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{SECT 0 {PARA 256 "" 0 "" {TEXT -1 10 "Hot Wheels" }}{PARA 257 "" 0 "
" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 22 "Jim Herod & Pam Poppe
*" }}{PARA 257 "" 0 "" {TEXT -1 12 "Georgia Tech" }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 283 " A ramp is prepared \+
following a trajectory as though the ramp followed along a curve given
by [x(r) , y(r)] from [0 ,0] to [4 ,-4]. A cart is allowed to roll do
wn the ramp. We present an explanation for a formula for the length of
time it takes the cart to roll along this curve." }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 206 " The explanation use
s the assumption that the work done by gravity in pulling the cart dow
n the curve must equal the change in kinetic energy. To use this assum
ption, take the work to be represented by " }{XPPEDIT 18 0 "m*g*Delta*
y" "6#**%\"mG\"\"\"%\"gGF%%&DeltaGF%%\"yGF%" }{TEXT -1 55 " and the c
hange in kinetic energy to be represented by" }}{PARA 0 "" 0 "" {TEXT
-1 62 " \+
" }{XPPEDIT 18 0 "m*v^2/2" "6#*(%\"mG\"\"\"*$%\"vG\"\"#F%F(!\"\"" }
{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 233 "Here, m is the mass o
f the cart, v is the velocity of the cart, g is the force of gravity, \+
and [x(r) , y(r)], 0 < r < E, is the path taken by the cart. (Of cour
se, we have also assumed that the cart moves in a frictionless manner.
) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 " \+
From the asumptions, we have a relationship between the velocity ach
ieved from r = 0:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 38 " " }{XPPEDIT 18 0 "m
*v^2/2=m*g*(y(0)-y(r))" "6#/*(%\"mG\"\"\"*$%\"vG\"\"#F&F)!\"\"*(F%F&%
\"gGF&,&-%\"yG6#\"\"!F&-F/6#%\"rGF*F&" }}{PARA 0 "" 0 "" {TEXT -1 20 "
or, taking y(0) = 0," }}{PARA 0 "" 0 "" {TEXT -1 37 " \+
" }{XPPEDIT 18 0 "v^2= -2*g*y" "6#/*$%\"vG\"\"#,$*
(F&\"\"\"%\"gGF)%\"yGF)!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 3 "or," }}
{PARA 0 "" 0 "" {TEXT -1 37 " " }
{XPPEDIT 18 0 "v = sqrt(-2*g*y)" "6#/%\"vG-%%sqrtG6#,$*(\"\"#\"\"\"%\"
gGF+%\"yGF+!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 3 "or," }}{PARA 0 "" 0 "
" {TEXT -1 36 " " }{XPPEDIT 18 0 "d
s/dt=sqrt(-2*g*y)" "6#/*&%#dsG\"\"\"%#dtG!\"\"-%%sqrtG6#,$*(\"\"#F&%\"
gGF&%\"yGF&F(" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 2 "or" }}{PARA 0 "" 0 "" {TEXT -1 35 " \+
" }{XPPEDIT 18 0 "ds/dr*dr/dt=sqrt(-2*g*y)" "6#/**%#dsG\"
\"\"%#drG!\"\"F'F&%#dtGF(-%%sqrtG6#,$*(\"\"#F&%\"gGF&%\"yGF&F(" }
{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 188 "It follows that the time it takes to roll down the ramp \+
[x(r) , y(r) ] from r = 0 to r = E can be expressed in terms of an int
egral involving the parameters of the curve. We take the curve" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 " \+
" }{XPPEDIT 18 0 "T = int(1,t=0..T)" "6#/%\"T
G-%$intG6$\"\"\"/%\"tG;\"\"!F$" }}{PARA 0 "" 0 "" {TEXT -1 8 " \+
" }}{PARA 0 "" 0 "" {TEXT -1 39 " \+
= " }{XPPEDIT 18 0 "int(diff(s,r)/sqrt(-2*g*y(r)),r=0..E)" "6#-%$intG6
$*&-%%diffG6$%\"sG%\"rG\"\"\"-%%sqrtG6#,$*(\"\"#F,%\"gGF,-%\"yG6#F+F,!
