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onclick="return false;"><i class="fa fa-star-o"></i></a> </h1> <ol class="breadcrumb"> <li><a href="../../../mathsanalysis.html"><i class="fa fa-home"></i></a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><a href="../540/geometry-trigonometry.html">Geometry & Trigonometry</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">Kinematics</span></li> <span class="pull-right" style="color: #555" title="Suggested study time: 30 minutes"><i class="fa fa-clock-o"></i> 30'</span> </ol> <article id="main-article"> <p><img alt="" src="../../files/vectors/kinematics/main.jpg" style="float: left; width: 100px; height: 100px;"></p> <p>This page is all about kinematics for vectors. You can describe the position of a <strong><em>moving </em></strong> object using vectors. This is very much like the equation of a straight line except that the parameter is <strong><em>t</em></strong><strong><em> </em></strong>for time. The equation below is an example: the objects initial position is (2,3) and the velocity is given as the vector <span class="math-tex">\(\left( \begin{matrix} -1 \\ 4 \end{matrix} \right) \)</span>.</p> <p style="text-align: center;"><span class="math-tex">\(\textbf{r}=\left( \begin{matrix} 2 \\ 3 \end{matrix} \right) +t\left( \begin{matrix} -1 \\ 4 \end{matrix} \right) \)</span></p> <hr class="hidden-separator"> <div class="panel panel-turquoise panel-has-colored-body"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Key Concepts</p> </div> </div> <div class="panel-body"> <div> <p>On this page, you should learn about</p> <ul> <li>applications of vector lines to kinematics</li> <li>speed and velocity</li> <li>collisions</li> </ul> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-yellow panel-has-colored-body"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Essentials</p> </div> </div> <div class="panel-body"> <p>The following videos will help you understand all the concepts from this page</p> <div class="panel panel-yellow panel-has-colored-body panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Torpedo Challenge</p> </div> </div> <div class="panel-body"> <p>This game will help you get a good understand kinematics with vectors. The parameter <strong><em>t</em></strong> is controlled by the slider</p> <ul> <li>The aim of the game is to fire a torpedo to sink the submarine.</li> <li>The submarine is initially positioned at (1,1) and you are at (3, 0)</li> <li>Once the submarine is launched, it takes 1 second to calculate and enter the direction of your torpedo shot.</li> <li>What direction should you fire your torpedo? Press play to see what happens.</li> <li>Change the direction of the torpedo by entering different numbers in red.</li> <li>Can you make them collide after 1 second, 2 seconds, 3 seconds?</li> <li>How do you make the bomb travel faster?</li> <li>You can recreate a different situation if you click ‘change the submarine’.</li> </ul> <p style="text-align: center;"><iframe height="581px" scrolling="no" src="https://www.geogebra.org/material/iframe/id/tYFAwdfy/width/709/height/581/border/888888/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/false/rc/false/ld/false/sdz/false/ctl/false" style="border:0px;" title="Torpedo Challenge" width="709px"></iframe></p> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-orange panel-has-colored-body panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Velocity Vectors</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="322"> <p>In the following video we are going to look we will try and gain a conceptual understanding of velocity vectors. One of the key ideas of this topic is to decide if objects collide. It is not enough to consider if their paths cross. We need to think about whether they occupy the <strong>same position</strong> at the <strong>same moment in time</strong>.</p> <hr class="hidden-separator"> <p>To get you started, you might like to play <a href="http://www.geogebra.org/m/SkeytGs5" target="_blank">this game</a> to give you an idea about what is going on. Try to hit the submarine with the torpedo!</p> <hr class="hidden-separator"> <p>Now let's consider the example below:</p> <p><em>A submarine is initially positioned at (0, 5) travels with velocity <span class="math-tex">\(\left( \begin{matrix} 4 \\ -3 \end{matrix} \right) \\ \)</span>ms-1 .</em></p> <p><em>One second later a torpedo is fired from (3, 0) with velocity <span class="math-tex">\(\left( \begin{matrix} 5 \\ 1 \end{matrix} \right) \\ \)</span>ms-1 .</em></p> <p><em>Does the torpedo manage to shoot the submarine?