Wagon collision - Revision history

http://mathpuzzlewiki.com/index.php?action=history&feed=atom&title=Wagon_collision Wagon collision - Revision history 2026-04-04T20:41:14Z Revision history for this page on the wiki MediaWiki 1.44.0 http://mathpuzzlewiki.com/index.php?title=Wagon_collision&diff=1376&oldid=prev Oscarlevin: /* Help */ 2017-07-25T02:19:31Z

<p><span class="autocomment">Help</span></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 20:19, 24 July 2017</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l7">Line 7:</td> <td colspan="2" class="diff-lineno">Line 7:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Help==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Help==</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>{{Solution| Let <m>x</m> denote the distance between City A and a vehicle (so either a wagon or a car) traveling on one of the roads. Similarly, let <m>y</m> denote the distance between City A and the other vehicle traveling on the other road. We can imagine these values together representing points in the unit square <m>I {{=}} \{(x,y) <del style="font-weight: bold; text-decoration: none;">\suchthat </del>0 \leq x,y \leq 1\}</m>. In the case of the cars, both vehicles start at City A and so <m>x{{=}}y{{=}}0</m> initially. As both cars traverse their respective roads to City B a continuous curve is drawn out linking the bottom left corner of <del style="font-weight: bold; text-decoration: none;">$</del>I<del style="font-weight: bold; text-decoration: none;">$ </del>to the top right corner. In the case of the wagons, both vehicles start at different cities, say <m>x{{=}}1</m> and <m>y {{=}} 0</m>. Thus, as both wagons move in opposite directions <m>x</m> is tending towards 0 (to City A) and <m>y</m> is tending towards 1 (to City B), and so a continuous curve is traced out from the bottom right corner to the top left corner. Since both curves link different corners they must intersect at some point in $I$. At this point of intersection each wagon must be at the exact position each car was at on their respective roads. Since the length of the rope attaching both cars was less than 2R and each wagon has radius exactly R both wagons must collide at this point.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>{{Solution| Let <m>x</m> denote the distance between City A and a vehicle (so either a wagon or a car) traveling on one of the roads. Similarly, let <m>y</m> denote the distance between City A and the other vehicle traveling on the other road. We can imagine these values together representing points in the unit square <m>I {{=}} \{(x,y) <ins style="font-weight: bold; text-decoration: none;">: </ins>0 \leq x,y \leq 1\}</m>. In the case of the cars, both vehicles start at City A and so <m>x{{=}}y{{=}}0</m> initially. As both cars traverse their respective roads to City B a continuous curve is drawn out linking the bottom left corner of <ins style="font-weight: bold; text-decoration: none;">\(</ins>I<ins style="font-weight: bold; text-decoration: none;">\) </ins>to the top right corner. In the case of the wagons, both vehicles start at different cities, say <m>x{{=}}1</m> and <m>y {{=}} 0</m>. Thus, as both wagons move in opposite directions <m>x</m> is tending towards 0 (to City A) and <m>y</m> is tending towards 1 (to City B), and so a continuous curve is traced out from the bottom right corner to the top left corner. Since both curves link different corners they must intersect at some point in $I$. At this point of intersection each wagon must be at the exact position each car was at on their respective roads. Since the length of the rope attaching both cars was less than <ins style="font-weight: bold; text-decoration: none;">\(</ins>2R<ins style="font-weight: bold; text-decoration: none;">\) </ins>and each wagon has radius exactly <ins style="font-weight: bold; text-decoration: none;">\(</ins>R<ins style="font-weight: bold; text-decoration: none;">\) </ins>both wagons must collide at this point.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>}}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>}}</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> </table>

Oscarlevin http://mathpuzzlewiki.com/index.php?title=Wagon_collision&diff=1375&oldid=prev Oscarlevin: /* Help */ 2017-07-25T02:18:17Z

