Acceleration and Velocity

6 Key excerpts on "Acceleration and Velocity"

• 

  eBook - ePub

  Essential Calculus with Applications

  • Richard A. Silverman(Author)
  • 2013(Publication Date)
  • Dover Publications

    (Publisher)

  Chapter 3 DIFFERENTIATION AS A TOOL 3.1 VELOCITY AND ACCELERATION

  3.11. By a particle we mean an object whose actual size can be ignored in a given problem, and which can therefore be idealized as a point. There are problems in which the earth itself can be regarded as a particle, just as there are problems in which a pinhead is a complicated structure made up of vast numbers of tiny particles.

  Consider the motion of a particle along a straight line. Let s be the particle’s distance at time t from some fixed reference point, where s is positive if measured in a given direction along the line and negative if measured in the opposite direction. Then the particle’s motion is described by some distance function

  Here, for simplicity, we denote the dependent variable and the function by the same letter, a common practice. In the language of physics, (1) is the equation of motion of the particle.

  We now ask a key question: How fast is the particle going? There are two answers, depending on whether we ask about a given interval of time or about a given instant of time. In the first case, we get the average velocity , which is a difference quotient. In the second case, we get the instantaneous velocity , which is a derivative.

  3.12. Average velocity

  a. By the average velocity of the particle with equation of motion (1) , over the interval from t to t + Δt , we mean the function of two variables

  It is meaningless to ask for the average velocity at a given instant t without specifying the averaging time .

  b. To be useful, an average velocity should not be too “crude,” that is, Δt

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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

  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

  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

  BIOS Instant Notes in Sport and Exercise Biomechanics

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

    (Publisher)

  v)

  and is said to be the differential of displacement, s, with respect to time, t.

  Following similar reasoning, and given that the average acceleration is

  the instantaneous acceleration is given by the gradient of the tangent to the velocity curve at that instant in time and therefore the instantaneous acceleration, a, is said to be the differential of velocity, v, with respect to time, t. This is written mathematically as:

  As this term contains velocity, which is itself a rate of change of displacement with respect to time, acceleration is said to be the second differential of displacement with respect to time.

  The sign of the gradient

  In

  Fig. A5.1a

  , the average velocity between time t1 and t2 will be positive because s2 is greater than s1 and therefore subtracting s1 from s2 will produce a positive result. The gradient of the line between A and B is a positive gradient. Similarly, the gradient of the tangent to the curve in

  Fig. A5.1b

  is positive.

  Consider now

  Fig. A5.2

  . Here s2 is less than s1 therefore subtracting s1 from s2 will produce a negative gradient and the velocities will also be negative. 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

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  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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