving body generally varies in magnitude and direction. The Velocity and Acceleration vectors may also vary. The acceleration vector is of particular interest because acceleration is produced by forces, as described by Newton’s second law of motion. The Velocity and Acceleration vectors are obtained from the position vector by the methods of differential calculus. The application of calculus to vectors is introduced in this chapter and is in fact the main theme of the remainder of this book.

  The study of moving bodies, without regard to the causes of motion, is called kinematics ; while the study of the effects of forces on moving bodies is called dynamics .

  Sections 3.1 and 3.2 introduce the concept of a vector function of time and show how a vector function can be differentiated from first principles. Rules for differentiating sums and products of vector functions are stated (but not derived) in Section 3.3 . important examples of particle motion such as projectile motion and motion in a circle are considered. The angular velocity vector is introduced in Section 3.4 . The final sections describe applications to relative motion, including the derivation of inertial forces in accelerating and rotating frames of reference.

  3.1 Introducing Vector Functions

  We begin with a brief review of ordinary scalar functions of a single scalar variable in order to establish some notation and definitions.

  3.1.1 Scalar functions - a review

  A scalar function f is defined by a rule and a domain. The rule specifies a unique scalar function value f (x) for each value x of the independent scalar variable. The rule is usually in the form of an equation, such as f (x) = x + 2. The set of values of the independent variable x over which the rule is to be used is called the domain of the function. Thus we have, for example, the scalar function f

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  eBook - ePub

  An Elementary Treatise on Theoretical Mechanics

  • Sir James H. Jeans(Author)
  • 2013(Publication Date)
  • Dover Publications

    (Publisher)

  9. A person traveling eastward at the rate of 3 miles an hour finds that the wind seems to blow directly from the north; on doubling his speed it appears to come from the northeast. Find the direction of the wind and its velocity.

  ACCELERATION

  11. Acceleration is rate of increase of velocity. If we find that the velocity of a moving point increases by an amount ƒ in a second, no matter which second is selected, we say that the motion of the point has a uniform acceleration ƒ per second. For instance, a stone or other body falling under gravity is found to increase its velocity by a certain constant velocity ƒ per second, where f denotes a velocity of about 32 feet per second. Thus we say that a falling stone has a uniform acceleration of ƒ per second, or of about 32 feet per second per second.

  Generally, however, an acceleration will not be uniform; the rate of increase of velocity will be different at different stages of the journey. To find the acceleration at any instant, we observe the change in velocity during an infinitesimal interval dt of time. If dv is the increase of velocity, we say that is the acceleration at the instant at which dt is taken. An acceleration will of course have sign as well as magnitude, for the velocity may be either increasing or decreasing. When the velocity is decreasing, the acceleration is reckoned with a negative sign. A negative acceleration is spoken of as a retardation. Thus a retardation ƒ means that the velocity is diminished by an amount ƒ per unit of time.

    EXAMPLES  

  1 . A workman fell from the top of a building and struck the ground in 4 seconds. With what velocity did he strike the ground, the acceleration due to gravity being 32 feet per second per second?

  2 . A train has at a given instant a velocity of 30 miles an hour, and moves with an acceleration of 1 foot per second per second. Find its velocity after 20 seconds.

  3.

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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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  eBook - ePub

  Describing Motion

  The Physical World

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

    (Publisher)

  −1 .

  Note that the constant downward acceleration does not prevent the particle from moving upwards. It simply means that an upward velocity would decrease at a steady rate. (Similarly, any downward velocity would increase at a steady rate due to a constant downward acceleration.)

  One final point can be made from Figure 2.19 . Notice that the arrows representing υ 1 and υ 2 differ only slightly in length (by much less than the length of the arrow for Δυ ). This means that the speed of the ball changes at a slower rate than 9.8 m s−2 . If this seems puzzling, you should realize that the acceleration of the ball is partly due to a change in speed and partly due to a change in the direction of motion. In Chapter 3 , you will meet the case of uniform circular motion, in which the speed of the particle remains fixed but acceleration occurs because the direction of motion continuously changes.

  Question 2.16 At a certain time a particle undergoing constant acceleration has velocity υ 1 = (10,0)ms−1 . One second later, the particle’s velocity is υ 2 =(0, 10) m s−1 . Find the following for this one-second interval:

  (a) the change in velocity; (b) the acceleration; (c) the magnitude of the acceleration; (d) the direction of the acceleration;

  (e) the change in the speed, Δυ ;

  (f) the magnitude of the change in the velocity, |Δυ|. What is the physical difference between Δ|υ| and |Δ|υ|?