<p><span class="autocomment">Help</span></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 20:18, 24 July 2017</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l7">Line 7:</td> <td colspan="2" class="diff-lineno">Line 7:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Help==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Help==</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>{{Solution| Let <m>x</m> denote the distance between City A and a vehicle (so either a wagon or a car) traveling on one of the roads. Similarly, let <m>y</m> denote the distance between City A and the other vehicle traveling on the other road. We can imagine these values together representing points in the unit square <m>I {{=}} \{(x,y) \suchthat 0 \leq x,y \leq 1\}</m>. In the case of the cars, both vehicles start at City A and so <m>x{{=}}y{{=}}0</m> initially. As both cars traverse their respective roads to City B a continuous curve is drawn out linking the bottom left corner of $I$ to the top right corner. In the case of the wagons, both vehicles start at different cities, say <m>x=1</m> and <m>y {{=}} 0</m>. Thus, as both wagons move in opposite directions <m>x</m> is tending towards 0 (to City A) and <m>y</m> is tending towards 1 (to City B), and so a continuous curve is traced out from the bottom right corner to the top left corner. Since both curves link different corners they must intersect at some point in $I$. At this point of intersection each wagon must be at the exact position each car was at on their respective roads. Since the length of the rope attaching both cars was less than 2R and each wagon has radius exactly R both wagons must collide at this point.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>{{Solution| Let <m>x</m> denote the distance between City A and a vehicle (so either a wagon or a car) traveling on one of the roads. Similarly, let <m>y</m> denote the distance between City A and the other vehicle traveling on the other road. We can imagine these values together representing points in the unit square <m>I {{=}} \{(x,y) \suchthat 0 \leq x,y \leq 1\}</m>. In the case of the cars, both vehicles start at City A and so <m>x{{=}}y{{=}}0</m> initially. As both cars traverse their respective roads to City B a continuous curve is drawn out linking the bottom left corner of $I$ to the top right corner. In the case of the wagons, both vehicles start at different cities, say <m>x<ins style="font-weight: bold; text-decoration: none;">{{</ins>=<ins style="font-weight: bold; text-decoration: none;">}}</ins>1</m> and <m>y {{=}} 0</m>. Thus, as both wagons move in opposite directions <m>x</m> is tending towards 0 (to City A) and <m>y</m> is tending towards 1 (to City B), and so a continuous curve is traced out from the bottom right corner to the top left corner. Since both curves link different corners they must intersect at some point in $I$. At this point of intersection each wagon must be at the exact position each car was at on their respective roads. Since the length of the rope attaching both cars was less than 2R and each wagon has radius exactly R both wagons must collide at this point.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>}}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>}}</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> </table>

Oscarlevin http://mathpuzzlewiki.com/index.php?title=Wagon_collision&diff=1374&oldid=prev Oscarlevin: /* Help */ 2017-07-25T02:17:41Z