</em></p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/264948030"></iframe></div> <h4><span></span><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span><span></span> Notes from the video</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/vectors/lines/velocity-vectors.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/lines/velocity-vectors.pdf" width="640"></iframe></p> </section> </div> <p> </p> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> <p> </p> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-violet"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Summary</p> </div> </div> <div class="panel-body"> <div> <p><iframe align="middle" frameborder="1" height="480" scrolling="yes" src="../../files/vectors/kinematics/kinematics_revision-notes.pdf" width="640"></iframe></p> <p>Print from <a href="../../files/vectors/kinematics/kinematics_revision-notes.pdf" target="_blank">here</a></p> </div> </div> <div class="panel-footer"> <div> <p>text</p> </div> </div> </div> <div class="panel panel-has-colored-body panel-green"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Test Yourself</p> </div> </div> <div class="panel-body"> <p>Here is a quiz that practises the skills from this page</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#67aef82d"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="67aef82d"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> Velocity Vectors <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="11-184-669" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>The position of an object is given by <span class="math-tex">\(\textbf{r}=\left( \begin{matrix} 2 \\ 3 \end{matrix} \right) +t\left( \begin{matrix} -3 \\ 4 \end{matrix} \right) \)</span></p><p>What is the velocity?</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\left( \begin{matrix} 2 \\ 3 \end{matrix} \right) \)</span></label></p><p><label class="radio"><input type="radio"> 5</label></p><p><label class="radio"><input type="radio"> -5</label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\left( \begin{matrix} -3 \\ 4 \end{matrix} \right) \)</span></label><label class="radio"><span class="radio"><span class="radio"></span></span></label><label class="radio"></label><label class="radio"> </label></p></div><div class="q-explanation"><p>Velocity is a vector</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><em>Distances are given in metres and time in seconds</em></p><p>The position of an object is given by <span class="math-tex">\(\textbf{r}=\left( \begin{matrix} -4 \\ 3 \end{matrix} \right) +t\left( \begin{matrix} 5 \\ -12 \end{matrix} \right) \)</span></p><p>What is the speed?</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>5ms<sup>-1</sup></span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>13ms<sup>-1</sup></span></label> </p><p><label class="radio"> <input type="radio"> <span>-5ms<sup>-1</sup></span></label> </p><p><label class="radio"> <input type="radio"> <span>-13ms<sup>-1</sup></span></label> </p></div><div class="q-explanation"><p>Use Pythagoras' Theorem on velocity vector</p><p>speed = <span class="math-tex">\(\sqrt { ({ 5 }^{ 2 }+{ (-12) }^{ 2 } } \)</span>=13ms<sup>-1</sup></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The object is travelling at 7ms<sup>-1</sup> . What is the value of <strong><em>a </em></strong>if <strong><em>a</em></strong>>0</p><p><span class="math-tex">\(\textbf{r}=\left( \begin{matrix} 1\\-2 \\ 1 \end{matrix} \right) +t\left( \begin{matrix} 2 \\ -3\\a \end{matrix} \right) \)</span></p></div><div class="q-answer"><p>a= <input type="text" style="height: auto;" data-c="6"> <span class="review"></span></p></div><div class="q-explanation"><p>speed = <span class="math-tex">\(\sqrt { { 2 }^{ 2 }+{ (-3) }^{ 2 } +{a}^{2}} \)</span></p><p><span class="math-tex">\(\sqrt { 13 +{a}^{2}} =7\)</span></p><p>13 + a<sup>2</sup> = 49</p><p>a<sup>2</sup> = 36</p><p>a = 6</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Which of the following objects can be described with position vector</p><p><span class="math-tex">\(\textbf{r}=\left( \begin{matrix} 2 \\ 1 \end{matrix} \right) +t\left( \begin{matrix} 1 \\ 2 \end{matrix} \right) \)</span></p><p><img alt="" class="gifffer" data-gifffer="/media/mathsanalysis/files/vectors/kinematics/quiz4.gif" style="width: 468px; height: 568px;"></p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>D</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>B</span></label> </p><p><label class="radio"> <input type="radio"> <span>C</span></label> </p><p><label class="radio"> <input type="radio"> <span>A</span></label> </p></div><div class="q-explanation"><p>When t = 