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Unit_8 - GREENBERG VECTOR GEOMETRY UNIT 8 VELOCITY...

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UNIT 8 VELOCITY, ACCELERATION AND PROJECTILES INTRO: In this Unit we will study velocity and acceleration along trajectories, and, in partic- ular, projectile problems. 1. Vector Functions In Units 1 and 2 we studied trajectories given by parametric equations, ~ r ( t ) = < x ( t ) , y ( t ) , z ( t ) > . Since ~ r ( t ) is a function of t which gives a vector at each t value, namely < x ( t ) , y ( t ) , z ( t ) > , we call it a vector function , or more accurately, a vector-valued function . The differentiation and integration of such vector-valued functions is defined and carried out component-wise, so that d dt < x ( t ) , y ( t ) , z ( t ) > = < x 0 ( t ) , y 0 ( t ) , z 0 ( t ) > and Z < x ( t ) , y ( t ) , z ( t ) > dt = < Z x ( t ) dt + c 1 , Z y ( t ) dt + c 2 , Z z ( t ) dt + c 3 > . Note that the indefinite integral of a vector-valued function introduces a different arbitrary constant in each component. One may check that this is valid by differentiating the result, in which case all of the constants disappear, as they should. Because the same trajectory can be parametrized many ways, there is not, in general, any physical significance to the derivative, or rather, to its magnitude. That is to say, the equations ~ q ( t ) = < t, t 2 + 1 , 2 t 3 > and ~ r ( t ) = < t 3 , t 6 + 1 , 2 t 9 > describe precisely the same trajectory, but their derivatives ~ q 0 ( t ) = < 1 , 2 t, 6 t 2 > and ~ r 0 ( t ) = < 3 t 2 , 6 t 5 , 18 t 8 > are quite different. However, when the parameter t is the time, then the derivatives of the trajectory ~ r ( t ) give the velocity ~v ( t ) and the acceleration ~a ( t ) of an object traveling along the curve. Note on Notation : When t is just a parameter, then writing the parametric equations ~ F ( t ) = < f ( t ) , g ( t ) , h ( t ) > is not a problem. However, when the parametric equations are intended to describe a physical trajectory in the x-y plane or in 3-space, and t is the time, then parametric equations for the trajectory such as ~ r ( t ) = < t, sin t > have a problem with units. In fact, if ~ r ( t ) is expected to give the location of an object in the x-y plane at time t , then the components of ~ r ( t ) must have units of length, say feet. But if ~ r ( t ) = < t, sin t > and time is in seconds, the first component of ~ r will have units of seconds, and the second component will make even less sense, since the sine function is unitless, and, moreover, the argument of a trig function must be in radians or degrees, certainly not seconds of time. To get around this, an engineering or science application will generally write this same trajectory as ~ r ( t ) = < at, b sin ωt > and then specify that a is the constant 1 ft/sec, b is the constant 1 ft, and ω is the constant 1 rad/sec, or 1 sec - 1 . For pedagogical reasons, we prefer not to have all these constants in our mathematics VECTOR GEOMETRY 96 GREENBERG
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lessons and quizzes, so we have made the physically illogical decision to allow these violations of unit sense in the next three Units in order to reduce the number of constants in examples and exam questions. Students, however, do need to understand that when a trig function is evaluated at a value of its parameters, the argument of the trig function must be taken in radians. More generally, except when its argument is explicitly stated as degrees, one should always assume the argument of a trig function is in radians.
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