<p><span class="autocomment">Help</span></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 20:17, 24 July 2017</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l7">Line 7:</td> <td colspan="2" class="diff-lineno">Line 7:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Help==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Help==</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>{{Solution| Let <m>x</m> denote the distance between City A and a vehicle (so either a wagon or a car) traveling on one of the roads. Similarly, let <m>y</m> denote the distance between City A and the other vehicle traveling on the other road. We can imagine these values together representing points in the unit square <m>I {{=}} \{(x,y) <del style="font-weight: bold; text-decoration: none;">{{|}} </del>0 \leq x,y \leq 1\}</m>. In the case of the cars, both vehicles start at City A and so <m>x{{=}}y{{=}}0</m> initially. As both cars traverse their respective roads to City B a continuous curve is drawn out linking the bottom left corner of $I$ to the top right corner. In the case of the wagons, both vehicles start at different cities, say <m>x=1</m> and <m>y {{=}} 0</m>. Thus, as both wagons move in opposite directions <m>x</m> is tending towards 0 (to City A) and <m>y</m> is tending towards 1 (to City B), and so a continuous curve is traced out from the bottom right corner to the top left corner. Since both curves link different corners they must intersect at some point in $I$. At this point of intersection each wagon must be at the exact position each car was at on their respective roads. Since the length of the rope attaching both cars was less than 2R and each wagon has radius exactly R both wagons must collide at this point.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>{{Solution| Let <m>x</m> denote the distance between City A and a vehicle (so either a wagon or a car) traveling on one of the roads. Similarly, let <m>y</m> denote the distance between City A and the other vehicle traveling on the other road. We can imagine these values together representing points in the unit square <m>I {{=}} \{(x,y) <ins style="font-weight: bold; text-decoration: none;">\suchthat </ins>0 \leq x,y \leq 1\}</m>. In the case of the cars, both vehicles start at City A and so <m>x{{=}}y{{=}}0</m> initially. As both cars traverse their respective roads to City B a continuous curve is drawn out linking the bottom left corner of $I$ to the top right corner. In the case of the wagons, both vehicles start at different cities, say <m>x=1</m> and <m>y {{=}} 0</m>. Thus, as both wagons move in opposite directions <m>x</m> is tending towards 0 (to City A) and <m>y</m> is tending towards 1 (to City B), and so a continuous curve is traced out from the bottom right corner to the top left corner. Since both curves link different corners they must intersect at some point in $I$. At this point of intersection each wagon must be at the exact position each car was at on their respective roads. Since the length of the rope attaching both cars was less than 2R and each wagon has radius exactly R both wagons must collide at this point.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>}}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>}}</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> </table>

Oscarlevin http://mathpuzzlewiki.com/index.php?title=Wagon_collision&diff=1373&oldid=prev Oscarlevin: /* Help */ 2017-07-25T02:17:19Z

<p><span class="autocomment">Help</span></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 20:17, 24 July 2017</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l7">Line 7:</td> <td colspan="2" class="diff-lineno">Line 7:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Help==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Help==</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>{{Solution| Let <m>x</m> denote the distance between City A and a vehicle (so either a wagon or a car) traveling on one of the roads. Similarly, let <m>y</m> denote the distance between City A and the other vehicle traveling on the other road. We can imagine these values together representing points in the unit square <m>I {{=}} \{(x,y) <del style="font-weight: bold; text-decoration: none;">: </del>0 \leq x,y \leq 1\}</m>. In the case of the cars, both vehicles start at City A and so <m>x{{=}}y{{=}}0</m> initially. As both cars traverse their respective roads to City B a continuous curve is drawn out linking the bottom left corner of $I$ to the top right corner. In the case of the wagons, both vehicles start at different cities, say <m>x=1</m> and <m>y {{=}} 0</m>. Thus, as both wagons move in opposite directions <m>x</m> is tending towards 0 (to City A) and <m>y</m> is tending towards 1 (to City B), and so a continuous curve is traced out from the bottom right corner to the top left corner. Since both curves link different corners they must intersect at some point in $I$. At this point of intersection each wagon must be at the exact position each car was at on their respective roads. Since the length of the rope attaching both cars was less than 2R and each wagon has radius exactly R both wagons must collide at this point.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>{{Solution| Let <m>x</m> denote the distance between City A and a vehicle (so either a wagon or a car) traveling on one of the roads. Similarly, let <m>y</m> denote the distance between City A and the other vehicle traveling on the other road. We can imagine these values together representing points in the unit square <m>I {{=}} \{(x,y) <ins style="font-weight: bold; text-decoration: none;">{{|}} </ins>0 \leq x,y \leq 1\}</m>. In the case of the cars, both vehicles start at City A and so <m>x{{=}}y{{=}}0</m> initially. As both cars traverse their respective roads to City B a continuous curve is drawn out linking the bottom left corner of $I$ to the top right corner. In the case of the wagons, both vehicles start at different cities, say <m>x=1</m> and <m>y {{=}} 0</m>. Thus, as both wagons move in opposite directions <m>x</m> is tending towards 0 (to City A) and <m>y</m> is tending towards 1 (to City B), and so a continuous curve is traced out from the bottom right corner to the top left corner. Since both curves link different corners they must intersect at some point in $I$. At this point of intersection each wagon must be at the exact position each car was at on their respective roads. Since the length of the rope attaching both cars was less than 2R and each wagon has radius exactly R both wagons must collide at this point.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>}}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>}}</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> </table>