0, objects is at (2,1)</p><p>Velocity = <span class="math-tex">\(\left( \begin{matrix} 1 \\ 2 \end{matrix} \right) \)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The initial position of the object below is (<strong>a</strong>,<strong>b</strong>)</p><p><strong>r </strong>= 3<strong><em>i</em></strong> + 6<strong><em>j</em></strong> + t(2<strong><em>i</em></strong> - <strong><em>j</em></strong>)</p><p>Find <strong><em>a </em></strong> and <strong><em>b</em></strong></p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p><p>b = <input type="text" style="height: auto;" data-c="6"> <span class="review"></span></p></div><div class="q-explanation"></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Find <strong><em>a </em></strong>that makes the paths of the two objects parallel</p><p><span class="math-tex">\(\textbf{r}_{A}=\left( \begin{matrix} 0\\-2 \\ 2 \end{matrix} \right) +t\left( \begin{matrix} a \\ 3\\-1 \end{matrix} \right) \)</span> , <span class="math-tex">\(\textbf{r}_{B}=\left( \begin{matrix} 3\\1 \\ 5 \end{matrix} \right) +t\left( \begin{matrix} 6 \\ -9\\3 \end{matrix} \right) \)</span></p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="-2"> <span class="review"></span></p></div><div class="q-explanation"><p>For objects must be parallel <span class="math-tex">\(\left( \begin{matrix} a \\ 3\\-1 \end{matrix} \right)=k\left( \begin{matrix} 6 \\ -9\\3 \end{matrix} \right)\\
k=-3\\\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The two objects are travelling in perpendicular directions</p><p><strong>r<sub>A</sub></strong> = 3<strong><em>i</em></strong> - <strong><em>j</em></strong> +t(a<strong><em>i</em></strong> + 6<strong><em>j</em></strong>)</p><p><strong>r<sub>B</sub></strong> = 2<strong><em>i</em></strong> - 3<strong><em>j</em></strong> +t(3<strong><em>i</em></strong> - 2<strong><em>j</em></strong>)</p><p>Find the value of <strong><em>a</em></strong></p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="4"> <span class="review"></span></p></div><div class="q-explanation"><p>Scalar product of the velocities=0</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The two objects below pass through the point (2,3). The second one passes the point (2,3) <strong><em>a</em></strong> seconds after the first one.</p><p><span class="math-tex">\(\textbf{r}_{A}=\left( \begin{matrix} 1 \\2 \end{matrix} \right) +t\left( \begin{matrix} 1 \\ 1 \end{matrix} \right) \)</span> , <span class="math-tex">\(\textbf{r}_{B}=\left( \begin{matrix} 12 \\-2 \end{matrix} \right) +t\left( \begin{matrix} -2 \\ 1 \end{matrix} \right) \)</span> </p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="4"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\left( \begin{matrix} 2 \\ 3 \end{matrix} \right) =\left( \begin{matrix} 1 \\2 \end{matrix} \right) +1\left( \begin{matrix} 1 \\ 1 \end{matrix} \right) =\left( \begin{matrix} 12 \\-2 \end{matrix} \right) +5\left( \begin{matrix} -2 \\ 1 \end{matrix} \right) \)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>A toy airplane sets off from ground level. Its position can be given by</p><p><span class="math-tex">\(\textbf{r}=\left( \begin{matrix} 10\\-8 \\ 0 \end{matrix} \right) +t\left( \begin{matrix} 3 \\ -4\\2 \end{matrix} \right) \)</span></p><p><em>where distance is measured in metres and time in seconds.</em></p><p>How many seconds before it reaches 15m</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="7.5"> <span class="review"></span></p></div><div class="q-explanation"><p>The z coordinate gives height</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div> </div> </div> <div class="modal-footer slide-quiz-actions"> <div class=""> <div class="pull-left pull-xs-none mb-xs-3"> <button class="btn btn-default d-xs-none btn-prev"> <i class="fa fa-arrow-left"></i> Prev </button> </div> <div class="pull-right pull-xs-none"> <button class="btn btn-success btn-xs-block text-xs-center btn-results" style="display: none"> <i class="fa fa-bar-chart"></i> Check Results </button> <button class="btn btn-default d-xs-none btn-next"> Next <i class="fa fa-arrow-right"></i> </button> <button class="btn btn-default btn-xs-block text-xs-center btn-close" data-dismiss="modal" style="display: none"> Close </button> </div> </div> </div> </div> </div></div> </div> <div class="panel-footer"> <div> <p>text</p> </div> </div> </div> <div class="panel panel-has-colored-body panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Exam-style Questions</p> </div> </div> <div class="panel-body"> <div> <div class="panel panel-has-colored-body panel-default panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 1</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="352"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"></p> <p><img alt="" src="../../files/vectors/kinematics/nearmiss.jpg" style="width: 400px; height: 209px;"></p> <p>During an air show, two planes, A and B, perform a manoeuvre in which their paths cross in a <em>near miss</em>. The two planes are flying at the same altitude.