# Elementary differential equations 7th edition - Boyce W.E

ISBN 0-471-31999-6

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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

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:

u (t) = — = lim-------------------------------

dt h^0 h

• A coordinate frame is a triple of vectors, denoted by {i, j, k}, 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 = tqi + 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)i + y(t)j + z(t)k If R is differentiable, then

R' (t) = X (t )i + 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 )i + f (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.

Newton’s First Law. A body remains in a state of rest, or in a state of uniform motion in a straight line if there is no net external force acting on it.

But the more interesting situation is when there is a net external force acting on the body.

Newton’s Second Law. For a body with acceleration a and constant mass m,

F = ma

where F is the sum of all external forces acting on the body.

Sometimes it’s easier to visualize Newton’s second law in terms of the x-, y-, and z-components of the position vector R of the moving body. If we project the acceleration vector a = R" and the forces onto the x-, y-, and z-axes, then for a body of mass m,

mx" = the sum of the forces in the x-direction

my1' = the sum of the forces in the y-direction

mZ' = the sum of the forces in the z-direction

We’ll look at motion in a plane with xmeasuring the horizontal distance and y measuring the vertical distance up from the ground. We don’t need the z-axis for our examples because the motion is entirely along a line or in a plane.

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Chapter 5

? Dunk Tank

What is the ball doing if 60 = 90°?

Figure 5.1 was done in the Tool, where Q0 is in radians.

Imagine your favorite professor seated over a dunk tank. Let’s construct a model that will help you find the secret to hitting the target and giving your teacher a swim!

You hurl a ball at the target from a height of 6 ft with speed V0 ft/sec and with a launch angle of Q0 degrees from the horizontal1. The target is centered 10 ft above the ground and 20 ft away. Let’s suppose that air resistance doesn't have much effect on the ball over its short path, so that gravity, acting downwards, is the only force acting on the ball.

Newton's second law says that

mR" = — mgj

where m is the ball’s mass, R(t) is the position of the ball at time t relative to your hand (which is 6 ft above the ground at the instant t = 0 of launch), and g = 32 ft/sec2 is the acceleration due to gravity. In coordinate terms,

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