Oscarlevin http://mathpuzzlewiki.com/index.php?title=Wagon_collision&diff=1372&oldid=prev Oscarlevin: /* Help */ 2017-07-25T02:16:53Z

<p><span class="autocomment">Help</span></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 20:16, 24 July 2017</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l7">Line 7:</td> <td colspan="2" class="diff-lineno">Line 7:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Help==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Help==</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>{{Solution| Let <m>x</m> denote the distance between City A and a vehicle (so either a wagon or a car) traveling on one of the roads. Similarly, let <m>y</m> denote the distance between City A and the other vehicle traveling on the other road. We can imagine these values together representing points in the unit square <m>I {{=}} \{(x,y) <del style="font-weight: bold; text-decoration: none;">\| </del>0 \leq x,y \leq 1\}</m>. In the case of the cars, both vehicles start at City A and so <m>x{{=}}y{{=}}0</m> initially. As both cars traverse their respective roads to City B a continuous curve is drawn out linking the bottom left corner of $I$ to the top right corner. In the case of the wagons, both vehicles start at different cities, say <m>x=1</m> and <m>y {{=}} 0</m>. Thus, as both wagons move in opposite directions <m>x</m> is tending towards 0 (to City A) and <m>y</m> is tending towards 1 (to City B), and so a continuous curve is traced out from the bottom right corner to the top left corner. Since both curves link different corners they must intersect at some point in $I$. At this point of intersection each wagon must be at the exact position each car was at on their respective roads. Since the length of the rope attaching both cars was less than 2R and each wagon has radius exactly R both wagons must collide at this point.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>{{Solution| Let <m>x</m> denote the distance between City A and a vehicle (so either a wagon or a car) traveling on one of the roads. Similarly, let <m>y</m> denote the distance between City A and the other vehicle traveling on the other road. We can imagine these values together representing points in the unit square <m>I {{=}} \{(x,y) <ins style="font-weight: bold; text-decoration: none;">: </ins>0 \leq x,y \leq 1\}</m>. In the case of the cars, both vehicles start at City A and so <m>x{{=}}y{{=}}0</m> initially. As both cars traverse their respective roads to City B a continuous curve is drawn out linking the bottom left corner of $I$ to the top right corner. In the case of the wagons, both vehicles start at different cities, say <m>x=1</m> and <m>y {{=}} 0</m>. Thus, as both wagons move in opposite directions <m>x</m> is tending towards 0 (to City A) and <m>y</m> is tending towards 1 (to City B), and so a continuous curve is traced out from the bottom right corner to the top left corner. Since both curves link different corners they must intersect at some point in $I$. At this point of intersection each wagon must be at the exact position each car was at on their respective roads. Since the length of the rope attaching both cars was less than 2R and each wagon has radius exactly R both wagons must collide at this point.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>}}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>}}</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> </table>

Oscarlevin http://mathpuzzlewiki.com/index.php?title=Wagon_collision&diff=1371&oldid=prev Oscarlevin: /* Help */ 2017-07-25T02:16:38Z

