Variable Acceleration

3 Key excerpts on "Variable Acceleration"

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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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  Infinite Powers

  The Story of Calculus - The Language of the Universe

  • Steven Strogatz(Author)
  • 2019(Publication Date)
  • Atlantic Books

    (Publisher)

  ROM A TWENTY-FIRST-CENTURY vantage point, calculus is often seen as the mathematics of change. It quantifies change using two big concepts: derivatives and integrals. Derivatives model rates of change and are the main topic of this chapter. Integrals model the accumulation of change and will be discussed in chapters 7 and 8 .

  Derivatives answer questions like “How fast?” “How steep?” and “How sensitive?” These are all questions about rates of change in one form or another. A rate of change means a change in a dependent variable divided by a change in an independent variable. In symbols, a rate of change always takes the form Δy /Δx , a change in y divided by a change in x . Sometimes other letters are used, but the structure is the same. For example, when time is the independent variable, it’s customary and clearer to write the rate of change as Δy /Δt , where t denotes time.

  The most familiar example of a rate is a speed . When we say a car is going 100 kilometers an hour, that number qualifies as a rate of change because it defines speed as a Δy /Δt when it states how far the car goes (Δy = 100 kilometers) in a given amount of time (Δt = 1 hour).

  Likewise, acceleration is a rate. It’s defined as the rate of change of speed, usually written Δv /Δt , where v stands for velocity. When the American car manufacturer Chevrolet claims that one of its muscle cars, the V-8 Camaro SS, can go from 0 to 60 miles per hour in 4 seconds flat, they’re quoting acceleration as a rate: a change in speed (from 0 to 60 miles per hour) divided by a change in time (4 seconds).

  The slope of a ramp is a third example of a rate of change. It’s defined as the ramp’s vertical rise Δy divided by its horizontal run Δx . A steep ramp has a large slope. A wheelchair-accessible ramp is required by US law to have a slope less than 1 /12

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  eBook - ePub

  Doing Physics with Scientific Notebook

  A Problem Solving Approach

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

    (Publisher)

  y = 0 m. Create an expression with the appropriate values and Solve it for the time.

  Discard the negative solution (it actually tells you how much earlier the ball would have to be thrown from the ground to follow the same trajectory and land at the same time) so the ball hits the ground 5.582 7 s after it was thrown.

  The ball’s velocity as a function of time is given by Eq. (2.6d) . Create an expression with the appropriate values and Evaluate it.

  Since the velocity is negative, the ball is moving down at a speed of 32.748 m/ s.

  Varying Acceleration

  If the acceleration is not constant, then we have to use the techniques of calculus to solve the equations of motion and describe the motion. While the mathematics are more complicated, this lets us solve more realistic problems by including the effects of air resistance on falling objects or the dependence of gravity on height.

  Displacement, Velocity, and Acceleration

  Let’s start with the same three basic kinematic definitions, now using calculus.

  Displacement

  During an infinitesimal time interval dt, an object’s change of position is dx. The displacement is the net change in position.

  Velocity

  The instantaneous velocity is the rate of change of the position. This is what your speedometer reads, the velocity at this instant. To find the instantaneous velocity, take the derivative of the position with respect to time.

  (2.7)

  If you know the position as a function of time, then you differentiate to find the velocity as a function of time.

  Example 2.9

  Engage!

  As the starship Enterprise goes into warp from rest, its position is given by

  where dw is the ship’s displacement while it is going to warp. How fast is the Enterprise moving when it enters warp at time tw

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