# DESCRIPTION # Problem from Calculus, multi-variable, Hughes-Hallett et al., # originally from 5ed (with updates) # WeBWorK problem written by Gavin LaRose, # ENDDESCRIPTION ## KEYWORDS('acceleration', 'velocity', 'vector function', 'arc length ') ## Tagged by glr 06/07/10 ## DBsubject('Calculus') ## DBchapter('Vector Functions') ## DBsection('Motion in Space: Velocity and Acceleration') ## Date('') ## Author('Gavin LaRose') ## Institution('University of Michigan') ## TitleText1('Calculus') ## EditionText1('5') ## AuthorText1('Hughes-Hallett') ## Section1('17.2') ## Problem1('35') ## Textbook tags ## HHChapter1('Parameterization and Vector Fields') ## HHSection1('Motion, Velocity and Acceleration') DOCUMENT(); loadMacros( "PGstandard.pl", "PGchoicemacros.pl", "MathObjects.pl", "parserPopUp.pl", # "parserNumberWithUnits.pl", # "parserFormulaWithUnits.pl", # "parserFormulaUpToConstant.pl", # "PGcourse.pl", ); Context("Interval"); Context()->strings->add( none=>{} ); $interv = List( String("none") ); Context("Vector"); Context()->flags->set( ijk=>1 ); Context()->constants->set( i => {TeX => "\,\mathit{\vec i}"}, j => {TeX => "\,\mathit{\vec j}"}, k => {TeX => "\,\mathit{\vec k}"} ); Context()->variables->are( t=>'Real' );$showPartialCorrectAnswers = 1; $zv = random(2,4,1);$z1 = random(6,20,2); $updown = PopUp( [ "?", "yes", "no" ], "no" );$timeToTop = Compute( "$z1/$zv" ); $velocity = Vector( Compute("-sin($z1/$zv)"), Compute("cos($z1/$zv)"), Compute("$zv") ); $xt = Compute( "cos($z1/$zv) - sin($z1/$zv)*(t-$z1/$zv)" );$yt = Compute( "sin($z1/$zv) + cos($z1/$zv)*(t-$z1/$zv)" ); $zt = Compute( "$zv*$z1/$zv + $zv*(t-$z1/$zv)" ); Context()->texStrings; TEXT(beginproblem()); BEGIN_TEXT Suppose $$\vec{r}(t)=\cos t\,\vec i + \sin t\, \vec j + zv t\,\vec k$$ represents the position of a particle on a helix, where $$z$$ is the height of the particle above the ground.$PAR ${BBOLD}(a)$EBOLD Is the particle ever moving downward? \{ $updown->menu() \}$BR If the particle is moving downward, when is this? When $$t$$ is in \{ ans_rule(35) \} $BR${BITALIC}(Enter ${BBOLD}none$EBOLD if it is never moving downward; otherwise, enter an interval or comma-separated list of intervals, e.g., ${BBOLD}(0,3], [4,5]$EBOLD.)$EITALIC$PAR ${BBOLD}(b)$EBOLD When does the particle reach a point $z1 units above the ground?$BR When $$t =$$ \{ ans_rule(35) \} $PAR${BBOLD}(c)$EBOLD What is the velocity of the particle when it is$z1 units above the ground? $BR $$\vec v =$$ \{ ans_rule(45) \}$PAR ${BBOLD}(d)$EBOLD When it is $z1 units above the ground, the particle leaves the helix and moves along the tangent. Find parametric equations for this tangent line (pick $$t$$ so that it is continuous through the time when the particle leaves the helix).$BR $$x(t) =$$ \{ ans_rule(35) \}, $BR $$y(t) =$$ \{ ans_rule(35) \},$BR $$z(t) =$$ \{ ans_rule(35) \} END_TEXT Context()->normalStrings; ANS( $updown->cmp() ); ANS($interv->cmp() ); ANS( $timeToTop->cmp() ); ANS($velocity->cmp() ); ANS( $xt->cmp() ); ANS($yt->cmp() ); ANS( $zt->cmp() ); ($tn,$td) = reduce($z1, $zv );$tshow = ( $td == 1 ) ?$tn : "\frac{$tn}{$td}"; ($tn,$td) = reduce( 2*$tn,$td ); $twotshow = ($td == 1 ) ? $tn : "\frac{$tn}{$td}"; Context()->texStrings; SOLUTION(EV3(<<'END_SOLUTION'));$PAR SOLUTION $PAR${BBOLD}(a)$EBOLD No. The height of the particle is given by $$zv t$$; the vertical velocity is the derivative $$d(zv t)/dt=zv$$. Because this is a positive constant, the vertical component of the velocity vector is upward at a constant speed of $$zv$$.$PAR ${BBOLD}(b)$EBOLD When $$zv t = z1$$, so $$t=tshow$$. $PAR${BBOLD}(c)$EBOLD The velocity vector is given by $\vec{v}(t) = \frac{d\vec r}{dt} = \frac{dx}{dt}\vec i + \frac{dy}{dt}\vec j + \frac{dz}{dt}\vec k,$ or $\vec v(t)= -\sin(t)\vec i + \cos(t)\vec j + zv \vec k.$ From (b), the particle is at$z1 units above the ground when $$t=tshow$$, so at $$t=tshow$$, $\vec{v}(tshow) = -\sin(tshow)\vec i + \cos(tsho)\vec j + zv\vec k.$ $PAR${BBOLD}(d)\$EBOLD At this point, $$t=tshow$$, the particle is located at $\vec r(tshow)=(\cos(tshow), \sin(tshow), z1).$ The tangent vector to the helix at this point is given by the velocity vector found in part (c), so, the equation of the tangent line is $\vec{r_{tan}}(t) = (\cos(tshow) - \sin(tshow)\,(t - tshow)\vec i + (\sin(tshow) + \cos(tshow)\,(t - tshow)\vec j + (z1 + zv\,(t - tshow))\vec k.$ END_SOLUTION Context()->normalStrings; ENDDOCUMENT();