<p><span class="autocomment">Help</span></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 20:16, 24 July 2017</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l7">Line 7:</td> <td colspan="2" class="diff-lineno">Line 7:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Help==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Help==</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>{{Solution| Let <m>x</m> denote the distance between City A and a vehicle (so either a wagon or a car) traveling on one of the roads. Similarly, let <m>y</m> denote the distance between City A and the other vehicle traveling on the other road. We can imagine these values together representing points in the unit square <m>I {{=}} \{(x,y)| 0 \leq x,y \leq 1\}</m>. In the case of the cars, both vehicles start at City A and so <m>x{{=}}y{{=}}0</m> initially. As both cars traverse their respective roads to City B a continuous curve is drawn out linking the bottom left corner of $I$ to the top right corner. In the case of the wagons, both vehicles start at different cities, say <m>x=1</m> and <m>y {{=}} 0</m>. Thus, as both wagons move in opposite directions <m>x</m> is tending towards 0 (to City A) and <m>y</m> is tending towards 1 (to City B), and so a continuous curve is traced out from the bottom right corner to the top left corner. Since both curves link different corners they must intersect at some point in $I$. At this point of intersection each wagon must be at the exact position each car was at on their respective roads. Since the length of the rope attaching both cars was less than 2R and each wagon has radius exactly R both wagons must collide at this point.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>{{Solution| Let <m>x</m> denote the distance between City A and a vehicle (so either a wagon or a car) traveling on one of the roads. Similarly, let <m>y</m> denote the distance between City A and the other vehicle traveling on the other road. We can imagine these values together representing points in the unit square <m>I {{=}} \{(x,y) <ins style="font-weight: bold; text-decoration: none;">\</ins>| 0 \leq x,y \leq 1\}</m>. In the case of the cars, both vehicles start at City A and so <m>x{{=}}y{{=}}0</m> initially. As both cars traverse their respective roads to City B a continuous curve is drawn out linking the bottom left corner of $I$ to the top right corner. In the case of the wagons, both vehicles start at different cities, say <m>x=1</m> and <m>y {{=}} 0</m>. Thus, as both wagons move in opposite directions <m>x</m> is tending towards 0 (to City A) and <m>y</m> is tending towards 1 (to City B), and so a continuous curve is traced out from the bottom right corner to the top left corner. Since both curves link different corners they must intersect at some point in $I$. At this point of intersection each wagon must be at the exact position each car was at on their respective roads. Since the length of the rope attaching both cars was less than 2R and each wagon has radius exactly R both wagons must collide at this point.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>}}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>}}</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> </table>

Oscarlevin http://mathpuzzlewiki.com/index.php?title=Wagon_collision&diff=1370&oldid=prev Oscarlevin: /* Help */ 2017-07-25T02:16:13Z

<p><span class="autocomment">Help</span></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 20:16, 24 July 2017</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l7">Line 7:</td> <td colspan="2" class="diff-lineno">Line 7:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Help==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Help==</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>{{Solution| Let <m>x</m> denote the distance between City A and a vehicle (so either a wagon or a car) traveling on one of the roads. Similarly, let <m>y</m> denote the distance between City A and the other vehicle traveling on the other road. We can imagine these values together representing points in the unit square <m>I = \{(x,y)| 0 \leq x,y \leq 1\}</m>. In the case of the cars, both vehicles start at City A and so <m>x{{=}}y{{=}}0</m> initially. As both cars traverse their respective roads to City B a continuous curve is drawn out linking the bottom left corner of $I$ to the top right corner. In the case of the wagons, both vehicles start at different cities, say <m>x=1</m> and <m>y = 0</m>. Thus, as both wagons move in opposite directions <m>x</m> is tending towards 0 (to City A) and <m>y</m> is tending towards 1 (to City B), and so a continuous curve is traced out from the bottom right corner to the top left corner. Since both curves link different corners they must intersect at some point in $I$. At this point of intersection each wagon must be at the exact position each car was at on their respective roads. Since the length of the rope attaching both cars was less than 2R and each wagon has radius exactly R both wagons must collide at this point.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>{{Solution| Let <m>x</m> denote the distance between City A and a vehicle (so either a wagon or a car) traveling on one of the roads. Similarly, let <m>y</m> denote the distance between City A and the other vehicle traveling on the other road. We can imagine these values together representing points in the unit square <m>I <ins style="font-weight: bold; text-decoration: none;">{{</ins>=<ins style="font-weight: bold; text-decoration: none;">}} </ins>\{(x,y)| 0 \leq x,y \leq 1\}</m>. In the case of the cars, both vehicles start at City A and so <m>x{{=}}y{{=}}0</m> initially. As both cars traverse their respective roads to City B a continuous curve is drawn out linking the bottom left corner of $I$ to the top right corner. In the case of the wagons, both vehicles start at different cities, say <m>x=1</m> and <m>y <ins style="font-weight: bold; text-decoration: none;">{{</ins>=<ins style="font-weight: bold; text-decoration: none;">}} </ins>0</m>. Thus, as both wagons move in opposite directions <m>x</m> is tending towards 0 (to City A) and <m>y</m> is tending towards 1 (to City B), and so a continuous curve is traced out from the bottom right corner to the top left corner. Since both curves link different corners they must intersect at some point in $I$. At this point of intersection each wagon must be at the exact position each car was at on their respective roads. Since the length of the rope attaching both cars was less than 2R and each wagon has radius exactly R both wagons must collide at this point.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>}}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>}}</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> </table>