  4 Projectile motion

  4.1 Introducing projectile motion

  The term projectile is used to describe any object that is launched into the air near the Earth’s surface and which thereafter moves in unpowered flight in such a way that its motion is determined by the effects of gravity and air resistance. Tennis balls, golf balls and footballs all provide good examples of projectiles, as do javelins, high jumpers and long jumpers. Some examples of projectiles being launched are shown in Figure 2.20

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  eBook - ePub

  3D Math Primer for Graphics and Game Development

  • Fletcher Dunn, Ian Parberry(Authors)
  • 2011(Publication Date)
  • A K Peters/CRC Press

    (Publisher)

  The accelerations experienced by an object can vary as a function of time, and indeed we can continue this process of differentiation, resulting in yet another function of time, which some people call the “jerk” function. We stick with the position function and its first two derivatives in this book. Furthermore, it’s very instructive to consider situations in which the acceleration is constant (or at least has constant magnitude). This is precisely what we’re going to do in the next few sections.

  Section 11.6 considers objects under constant acceleration, such as objects in free fall and projectiles. This will provide an excellent backdrop to introduce the integral, the complement to the derivative, in Section 11.7 . Then Section 11.8 examines objects traveling in a circular path, which experience an acceleration that has a constant magnitude but a direction that changes continually and always points towards the center of the circle.

  11.6   Motion under Constant Acceleration

  Let’s look now at the trajectory an object takes when it accelerates at a constant rate over time. This is a simple case, but a common one, and an important one to fully understand. In fact, the equations of motion we present in this section are some of the most important mechanics equations to know by heart, especially for video game programming.

  Before we begin, let’s consider an even simpler type of motion—motion with constant velocity. Motion with constant velocity is a special case of motion with constant acceleration—the case where the acceleration is constantly zero. The motion of a particle with constant velocity is an intuitive linear equation, essentially the same as Equation (9.1), the equation of a ray. In one dimension, the position of a particle as a function of time is

  x ( t ) =

  x 0

  + υ t ,

  (11.14)

  where x 0 is the position of the particle at time t = 0, and υ is the constant velocity.

  Now let’s consider objects moving with constant acceleration. We’ve already mentioned at least one important example: when they are in free fall, accelerating due to gravity. (We’ll ignore wind resistance and all other forces.) Motion in free fall is often called projectile motion . We start out in one dimension here to keep things simple. Our goal is a formula x (t ) for the position of a particle at a given time.

  Take our example of illegal ball-bearing-bombing off of Willis Tower. Let’s set a reference frame where x increases in the downward direction, and x 0 = 0. In other words, x (t ) measures the distance the object has fallen from its drop height at time t . We also assume for now that initial velocity is υ 0

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  eBook - ePub

  BIOS Instant Notes in Sport and Exercise Biomechanics

  • Paul Grimshaw, Neil Fowler, Adrian Lees, Adrian Burden(Authors)
  • 2007(Publication Date)
  • Routledge

    (Publisher)

  Because velocity is a vector quantity, its sign tells us about the direction of travel. In Fig. A5.1, the object is moving away from the reference point (i.e., its displacement is increasing from zero). In Fig. A5.2, however, the object’s displacement is decreasing: it is getting closer to the origin. The negative sign of the velocity tells us that the object is now moving in the opposite direction. Acceleration may also be either positive or negative but whilst the sign of the velocity is only dependent upon the direction of motion, the sign of an object’s acceleration is dependent upon whether the object is accelerating or decelerating. For example, a ball thrown vertically into the air will be moving in a positive direction but as it is slowing down its acceleration will be negative (i.e., decelerating). When the ball reaches the apex of its flight and falls back to earth the magnitude of its velocity will now be increasing but in a negative direction (i.e., its velocity is negative) but its acceleration will still be negative (Fig. A5.3). Fig. A5.2. An example of a negative gradient and negative tangent Fig. A5.3. The displacement, velocity, and acceleration profiles for a projectile Points of maxima, minima and inflection Sometimes, when plotting the motion of an object on a displacement time graph, we see localized points of maximum (point A, Fig. A5.4a) or minimum (point B, Fig. A5.4a) displacement (localized maxima and minima). At these points the gradient of the curve is neither positive nor negative because the tangent is horizontal. Here the velocity must be zero. Points of inflection may also occur. Points of inflection occur when the curve moves from a concave to convex (point C, Fig. A5.4b) or from convex to concave (point D, Fig. A5.4b). These represent localized maximum and minimum gradients respectively and hence points of maximum or minimum velocity

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