</p> <p><span class="math-tex">\(\textbf{ r }_{ A }=\left( \begin{matrix} 150 \\ 320 \end{matrix} \right) +t\left( \begin{matrix} 200 \\ 300 \end{matrix} \right) \)</span></p> <p><span class="math-tex">\(\textbf{ r }_{ B }=\left( \begin{matrix} 875 \\ 110 \end{matrix} \right) +t\left( \begin{matrix} -100 \\ 400 \end{matrix} \right) \)</span></p> <p>t = time in seconds. Distances are given in metres.</p> <p>a) Show that the two planes cross paths, but the planes do not collide</p> <p>b) Find the distance between the planes when t = 0.</p> <p>c) Show that the distance <strong>d </strong>between A and B at any time t can be given by the expression</p> <p style="margin-left: 40px;">d = <span class="math-tex">\(\sqrt { 100000{ t }^{ 2 }-477000t+569725 } \)</span></p> <p>d) To the nearest metre, find the closest distance that the two planes get to one another.</p> <h4> </h4> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><img alt="" class="gifffer" data-gifffer="/media/mathsanalysis/files/vectors/kinematics/closest_distance.gif" style="width: 625px; height: 445px;"></p> <p>a) To decide if the planes collide, find time, t<sub>1</sub> when x<sub>A</sub>=x<sub>B</sub> . Then find time, t<sub>2</sub> when y<sub>A</sub>=y<sub>B</sub>. Show t<sub>1</sub> <span class="math-tex">\(\neq \)</span> t<sub>2</sub></p> <p>b) Use Pythagoras' Theorem to find the distance between the points</p> <p>d) Use you calculator!</p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/vectors/kinematics/esq_vectors_kinematics1hl.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/kinematics/esq_vectors_kinematics1hl.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 2</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="353"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"></p> <p><em>Distances in this question are given in metres</em></p> <p>Two brothers, Orville and Wilbur are testing their model airplanes. The position of Orville's airplane <strong><em>t</em></strong> seconds after taking off from ground level is given by</p> <p><span class="math-tex">\(\textbf{ r }=\left( \begin{matrix} 12 \\ -19\\0 \end{matrix} \right) +t\left( \begin{matrix} -4 \\ 4\\3 \end{matrix} \right) \)</span></p> <p>a) Find the height of the plane after 4 seconds.</p> <p>b) Wilbur's airplane takes off <strong>after</strong> Orville's airplane <strong><em>s</em></strong> seconds after taking off is given by</p> <p><span class="math-tex">\(\textbf{ r }=\left( \begin{matrix} -26 \\ 25\\0 \end{matrix} \right) +s\left( \begin{matrix} 2 \\ -4\\8 \end{matrix} \right) \)</span></p> <p>Find the angle between the two paths.</p> <p>c) The two airplanes collide at (-20,13,24). How long after Orville’s airplane takes off does Wilbur’s airplane take off?</p> <p>d) Find the speed of the two airplanes at the moment of the collision.</p> <h4> </h4> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden">d) the speed of the two airplanes does not change. Use the velocity vectors.</section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/vectors/kinematics/esq_vectors_kinematics2hl.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/vectors/kinematics/esq_vectors_kinematics2hl.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 3</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="350"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p><em>All distances in this question are in km.</em></p> <p>An interceptor missile, M<sub>1 </sub>is positioned at the origin. A missile, M<sub>2</sub> is launched from (-20,7) with velocity <span class="math-tex">\(\left( \begin{matrix} 3 \\ 1 \end{matrix} \right) \)</span> kms<sup>-1</sup> . M<sub>1</sub> is capable of twice the speed of M<sub>2</sub> . How many seconds later must the interceptor missile, M<sub>1</sub> be launched if it is to travel the <strong>shortest possible distance</strong>?</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden">To travel the shortest possible distance, M<sub>1</sub> must travel in a direction perpendicular to M<sub>2</sub> .</section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/vectors/kinematics/esq_vectors_kinematics3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" 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