Oscarlevin http://mathpuzzlewiki.com/index.php?title=Wagon_collision&diff=1369&oldid=prev Oscarlevin: /* Help */ 2017-07-25T02:15:45Z

<p><span class="autocomment">Help</span></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 20:15, 24 July 2017</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l7">Line 7:</td> <td colspan="2" class="diff-lineno">Line 7:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Help==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Help==</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>{{Solution| Let <m>x</m> denote the distance between City A and a vehicle (so either a wagon or a car) traveling on one of the roads. Similarly, let <m>y</m> denote the distance between City A and the other vehicle traveling on the other road. We can imagine these values together representing points in the unit square <m>I<del style="font-weight: bold; text-decoration: none;">{{</del>=<del style="font-weight: bold; text-decoration: none;">}}</del>\{(x,y)<del style="font-weight: bold; text-decoration: none;">\ </del>|<del style="font-weight: bold; text-decoration: none;">\ </del>0 \leq x,y \leq 1\}</m>. In the case of the cars, both vehicles start at City A and so <m>x{{=}}y{{=}}0</m> initially. As both cars traverse their respective roads to City B a continuous curve is drawn out linking the bottom left corner of $I$ to the top right corner. In the case of the wagons, both vehicles start at different cities, say <m>x=1</m> and <m>y<del style="font-weight: bold; text-decoration: none;">{{</del>=<del style="font-weight: bold; text-decoration: none;">}}</del>0</m>. Thus, as both wagons move in opposite directions <m>x</m> is tending towards 0 (to City A) and <m>y</m> is tending towards 1 (to City B), and so a continuous curve is traced out from the bottom right corner to the top left corner. Since both curves link different corners they must intersect at some point in $I$. At this point of intersection each wagon must be at the exact position each car was at on their respective roads. Since the length of the rope attaching both cars was less than 2R and each wagon has radius exactly R both wagons must collide at this point.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>{{Solution| Let <m>x</m> denote the distance between City A and a vehicle (so either a wagon or a car) traveling on one of the roads. Similarly, let <m>y</m> denote the distance between City A and the other vehicle traveling on the other road. We can imagine these values together representing points in the unit square <m>I = \{(x,y)| 0 \leq x,y \leq 1\}</m>. In the case of the cars, both vehicles start at City A and so <m>x{{=}}y{{=}}0</m> initially. As both cars traverse their respective roads to City B a continuous curve is drawn out linking the bottom left corner of $I$ to the top right corner. In the case of the wagons, both vehicles start at different cities, say <m>x=1</m> and <m>y = 0</m>. Thus, as both wagons move in opposite directions <m>x</m> is tending towards 0 (to City A) and <m>y</m> is tending towards 1 (to City B), and so a continuous curve is traced out from the bottom right corner to the top left corner. Since both curves link different corners they must intersect at some point in $I$. At this point of intersection each wagon must be at the exact position each car was at on their respective roads. Since the length of the rope attaching both cars was less than 2R and each wagon has radius exactly R both wagons must collide at this point.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>}}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>}}</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> </table>

