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| <section class="Post" data-answers="1" data-icon="correct"> | ||
| <h2 class="Collapse">Remember?<span data-print="PreTest"></span></h2><div class="Collapse"> | ||
| <div id="PreTest"><ol> | ||
| <li>A ball is launched from the ground with an initial velocity of 36.0 m/s directed at a 52.0° angle above horizontal. Determine the “hang time” and the “range” of the ball.</li> | ||
| </ol></div> | ||
| </div></section> | ||
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| <section class="Post" data-icon="slides"> | ||
| <h2 class="Collapse">Lesson Notes<span data-print="LessonNotes"></span></h2><div class="Collapse"> | ||
| <div id="LessonNotes"> | ||
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| <section class="Slide Center"> | ||
| <h1 id="Title">Motion in a Uniform Electric Field</h1> | ||
| </section> | ||
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| <section class="Slide"><h3>Projecile Motion</h3> | ||
| <p><em class="Defn">Projectile motion</em> is a type of 2D motion where the force (and therefore acceleration) remains constant.</p> | ||
| <ul data-cue="+"> | ||
| <li>We studied projectile motion in Physics 20, where the constant acceleration is caused by gravity.</li> | ||
| <li data-cue="+">The motion in the <span class="TeX">y</span>-direction (<span class="TeX">{\vec{\bf a}}_y = \vec{\bf g}</span>) is <em>uniform accelerated motion</em>.</li> | ||
| <li data-cue="+">The motion in the <span class="TeX">x</span>-direction (<span class="TeX">{\vec{\bf a}}_x = 0</span>) is <em>uniform motion</em>.</li> | ||
| </ul> | ||
| <div data-cue="+"> | ||
| <p>The equations for uniform accelerated motion / projectile motion are...</p> | ||
| <p class="TeX">{\vec{\bf a}} = {{{\vec{\bf v}}_f - {\vec{\bf v}}_i}\over\Delta t}</p> | ||
| <p class="TeX">\Delta{\vec{\bf d}} = {{{\vec{\bf v}}_i + {\vec{\bf v}}_f}\over 2}\Delta t</p> | ||
| <p class="TeX">{v_f}^2 = {v_i}^2 + 2{\vec{\bf a}}\cdot{\Delta\vec{\bf d}}</p> | ||
| <p class="TeX">{\Delta\vec{\bf d}} = {\vec{\bf v}}_i\Delta t + {1\over 2}{\vec{\bf a}}\left(\Delta t\right)^2</p> | ||
| <!--p><p class="TeX">{\Delta\vec{\bf d}} = {\vec{\bf v}}_f\Delta t - {1\over 2}{\vec{\bf a}}\left(\Delta t\right)^2</p></p--> | ||
| </div> | ||
| <ul data-cue="+"><li>The shape of the trajectory is a <em class="Defn">parabola</em> because when the acceleration is constant, the relationship between position and time is <em>quadratic</em>, as shown by the last equation above.</li></ul> | ||
| <table data-cue="+" class="Center"> | ||
| <thead> | ||
| <tr><i><th colspan="2"></th><th class="xy"><span class="TeX">x</span></th><th class="xy"><span class="TeX">y</span></th></i></tr> | ||
| </thead> | ||
| <tbody> | ||
| <tr> | ||
| <td><span class="TeX">\vec{\bf a}</span></td> | ||
| <td><span class="TeX">{\rm{m/s^2}}</span></td><td>0.00</td><td>–9.81</td></tr> | ||
| <tr> | ||
| <td><span class="TeX">{\vec{\bf v}}_i</span></td> | ||
| <td><span class="TeX">{\rm{m/s}}</span></td> | ||
| <td rowspan="2"><span class="TeX">36.0\ \cos(52.0°)\\=22.2</span></td><td><span class="TeX">36.0\ \sin(52.0°)\\=28.4</span></td> | ||
| </tr> | ||
| <tr> | ||
| <td><span class="TeX">{\vec{\bf v}}_f</span></td> | ||
| <td><span class="TeX">{\rm{m/s}}</span></td> | ||
| <td class="Reveal"><span>–28.4</span></td> | ||
| </tr> | ||
| <tr> | ||
| <td><span class="TeX">\Delta\vec{\bf d}</span></td> | ||
| <td><span class="TeX">{\rm{m}}</span></td> | ||
| <td class="Reveal"><span>128</span></td> | ||
| <td>0.00</td> | ||
| </tr> | ||
| <tr> | ||
| <td><span class="TeX">\Delta t</span></td> | ||
| <td><span class="TeX">{\rm{s}}</span></td> | ||
