# Elementary Differential Equations and Boundary Value Problems - Boyce W.E.

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How do you interpret this motion? Choose c = 2 and ø0 = 3, and set y(0) = y (0) = 0. Use ODE Architect to find y(t) from ODE (36) for the case T =

2, and display both y(t) and f (t). Note: d2 f/ dt2 can be written using a step function. How do the features of y(t) compare with those of f (t)? For example, what is the maximum magnitude of y(t), and when does it occur?

76 Exploration 4.4

2. Now suppose that the ground motion is given by the function f = e-at sin(^t). Choose some values of a in the range 0 < a < 0.5 and study how Seismo’s arm displacements change with the parameter a.

3. How do you think the results of Problem 2 would change if the period of the sinusoidal oscillation were different from 2? Try a few cases to check your predictions.

5 Models of Motion

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Overview

Key words See also

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Seventeen ski jumpers take off from an upward-tilted ski-jump.

How would you model the motion of a baseball thrown at a target, or the rise and fall of a whiffle ball, or the trajectory of a ski jumper? You need modeling principles to explain the effects of the surroundings on the motion of a body.

Building on the work of Galileo, Newton formulated the fundamental laws of motion that describe the forces acting on a body in terms of the body's acceleration and mass. Newton's second law of motion, for example, relates the mass and the acceleration of a moving body to the forces acting on it and ultimately leads to differential equations for the motion.

Bodies moving through the air near the surface of the earth (e.g., a whiffle ball, Indiana Newton jumping onto a boxcar, or a ski jumper) are subject to the forces of gravity and air resistance, so these forces will affect their motion.

Vectors; force; gravity; Newton's laws; acceleration; trajectory; air resistance; viscous drag; Newtonian drag; lift

Chapter 1 for more on modeling, and Chapter 2 for "The Juggler" and "The Sky Diver".

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Chapter 5

¦ Vectors

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A vector is a directed line segment and can be represented by an arrow with a

head and a tail. We use boldface letters to denote vectors.

Some terminology:

• The length of a vector v is denoted by |v|.

• Two vectors v and w are equivalent if they can be made to coincide by

translations. (Translations preserve length and direction of vectors.) So parallel vectors of equal length and pointing in the same direction are equivalent.

• The sum v + w of v and w is defined by the parallelogram law as follows: v + w is the diagonal vector of the parallelogram formed by v and w as shown in the margin figure.

• If r is any real number, then the product rv is the vector of length |r| |v| that points in the direction of v if r > 0 and in the direction opposite to v if r < 0.

• If a vector u = u(t) depends on a variable t, then the derivative du/dt [or u' (t)] is defined as the limit of a difference quotient:

, du r u(t + h) — u(t)

u'(0 = — = Hm

dt h^o h

• A coordinate frame is a triple of vectors, denoted by {i, j, li}, that are mutually orthogonal and all of unit length. Every vector can be uniquely written as the sum of vectors parallel to i, j, and k. So for each vector v there is a unique set of real numbers v1, v2, and v3 such that v = vii + v2j + v3k. Here v1,v2, and v3 are called the coordinates (or components) of v in the frame {i, j, k}.

Let’s see how to use vectors in a real-life situation. Suppose a particle of mass m moves in a manner described by the position vector

R = R(t) = x(t)j + y(t)j + z(t)k If R is differentiable, then

R' (t) = X (t )j + y/(t )j + z! (t )k

The vector R' ( t) = v(t) is the velocity vector of the particle at time t, and v (t) is tangent to the path of the particle’s motion at the point R(t). If R' (t) is differentiable, then

R''(t) = v' (t) = X'(t )j + / (t )j + z" (t )k The vector R''(t) = a(t) is the acceleration vector for the particle.

Forces and Newton’s Laws

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? “Check” your understanding by answering this question: If a particle moves at a constant speed around a circle, does the acceleration vector from the particle point to the inside of the circle or to the outside of the circle?

? If a particle’s acceleration vector is always tangent to its path, what is the path?

Next, let’s use vectors to express Newton’s laws of motion.

¦ Forces and Newton's Laws

Deceleration is just negative acceleration.

Our environment creates forces that act on bodies in a way that causes the bodies to accelerate or decelerate. Forces have magnitudes and directions and so can be represented by vectors. Newton formulated two laws that describe how the forces on a body relate to its motion.

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