Oscarlevin http://mathpuzzlewiki.com/index.php?title=Wagon_collision&diff=1112&oldid=prev Oscarlevin at 16:31, 6 July 2013 2013-07-06T16:31:43Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 10:31, 6 July 2013</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l7">Line 7:</td> <td colspan="2" class="diff-lineno">Line 7:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Help==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Help==</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>{{Solution| Let <m>x</m> denote the distance between City A and a vehicle (so either a wagon or a car) traveling on one of the roads. Similarly, let <m>y</m> denote the distance between City A and the other vehicle traveling on the other road. We can imagine these values together representing points in the unit square <m>I{{=}}\{(x,y)\ |\ 0 \leq x,y \leq 1\}<<del style="font-weight: bold; text-decoration: none;">\</del>m>. In the case of the cars, both vehicles start at City A and so <m>x{{=}}y{{=}}0</m> initially. As both cars traverse their respective roads to City B a continuous curve is drawn out linking the bottom left corner of $I$ to the top right corner. In the case of the wagons, both vehicles start at different cities, say <m>x<del style="font-weight: bold; text-decoration: none;">{{</del>=<del style="font-weight: bold; text-decoration: none;">}}</del>1</m> and <m>y{{=}}0</m>. Thus, as both wagons move in opposite directions <m>x</m> is tending towards 0 (to City A) and <m>y</m> is tending towards 1 (to City B), and so a continuous curve is traced out from the bottom right corner to the top left corner. Since both curves link different corners they must intersect at some point in $I$. At this point of intersection each wagon must be at the exact position each car was at on their respective roads. Since the length of the rope attaching both cars was less than 2R and each wagon has radius exactly R both wagons must collide at this point.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>{{Solution| Let <m>x</m> denote the distance between City A and a vehicle (so either a wagon or a car) traveling on one of the roads. Similarly, let <m>y</m> denote the distance between City A and the other vehicle traveling on the other road. We can imagine these values together representing points in the unit square <m>I{{=}}\{(x,y)\ |\ 0 \leq x,y \leq 1\}<<ins style="font-weight: bold; text-decoration: none;">/</ins>m>. In the case of the cars, both vehicles start at City A and so <m>x{{=}}y{{=}}0</m> initially. As both cars traverse their respective roads to City B a continuous curve is drawn out linking the bottom left corner of $I$ to the top right corner. In the case of the wagons, both vehicles start at different cities, say <m>x=1</m> and <m>y{{=}}0</m>. Thus, as both wagons move in opposite directions <m>x</m> is tending towards 0 (to City A) and <m>y</m> is tending towards 1 (to City B), and so a continuous curve is traced out from the bottom right corner to the top left corner. Since both curves link different corners they must intersect at some point in $I$. At this point of intersection each wagon must be at the exact position each car was at on their respective roads. Since the length of the rope attaching both cars was less than 2R and each wagon has radius exactly R both wagons must collide at this point.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>}}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>}}</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> </table>

