View of /trunk/NationalProblemLibrary/Rochester/setVecFunction3Motion/ur_vc_4_7.pg

    1 ## DESCRIPTION
    2 ## Calculus
    3 ## ENDDESCRIPTION
    4 
    5 ## KEYWORDS('vector' 'acceleration' 'multivariable' 'velocity')
    6 ## Tagged by tda2d
    7 
    8 ## DBsubject('Calculus')
    9 ## DBchapter('Vector Functions')
   10 ## DBsection('Motion in Space: Velocity and Acceleration')
   11 ## Date('')
   12 ## Author('')
   13 ## Institution('Rochester')
   14 ## TitleText1('')
   15 ## EditionText1('')
   16 ## AuthorText1('')
   17 ## Section1('')
   18 ## Problem1('')
   19 
   20 DOCUMENT();        # This should be the first executable line in the problem.
   21 
   22 loadMacros("PG.pl",
   23            "PGbasicmacros.pl",
   24            "PGchoicemacros.pl",
   25            "PGanswermacros.pl",
   26            "PGauxiliaryFunctions.pl");
   27 
   28 TEXT( beginproblem() );
   29 $showPartialCorrectAnswers = 1;
   30 
   31 $a = non_zero_random( -5, 5, 1 );
   32 $b = non_zero_random( -5, 5, 1 );
   33 $asquared = $a**2;
   34 
   35 $ans1 = "- $a * sin( $a * t ) + 1";
   36 $ans2 = "$a * cos( $a * t ) - $a";
   37 $ans3 = "( $b * t**2 ) / 2 + 1";
   38 $ans4 = "cos( $a * t ) + t";
   39 $ans5 = "sin( $a * t ) - $a * t + 1";
   40 $ans6 = "( $b * t**3 ) / 6 + t + 1";
   41 
   42 BEGIN_TEXT
   43 Given that the acceleration vector is \( \mathbf{a} \left( t \right) = \left(
   44 -$asquared \cos \left( $a t \right) \right) \mathbf{i} + \left( -$asquared \sin
   45 \left( $a t \right) \right) \mathbf{j} + \left( $b t \right) \mathbf{k} \), the
   46 initial velocity is \( \mathbf{v} \left( 0 \right) = \mathbf{i + k} \),
   47 and the initial position vector is \( \mathbf{r} \left( 0 \right) = \mathbf{i +
   48 j + k} \), compute:
   49  $PAR
   50 A. The velocity vector \( \mathbf{v} \left( t \right) = \) \{ans_rule(15)\} \(
   51 \mathbf{i} + \) \{ans_rule(15)\} \( \mathbf{j} + \) \{ans_rule(15)\} \(
   52 \mathbf{k} \)
   53  $PAR
   54 B. The position vector \( \mathbf{r} \left( t \right) = \) \{ans_rule(15)\} \(
   55 \mathbf{i} + \) \{ans_rule(15)\} \( \mathbf{j} + \) \{ans_rule(15)\} \( \mathbf{k} \)
   56  $PAR
   57 Note: the coefficients in your answers must be entered in the form of expressions in the variable
   58 \emph{t}; e.g. ${LQ}5 cos(2t)${RQ}
   59 END_TEXT
   60 
   61 ANS(fun_cmp($ans1, vars=>"t"));
   62 ANS(fun_cmp($ans2, vars=>"t"));
   63 ANS(fun_cmp($ans3, vars=>"t"));
   64 ANS(fun_cmp($ans4, vars=>"t"));
   65 ANS(fun_cmp($ans5, vars=>"t"));
   66 ANS(fun_cmp($ans6, vars=>"t"));
   67 
   68 ENDDOCUMENT();        # This should be the last executable line in the problem.