| <td class=Reveal colspan="2"><span>5.78</span></td></tr> | ||
| </tbody> | ||
| </table> | ||
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| <ul data-cue="+"> | ||
| <li>Because there is no <span class="TeX">x</span>-acceleration, the initial and final <span class="TeX">x</span>-velocities are the same.</li> | ||
| <li data-cue="+">Time is a <em>scalar</em>; it does not have separate <span class="TeX">x</span>- and <span class="TeX">y</span>-components.</li> | ||
| <li data-cue="+">Choose one of the kinematics equations to find <span class="TeX">{\vec{\bf v}}_{fy}</span>: | ||
| <p class="TeX">{v_f}^2 = {v_i}^2 + 2{\vec{\bf a}}\cdot{\Delta\vec{\bf d}}</p></li> | ||
| <li data-cue="+">Choose a different equation to find <span class="TeX">\Delta t</span>: | ||
| <p class="TeX">{\vec{\bf a}} = {{{\vec{\bf v}}_f - {\vec{\bf v}}_i}\over\Delta t}</p></li> | ||
| <li data-cue="+">Finally, find <span class="TeX">\Delta{\vec{\bf d}}_x</span>: | ||
| <p class="TeX">\Delta\vec{\bf d} = \vec{\bf v}_{avg}\Delta t</p></li> | ||
| </ul> | ||
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| <!--ul data-cue="+"> | ||
| <li>When a charge moves in a uniform electric field, it will have a constant acceleration, but it will not equal <span class="TeX">\vec{\bf g}</span>... | ||
| <p class="TeX">\vec{\bf a} = \frac{{\vec{\bf F}}_e}{m} = \frac{q\vec{\bf E}}{m}</p> | ||
| </li> | ||
| <li data-cue="+">Usually we will take the direction of the acceleration to be the <span class="TeX">-y</span>-direction.</li> | ||
| </ul--> | ||
| </section> | ||
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| <section class="Slide"><h3>Motion in a Uniform Electric Field</h3> | ||
| <p>If the electric field is uniform, a charge movingthrough the field will experience a <em>constant acceleration</em>, so its motion will be parabolic just like for projectile motion.</p> | ||
| <ul data-cue="+"> | ||
| <li>The constant acceleration is calculated using Newton’s 2<sup>nd</sup> Law:<p class="TeX">\vec{\bf a} = {{\vec{\bf F}}_e + {\vec{\bf F}}_g \over m} = {q{\vec{\bf E}} + m{\vec{\bf g}}\over m}</p> | ||
| <li data-cue="+">Typically, we would set up an experiment so that the electric and gravitational fields are both in the <i>y</i>-direction, so that <span class="TeX">{\vec{\bf a}}_x = 0</span>.</li> | ||
| <li data-cue="+">Often (but not always) the electric force is so strong that the gravity term has no effect on the answer to the precision of our calculations.</li> | ||
| </ul> | ||
| </section> | ||
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| <script type="text/javascript" data-init="notes"> | ||
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| </script> | ||
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| </div></div></section> | ||
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| <section class="Post" data-show="2024.9.25.10" data-icon="gdrv"> | ||
| <h2 class="Collapse">Practice & Review</h2><div class="Collapse"> | ||
| <p class="BtnGrid"> | ||
| <button data-icon="gdrv" data-gdrv="187EL4mH70qJ14-cCA1S7vhvh2GDdakVw7B0Kh29XM8I">Answer Key</button> | ||
| </p> | ||
| </div></section> | ||
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| <script type="text/javascript"> | ||
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| title: `${siteData.lesson}11.2 — Motion in a Uniform Electric Field`, | ||
| answerDate: "2024.9.25.10", | ||
| num: "11.2", | ||
| up: "p30/units/B", | ||
| prev: "p30/elec/Efield", | ||
| next: "p30/elec/volt", | ||
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| </script> |
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