Oscarlevin http://mathpuzzlewiki.com/index.php?title=Wagon_collision&diff=1111&oldid=prev Oscarlevin at 16:30, 6 July 2013 2013-07-06T16:30:24Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 10:30, 6 July 2013</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l7">Line 7:</td> <td colspan="2" class="diff-lineno">Line 7:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Help==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Help==</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>{{Solution| Let x denote the distance between City A and a vehicle (so either a wagon or a car) traveling on one of the roads. Similarly, let y denote the distance between City A and the other vehicle traveling on the other road. We can imagine these values together representing points in the unit square <del style="font-weight: bold; text-decoration: none;">$</del>I=\{(x,y)\ |\ 0 \leq x,y \leq 1\}<del style="font-weight: bold; text-decoration: none;">$</del>. In the case of the cars, both vehicles start at City A and so <del style="font-weight: bold; text-decoration: none;">$</del>x=y=0<del style="font-weight: bold; text-decoration: none;">$ </del>initially. As both cars traverse their respective roads to City B a continuous curve is drawn out linking the bottom left corner of $I$ to the top right corner. In the case of the wagons, both vehicles start at different cities, say <del style="font-weight: bold; text-decoration: none;">$</del>x=1<del style="font-weight: bold; text-decoration: none;">$ </del>and <del style="font-weight: bold; text-decoration: none;">$</del>y=0<del style="font-weight: bold; text-decoration: none;">$</del>. Thus, as both wagons move in opposite directions x is tending towards 0 (to City A) and y is tending towards 1 (to City B), and so a continuous curve is traced out from the bottom right corner to the top left corner. Since both curves link different corners they must intersect at some point in $I$. At this point of intersection each wagon must be at the exact position each car was at on their respective roads. Since the length of the rope attaching both cars was less than 2R and each wagon has radius exactly R both wagons must collide at this point.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>{{Solution| Let <ins style="font-weight: bold; text-decoration: none;"><m></ins>x<ins style="font-weight: bold; text-decoration: none;"></m> </ins>denote the distance between City A and a vehicle (so either a wagon or a car) traveling on one of the roads. Similarly, let <ins style="font-weight: bold; text-decoration: none;"><m></ins>y<ins style="font-weight: bold; text-decoration: none;"></m> </ins>denote the distance between City A and the other vehicle traveling on the other road. We can imagine these values together representing points in the unit square <ins style="font-weight: bold; text-decoration: none;"><m></ins>I<ins style="font-weight: bold; text-decoration: none;">{{</ins>=<ins style="font-weight: bold; text-decoration: none;">}}</ins>\{(x,y)\ |\ 0 \leq x,y \leq 1\}<ins style="font-weight: bold; text-decoration: none;"><\m></ins>. In the case of the cars, both vehicles start at City A and so <ins style="font-weight: bold; text-decoration: none;"><m></ins>x<ins style="font-weight: bold; text-decoration: none;">{{</ins>=<ins style="font-weight: bold; text-decoration: none;">}}</ins>y<ins style="font-weight: bold; text-decoration: none;">{{</ins>=<ins style="font-weight: bold; text-decoration: none;">}}</ins>0<ins style="font-weight: bold; text-decoration: none;"></m> </ins>initially. As both cars traverse their respective roads to City B a continuous curve is drawn out linking the bottom left corner of $I$ to the top right corner. In the case of the wagons, both vehicles start at different cities, say <ins style="font-weight: bold; text-decoration: none;"><m></ins>x<ins style="font-weight: bold; text-decoration: none;">{{</ins>=<ins style="font-weight: bold; text-decoration: none;">}}</ins>1<ins style="font-weight: bold; text-decoration: none;"></m> </ins>and <ins style="font-weight: bold; text-decoration: none;"><m></ins>y<ins style="font-weight: bold; text-decoration: none;">{{</ins>=<ins style="font-weight: bold; text-decoration: none;">}}</ins>0<ins style="font-weight: bold; text-decoration: none;"></m></ins>. Thus, as both wagons move in opposite directions <ins style="font-weight: bold; text-decoration: none;"><m></ins>x<ins style="font-weight: bold; text-decoration: none;"></m> </ins>is tending towards 0 (to City A) and <ins style="font-weight: bold; text-decoration: none;"><m></ins>y<ins style="font-weight: bold; text-decoration: none;"></m> </ins>is tending towards 1 (to City B), and so a continuous curve is traced out from the bottom right corner to the top left corner. Since both curves link different corners they must intersect at some point in $I$. At this point of intersection each wagon must be at the exact position each car was at on their respective roads. Since the length of the rope attaching both cars was less than 2R and each wagon has radius exactly R both wagons must collide at this point.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>}}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>}}</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> </table>

Oscarlevin