\"\"F7/F+;\"\"!%\"EG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 38 " = " }{XPPEDIT 18 0
"int(sqrt(diff(x,r)^2+diff(y,r)^2)/sqrt(-2*g*y(r)),r=0..E)" "6#-%$intG
6$*&-%%sqrtG6#,&*$-%%diffG6$%\"xG%\"rG\"\"#\"\"\"*$-F-6$%\"yGF0F1F2F2-
F(6#,$*(F1F2%\"gGF2-F66#F0F2!\"\"F>/F0;\"\"!%\"EG" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "Here, we \+
have used that " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 31 " " }{XPPEDIT 18 0 "s(tau)"
"6#-%\"sG6#%$tauG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "int(sqrt(diff(x,r
)^2+diff(y,r)^2),r=0..tau" "6#-%$intG6$-%%sqrtG6#,&*$-%%diffG6$%\"xG%
\"rG\"\"#\"\"\"*$-F,6$%\"yGF/F0F1/F/;\"\"!%$tauG" }}{PARA 0 "" 0 ""
{TEXT -1 8 "and that" }}{PARA 0 "" 0 "" {TEXT -1 38 " \+
v(t) = " }{XPPEDIT 18 0 "ds/dt" "6#*&%#dsG\"\"\"%#dtG!\"
\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT 262 13 "Illustration:" }{TEXT -1 55 " We take the trajectory des
cribed by the circular path " }}{PARA 0 "" 0 "" {TEXT -1 49 " \+
x(r) =4 +4 cos(r + " }{XPPEDIT 18 0 "Pi;" "6#%#Pi
G" }{TEXT -1 0 "" }{TEXT -1 19 "), y(r) = 4 sin(r +" }{XPPEDIT 18 0 "P
i;" "6#%#PiG" }{TEXT -1 0 "" }{TEXT -1 4 " ). " }}{PARA 0 "" 0 ""
{TEXT -1 166 "Using this trajectory, we find the length of time it tak
es to fall from [0 ,0] to [4 , -4]. First, we confirm that this path g
oes between the two points with a graph." }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 44 "plot([4+4*cos(r+Pi),4*sin(r+Pi),r=0..Pi/2]);" }}}
{PARA 0 "" 0 "" {TEXT -1 70 "Now we compute the time it takes for the \+
cart to drop along this ramp." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1
0 39 "x:=r->4+4*cos(r+Pi);\ny:=r->4*sin(r+Pi);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 69 "int(sqrt(diff(x(r),r)^2+diff(y(r),r)^2)/sqrt((
y(0)-y(r))),r=0..Pi/2);" }}}{PARA 0 "" 0 "" {TEXT -1 148 "It may be th
at the above integral is an improper integral and can not be evaluated
analytically, In this case, we evaluate the integral numerically." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }{TEXT -1 67 "The imaginary part of that number comes \+
from making approximations." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
6 "Re(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 11 "Assignment
." }}{PARA 0 "" 0 "" {TEXT -1 43 "I. Draw four curves from [0,0] to [4
, -4]." }}{PARA 0 "" 0 "" {TEXT -1 44 " (a) The first curve is a \+
straight line." }}{PARA 0 "" 0 "" {TEXT -1 59 " (b) The second cur
ve is a quadratic curve, concave up." }}{PARA 0 "" 0 "" {TEXT -1 60 " \+
(c) The third curve is a quadratic curve, concave down." }}{PARA
0 "" 0 "" {TEXT -1 77 " (d) The fourth curve is one of your choosi
ng different from those above." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 79 "II. Compute the length of time it takes t
o move along each of the above curves." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT 256 7 "Remark:" }{TEXT -1 152 " If you have
a function, f(x) whose graph lies in the plane, then the graph can be
drawn parametrically as [x(r), y(r)] where x(r) = r and y(r) = f(r).
" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 16 "HISTORICAL NOTE:" }}{PARA 0 "" 0
"" {TEXT -1 179 " The source for the ideas in this work sheet is a
famous problem in the history of calculus. The problem was posed by J
ohn Bernoulli in 1696 and is often described as follows:" }}{PARA 0 "
" 0 "" {TEXT -1 10 " " }}{PARA 0 "" 0 "" {TEXT -1 30 " \+
Consider two points " }{TEXT 257 1 "P" }{TEXT -1 5 " and " }{TEXT
258 1 "Q" }{TEXT -1 50 " in a plane connected by a thin wire that foll
ows " }}{PARA 0 "" 0 "" {TEXT -1 87 " a continuous curve. A \+
bead is threaded onto the wire and released from rest " }}{PARA 0 ""
0 "" {TEXT -1 19 " at point " }{TEXT 259 1 "P" }{TEXT -1 35 "
, sliding without friction to point" }{TEXT 260 2 " Q" }{TEXT -1 30 ".
What shape should the wire " }}{PARA 0 "" 0 "" {TEXT -1 64 " \+
have to minimize the time for the bead to reach point " }{TEXT 261
1 "Q" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 672 "The deadline for solving the puzzle was Easter, 1697. \+
Reportedly, Newton received the problem in January, 1697 and submitted
the solution the next day, convinced that it was a challenge to him f
rom the Continential mathematicians. The Bernoulli brothers, as well \+
as others, solved the problem, too. The tools for discovering the solu
tion to this problem were established and a part of the interest of ma
thematicians of the time. In a delightful book by George F. Simmons, \+
there is an account of this problem, of Newton's tackling this problem
after a hard day at the Mint, and of the difference in the nature of \+
the solutions by John and James Bernoulli. Simmon's book, " }{TEXT
263 52 "Calculus Gems: Brief Lives and Memorable Mathematics" }{TEXT
-1 52 ", is published by McGraw-Hill. (ISBN: G 07 057566 5)" }}}{PARA
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 204 "* During the p
reparation of this worksheet Dr. Poppe was a participant in the GIFT \+
Program at Georgia Tech. Her usual address is Brookwood High School, G
winnett County Public Schools, with email address " }}{PARA 258 "" 0 "
" {TEXT -1 21 "ppoppe@mindspring.com" }}}{MARK "0 0" 0 }{VIEWOPTS 1 1
0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }