<div id="pf1" class="pf w0 h0" data-page-no="1"><div class="pc pc1 w0 h0"><div class="t m0 x0 h1 y0 ff1 fs0 fc0 sc0 ls4 ws6">1. The initial position vector<span class="_0 blank"> </span><span class="ff2 fs1 v1">G</span></div><div class="t m0 x1 h2 y1 ff3 fs1 fc0 sc0 ls4">r</div><div class="t m0 x2 h3 y2 ff1 fs2 fc0 sc0 ls0">o<span class="fs0 ls4 ws6 v2"> satisfies <span class="_1 blank"> </span><span class="ff2 fs1 v1">G</span></span></div><div class="c x3 y3 w1 h4"><div class="t m0 x4 h5 y4 ff2 fs1 fc0 sc0 ls4">G</div></div><div class="t m0 x5 h5 y5 ff2 fs1 fc0 sc0 ls4">G</div><div class="t m0 x6 h2 y6 ff3 fs1 fc0 sc0 ls4 ws0">r r<span class="_2 blank"> </span>r<span class="_3 blank"></span><span class="ff4 ws1">\u2212 =</span></div><div class="t m0 x7 h6 y2 ff1 fs2 fc0 sc0 ls1">o<span class="ff4 fs1 ls2 v2">\u2206</span><span class="fs0 ls4 ws6 v2">, which results in </span></div><div class="t m0 x8 h7 y7 ff1 fs3 fc0 sc0 ls3">o<span class="fs4 ls4 ws2 v3">\u2022<span class="_4 blank"> </span>\u2022<span class="_5 blank"> </span>\u2022 \u2022<span class="_6 blank"> </span>\u2022<span class="_7 blank"> </span>\u2022<span class="_8 blank"> </span>\u2022<span class="_9 blank"> </span>\u2022</span></div><div class="t m0 x9 h8 y8 ff1 fs4 fc0 sc0 ls4 ws6">(3.0<span class="_a blank"> </span>j<span class="_b blank"> </span>4.0<span class="_c blank"> </span>k)m<span class="_b blank"> </span>(2.0<span class="_c blank"> </span>i<span class="_d blank"> </span>3<span class="_e blank"></span>.0<span class="_a blank"> </span>j<span class="_d blank"> </span>6<span class="_f blank"></span>.0<span class="_a blank"> </span>k)<span class="_c blank"> </span>m<span class="_b blank"> </span>(<span class="_10 blank"> </span>2.0<span class="_c blank"> </span> m<span class="_c blank"> </span>)<span class="_11 blank"> </span>i<span class="_b blank"> </span>(6.<span class="_c blank"> </span>0 m) j<span class="_12 blank"> </span>(<span class="_13 blank"> </span>10 m)<span class="_11 blank"> </span>k<span class="_14 blank"></span><span class="ff3 ws3">r r<span class="_15 blank"> </span>r<span class="_16 blank"></span><span class="ff4 ws4">=<span class="_17 blank"> </span>\u2212 \u2206<span class="_18 blank"> </span>=<span class="_19 blank"> </span>\u2212<span class="_1a blank"> </span>\u2212<span class="_19 blank"> </span>\u2212<span class="_2 blank"> </span>+<span class="_1b blank"> </span>=<span class="_13 blank"> </span>\u2212<span class="_1a blank"> </span>+<span class="_1c blank"> </span>+<span class="_1d blank"> </span>\u2212</span></span></div><div class="t m0 xa h9 y9 ff2 fs4 fc0 sc0 ls4 ws5">G G<span class="_15 blank"> </span>G<span class="_1e blank"> </span><span class="ff1 fs0 v4">.</span></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf2" class="pf w0 h0" data-page-no="2"><div class="pc pc2 w0 h0"><img class="bi xb ya w2 ha" alt="" src="https://files.passeidireto.com/ad0bf699-331c-402e-9de4-dbbf94f7e157/bg2.png"><div class="t m0 x0 hb yb ff1 fs0 fc0 sc0 ls4 ws6">2. (a) The position vector, according to Eq. 4-1, is <span class="_1f blank"> </span><span class="fs5 ws7 v5">\u2022 \u2022</span></div><div class="t m0 xc hc yc ff1 fs5 fc0 sc0 ls4 ws6">= (<span class="_12 blank"> </span>5.0 m) i<span class="_c blank"> </span> + (8.0 m)j<span class="_20 blank"></span><span class="ff3 ls5">r<span class="ff4 ls4 ws8">\u2212</span></span></div><div class="t m0 xd hd yd ff2 fs5 fc0 sc0 ls6">G<span class="ff1 fs0 ls4 v4">.</span></div><div class="t m0 x0 he ye ff1 fs0 fc0 sc0 ls4 ws6">(b) The magnitude is <span class="_21 blank"> </span><span class="fs6 ws9 v6">2<span class="_22 blank"> </span>2 2<span class="_23 blank"> </span>2<span class="_8 blank"> </span>2<span class="_24 blank"> </span>2</span></div><div class="t m0 xe hf yf ff1 fs7 fc0 sc0 ls4 ws6">|<span class="_18 blank"> </span>|<span class="_4 blank"> </span> + <span class="_d blank"> </span> + <span class="_d blank"> </span> <span class="_22 blank"> </span>(<span class="_10 blank"> </span>5.0<span class="_c blank"> </span> m<span class="_c blank"> </span>)<span class="_25 blank"> </span>(8.0<span class="_c blank"> </span> m<span class="_c blank"> </span>)<span class="_25 blank"> </span>(0<span class="_c blank"> </span> m<span class="_c blank"> </span>)<span class="_15 blank"> </span> 9<span class="_c blank"> </span>.4 m.<span class="_26 blank"></span><span class="ff3 wsa">r<span class="_2 blank"> </span>x y z<span class="_27 blank"></span><span class="ff4 wsb">=<span class="_28 blank"> </span>= <span class="ff5 wsc">\u2212<span class="_24 blank"> </span>+<span class="_5 blank"> </span>+ =</span></span></span></div><div class="t m0 xf h10 y10 ff2 fs7 fc0 sc0 ls4">G</div><div class="t m0 x0 h11 y11 ff1 fs0 fc0 sc0 ls4 ws6">(c) <span class="_29 blank"> </span>Many <span class="_29 blank"> </span>calculators <span class="_29 blank"> </span>have <span class="_29 blank"> </span>polar </div><div class="c x10 y12 w3 h12"><div class="t m0 x4 h13 y13 ff5 fs0 fc0 sc0 ls4">\u2194</div></div><div class="c x4 y14 w4 h14"><div class="t m0 x11 h11 y15 ff1 fs0 fc0 sc0 ls4 ws6"> rectangular <span class="_29 blank"> </span>conversion <span class="_29 blank"> </span>capabilities <span class="_29 blank"> </span>which <span class="_29 blank"> </span>make <span class="_29 blank"> </span>this </div><div class="t m0 x0 h11 y16 ff1 fs0 fc0 sc0 ls4 ws6">computation <span class="_29 blank"> </span>more <span class="_29 blank"> </span>efficient <span class="_29 blank"> </span>than <span class="_29 blank"> </span>what <span class="_29 blank"> </span>is <span class="_29 blank"> </span>shown <span class="_29 blank"> </span>below. <span class="_2a blank"> </span>Noting <span class="_2a blank"> </span>that <span class="_29 blank"> </span>the <span class="_29 blank"> </span>vector <span class="_29 blank"> </span>lies <span class="_29 blank"> </span>in <span class="_2a blank"> </span>the </div><div class="t m0 x0 h11 y17 ff3 fs0 fc0 sc0 ls4 wsd">xy<span class="ff1 ws6"> plane and using Eq. 3-6, we obtain: </span></div><div class="t m0 x12 h15 y18 ff1 fs8 fc0 sc0 ls7">1<span class="fs9 ls4 ws6 v7">8.0 m</span></div><div class="t m0 x2 h16 y19 ff1 fs9 fc0 sc0 ls4 ws6">tan<span class="_28 blank"> </span>58<span class="_2b blank"> </span> or 122</div><div class="t m0 x13 h16 y1a ff1 fs9 fc0 sc0 ls4 ws6">5.0<span class="_c blank"> </span> m</div><div class="t m1 x14 h17 y1b ff5 fsa fc0 sc0 ls4">\u03b8</div><div class="t m0 x15 h18 y18 ff5 fs8 fc0 sc0 ls8">\u2212<span class="ff6 fs9 ls4 wse v8">§ ·</span></div><div class="t m0 x16 h19 y1c ff5 fs9 fc0 sc0 ls4 wsf">=<span class="_2c blank"> </span>= \u2212<span class="_d blank"> </span>°<span class="_1a blank"> </span>°</div><div class="t m0 x17 h1a y1d ff6 fs9 fc0 sc0 ls4 wse">¨ ¸</div><div class="t m0 x18 h19 y1e ff5 fs9 fc0 sc0 ls4 ws10">\u2212</div><div class="t m0 x17 h1a y1f ff6 fs9 fc0 sc0 ls4 wse">© ¹</div><div class="t m0 x0 h11 y20 ff1 fs0 fc0 sc0 ls4 ws6">where t<span class="_c blank"> </span>he latter <span class="_c blank"> </span>possibility <span class="_c blank"> </span>(122° measured <span class="_c blank"> </span>counterclockwise from <span class="_c blank"> </span>the +<span class="ff3">x</span></div><div class="t m0 x0 h11 y21 ff1 fs0 fc0 sc0 ls4 ws6">direction) <span class="_c blank"> </span>is chosen <span class="_c blank"> </span>since <span class="_c blank"> </span>the <span class="_c blank"> </span>signs of <span class="_c blank"> </span>the <span class="_c blank"> </span>components <span class="_c blank"> </span>imply the <span class="_c blank"> </span>vector <span class="_c blank"> </span>is </div><div class="t m0 x0 h11 y22 ff1 fs0 fc0 sc0 ls4 ws6">in the second quadrant. </div><div class="t m0 x0 h11 y23 ff1 fs0 fc0 sc0 ls4 ws6">(d) The sketch is shown on the right. The vector is 122° counterclockwise<span class="_c blank"> </span> </div><div class="t m0 x0 h11 y24 ff1 fs0 fc0 sc0 ls4 ws6">from the +<span class="ff3 ls9">x</span> direction.</div><div class="t m0 x0 h11 y25 ff1 fs0 fc0 sc0 ls4 ws6">(e) <span class="_2d blank"> </span>The <span class="_2d blank"> </span>displacement is <span class="_12 blank"> </span><span class="ff3 fsb ws11 v0">r r<span class="_2e blank"> </span>r</span></div><div class="t m0 x19 h1b y26 ff5 fsb fc0 sc0 ls4">\u2032</div><div class="t m0 x14 h1b y27 ff5 fsb fc0 sc0 ls4 ws12">\u2206 =<span class="_2f blank"> </span>\u2212</div><div class="t m0 x1a h1c y28 ff2 fsb fc0 sc0 ls4 ws13">G<span class="_30 blank"> </span>G<span class="_31 blank"> </span>G <span class="ff1 fs0 ws14 v4">where </span><span class="v0">G</span></div></div><div class="c x1b y29 w5 h1d"><div class="t m0 x4 h1e y2a ff3 fsb fc0 sc0 ls4">r</div></div><div class="c x4 y14 w4 h14"><div class="t m0 x1c h11 y27 ff1 fs0 fc0 sc0 ls4 ws6"> is given <span class="_2d blank"> </span>in <span class="_2d blank"> </span>part <span class="_2d blank"> </span>(a) and </div><div class="t m0 x1d h1f y2b ff1 fsc fc0 sc0 ls4">\u2022</div><div class="t m0 x1e h20 y2c ff1 fsc fc0 sc0 ls4 ws6"> (3.0 <span class="_f blank"></span>m)i.<span class="_32 blank"></span><span class="ff3 lsa">r<span class="ff5 lsb v9">\u2032</span><span class="ff5 ls4">=</span></span></div><div class="t m0 x1f h21 y2d ff2 fsc fc0 sc0 lsc">G<span class="ff1 fs0 ls4 ws15 va">Therefore, <span class="fs7 ws16 v5">\u2022 \u2022</span></span></div><div class="t m0 x20 hf y2e ff1 fs7 fc0 sc0 ls4 ws6">(8.0 m)<span class="_c blank"> </span>i<span class="_33 blank"> </span>(8.0 m)j<span class="_34 blank"></span><span class="ff3 ws17">r<span class="_35 blank"></span><span class="ff5 ws18">\u2206 =<span class="_36 blank"> </span>\u2212</span></span></div><div class="t m0 x21 h10 y2f ff2 fs7 fc0 sc0 lsd">G<span class="ff1 fs0 ls4 v4">.</span></div><div class="t m0 x0 h22 y30 ff1 fs0 fc0 sc0 ls4 ws6">(f) The magnitude of the displacement is <span class="_37 blank"> </span><span class="fs8 ws19 v6">2 2</span></div><div class="t m0 x6 hf y31 ff1 fs7 fc0 sc0 ls4 ws6">|<span class="_38 blank"> </span>|<span class="_39 blank"> </span>(8.0 m<span class="_c blank"> </span>)<span class="_3a blank"> </span>(<span class="_3b blank"> </span>8.0<span class="_c blank"> </span> <span class="_c blank"> </span>m<span class="_c blank"> </span>)<span class="_3c blank"> </span>11 m.<span class="_3d blank"></span><span class="ff3 ws17">r<span class="_35 blank"></span><span class="ff5 ws1a">\u2206<span class="_12 blank"> </span>=<span class="_3e blank"> </span>+ \u2212<span class="_3f blank"> </span>=</span></span></div><div class="t m0 x22 h10 y32 ff2 fs7 fc0 sc0 ls4">G</div><div class="t m0 x0 h11 y33 ff1 fs0 fc0 sc0 ls4 ws6">(g) The angle for the displacement, using Eq. 3-6, is </div><div class="t m0 x23 h15 y34 ff1 fs8 fc0 sc0 lse">1<span class="fs9 ls4 ws6 v7">8.0 m</span></div><div class="t m0 x24 h16 y35 ff1 fs9 fc0 sc0 ls4 ws6">tan<span class="_40 blank"> </span> = <span class="_33 blank"> </span>45<span class="_41 blank"> </span> or 135</div><div class="t m0 x1a h16 y36 ff1 fs9 fc0 sc0 ls4 ws6">8.0<span class="_c blank"> </span> m</div><div class="t m0 x25 h18 y34 ff5 fs8 fc0 sc0 lsf">\u2212<span class="ff6 fs9 ls4 ws1b v8">§<span class="_24 blank"> </span>· </span><span class="fs9 ls4 ws1c vb">\u2212 °<span class="_1a blank"> </span>°</span></div><div class="t m0 x26 h1a y37 ff6 fs9 fc0 sc0 ls4 ws1d">¨ ¸</div><div class="t m0 x27 h19 y38 ff5 fs9 fc0 sc0 ls4 ws10">\u2212</div><div class="t m0 x26 h1a y39 ff6 fs9 fc0 sc0 ls4 ws1d">© ¹</div><div class="t m0 x0 h13 y3a ff1 fs0 fc0 sc0 ls4 ws6">where we c<span class="_c blank"> </span>hoose the former possib<span class="_c blank"> </span>ility (<span class="ff5 ls10">\u2212</span>45°, or 45° measured <span class="ff3 wsd">clockwise</span></div><div class="t m0 x0 h11 y3b ff1 fs0 fc0 sc0 ls4 ws6">from <span class="_11 blank"> </span>+<span class="ff3 ls11">x</span>) <span class="_11 blank"> </span>since <span class="_11 blank"> </span>the <span class="_42 blank"> </span>signs <span class="_11 blank"> </span>of <span class="_11 blank"> </span>the <span class="_11 blank"> </span>components <span class="_42 blank"> </span>imply <span class="_11 blank"> </span>the <span class="_11 blank"> </span>vector <span class="_11 blank"> </span>is <span class="_42 blank"> </span>in <span class="_11 blank"> </span>the </div><div class="t m0 x0 h23 y3c ff1 fs0 fc0 sc0 ls4 ws6">fourth quadrant. A sketch of <span class="_43 blank"> </span><span class="ff3 fsd ws1e v0">r<span class="_35 blank"></span><span class="ff5 ls12">\u2206<span class="ff2 ls13 v1">G</span><span class="ff1 fs0 ls4 ws6"> is shown on the right. </span></span></span></div></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf3" class="pf w0 h0" data-page-no="3"><div class="pc pc3 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x28 y3d w6 h24" alt="" src="https://files.passeidireto.com/ad0bf699-331c-402e-9de4-dbbf94f7e157/bg3.png"><div class="t m0 x1f h11 y3e ff1 fs0 fc0 sc0 ls4 ws6">3. (a) The magnitude of </div><div class="c x29 y3f w7 h25"><div class="t m0 x4 h26 y40 ff2 fse fc0 sc0 ls4">G</div></div><div class="c x29 y41 w8 h27"><div class="t m0 x4 h28 y42 ff3 fse fc0 sc0 ls4">r</div></div><div class="t m0 x14 h11 y3e ff1 fs0 fc0 sc0 ls4 ws6"> is </div><div class="t m0 x2a h29 y43 ff1 fsf fc0 sc0 ls4 ws1f">2<span class="_44 blank"> </span>2 2</div><div class="t m0 x2b h2a y44 ff1 fs7 fc0 sc0 ls4 ws6">|<span class="_12 blank"> </span>|<span class="_3c blank"> </span>(5.0 m<span class="_c blank"> </span>)<span class="_3a blank"> </span>(<span class="_3b blank"> </span>3<span class="_c blank"> </span>.0<span class="_c blank"> </span> m<span class="_c blank"> </span>)<span class="_3a blank"> </span>(2.0 <span class="_c blank"> </span>m)<span class="_38 blank"> </span>6.<span class="_c blank"> </span>2 m.<span class="_45 blank"></span><span class="ff3 ls14">r<span class="ff7 ls4 ws20">=<span class="_3e blank"> </span>+ \u2212<span class="_1b blank"> </span>+<span class="_46 blank"> </span>=</span></span></div><div class="t m0 x2c h10 y45 ff2 fs7 fc0 sc0 ls4">G</div><div class="t m0 x1f h11 y46 ff1 fs0 fc0 sc0 ls4 ws6">(b) <span class="_11 blank"> </span>A <span class="_42 blank"> </span>sketch <span class="_11 blank"> </span>is <span class="_42 blank"> </span>shown. <span class="_11 blank"> </span>The <span class="_42 blank"> </span>coordinate <span class="_11 blank"> </span>values <span class="_42 blank"> </span>are <span class="_11 blank"> </span>in </div><div class="t m0 x1f h11 y47 ff1 fs0 fc0 sc0 ls4 ws6">meters. </div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf4" class="pf w0 h0" data-page-no="4"><div class="pc pc4 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x2d y48 w9 h2b" alt="" src="https://files.passeidireto.com/ad0bf699-331c-402e-9de4-dbbf94f7e157/bg4.png"><div class="t m0 x0 h2c y49 ff8 fs10 fc0 sc0 ls4 ws6">5. Using Eq. 4-3 and Eq. 4-8, we have </div><div class="t m0 x2e h2d y4a ff8 fs11 fc0 sc0 ls4 ws21">avg</div><div class="t m0 x2f h2e y4b ff1 fs12 fc0 sc0 ls4 ws22">\u2022 \u2022<span class="_47 blank"> </span>\u2022<span class="_48 blank"> </span>\u2022<span class="_19 blank"> </span>\u2022<span class="_6 blank"> </span>\u2022</div><div class="t m0 x30 h2e y4c ff1 fs12 fc0 sc0 ls4 ws6">(<span class="_12 blank"> </span>2.0i<span class="_c blank"> </span> + 8.0j<span class="_30 blank"> </span>2.0k) m<span class="_2e blank"> </span>(5.0<span class="_c blank"> </span>i<span class="_49 blank"> </span>6.0j + 2.0k) m<span class="_5 blank"> </span><span class="ws23 vc">\u2022 \u2022<span class="_4a blank"> </span>\u2022</span></div><div class="t m0 x31 h2e y4d ff1 fs12 fc0 sc0 ls4 ws6">(<span class="_10 blank"> </span>0.70i<span class="_42 blank"> </span>+<span class="_c blank"> </span>1.40j<span class="_30 blank"> </span>0.40k) m/s.</div><div class="t m0 x20 h2e y4e ff1 fs12 fc0 sc0 ls4 ws24">10 s</div><div class="t m0 x8 h2f y4d ff3 fs12 fc0 sc0 ls15">v<span class="ff4 ls4 ws25 v1">\u2212<span class="_4b blank"> </span>\u2212<span class="_5 blank"> </span>\u2212 \u2212</span></div><div class="t m0 x32 h30 y4d ff4 fs12 fc0 sc0 ls4 ws26">=<span class="_4c blank"> </span>= \u2212<span class="_4d blank"> </span>\u2212</div><div class="t m0 x33 h31 y4f ff2 fs12 fc0 sc0 ls4">G</div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf5" class="pf w0 h0" data-page-no="5"><div class="pc pc5 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x34 y50 wa h32" alt="" src="https://files.passeidireto.com/ad0bf699-331c-402e-9de4-dbbf94f7e157/bg5.png"><div class="c x4 y14 w4 h14"><div class="t m0 x0 h2e y51 ff8 fs12 fc0 sc0 ls4 ws6">The total displacement is </div><div class="t m0 x35 h33 y52 ff8 fs13 fc0 sc0 ls4 ws27">1 2 3<span class="_4e blank"> </span><span class="fs14 ws28 v3">\u2022 \u2022 \u2022<span class="_4f blank"> </span>\u2022</span></div><div class="t m0 x2 h34 y53 ff8 fs14 fc0 sc0 ls4 ws6">(40.0 km)i<span class="_12 blank"> </span>(<span class="_e blank"></span>1<span class="_f blank"></span>5.3 km<span class="_f blank"></span>)<span class="_a blank"> </span>i<span class="_3b blank"> </span>(<span class="_e blank"></span>12.9 km<span class="_f blank"></span>)<span class="_1 blank"> </span>j<span class="_33 blank"> </span>(<span class="_a blank"> </span>50.0 <span class="_f blank"></span>km)<span class="_a blank"> </span>i</div><div class="t m0 x29 h34 y54 ff8 fs14 fc0 sc0 ls4 ws29">\u2022 \u2022</div><div class="t m0 x36 h34 y55 ff8 fs14 fc0 sc0 ls4 ws6">(5.30<span class="_f blank"></span> km)<span class="_f blank"></span> <span class="_c blank"> </span>i<span class="_3b blank"> </span>(<span class="_e blank"></span>12.9 km<span class="_f blank"></span>) j.</div><div class="t m0 x37 h35 y56 ff3 fs14 fc0 sc0 ls4 ws2a">r r<span class="_50 blank"> </span>r r<span class="_51 blank"></span><span class="ff9 ws2b">\u2206<span class="_0 blank"> </span>=<span class="_52 blank"> </span>\u2206<span class="_0 blank"> </span>+ \u2206<span class="_17 blank"> </span>+ \u2206<span class="_3b blank"> </span>=<span class="_53 blank"> </span>+<span class="_4b blank"> </span>+<span class="_54 blank"> </span>\u2212</span></div><div class="t m0 x1d h36 y55 ff9 fs14 fc0 sc0 ls4 ws2c">= +</div><div class="t m0 x37 h37 y57 ff2 fs14 fc0 sc0 ls4 ws2d">G<span class="_22 blank"> </span>G G<span class="_22 blank"> </span>G</div><div class="t m0 x0 h2e y58 ff8 fs12 fc0 sc0 ls4 ws6">The time for <span class="_c blank"> </span>the trip is <span class="_c blank"> </span>(40.0 + 20.0 <span class="_c blank"> </span>+ 50.0) min = <span class="_c blank"> </span>110 min, which is <span class="_c blank"> </span>equivalent to 1.83 <span class="_c blank"> </span>h. </div><div class="t m0 x0 h2e y59 ff8 fs12 fc0 sc0 ls4 ws6">Eq. 4-8 then yields </div><div class="t m0 x38 h38 y5a ff8 fs15 fc0 sc0 ls4 ws2e">avg</div><div class="t m0 x34 h39 y5b ff8 fs16 fc0 sc0 ls4 ws6">5.30 km<span class="_4a blank"> </span>12.9<span class="_c blank"> </span> km</div><div class="t m0 x16 h39 y5c ff1 fs16 fc0 sc0 ls4 ws2f">\u2022<span class="_55 blank"> </span>\u2022<span class="_56 blank"> </span>\u2022 \u2022</div><div class="t m0 x16 h39 y5d ff1 fs16 fc0 sc0 ls4 ws6">i <span class="_12 blank"> </span> <span class="_53 blank"> </span> j = (2.90 km/h)<span class="_11 blank"> </span>i + (7<span class="_c blank"> </span>.01 km/h) j.</div><div class="t m0 x24 h39 y5e ff1 fs16 fc0 sc0 ls4 ws6">1.83 h<span class="_46 blank"> </span>1.<span class="_c blank"> </span>83 h</div><div class="t m0 x39 h3a y5d ff3 fs16 fc0 sc0 ls16">v<span class="ff6 ls4 ws30 v3">§<span class="_36 blank"> </span>· §<span class="_57 blank"> </span>·</span></div><div class="t m0 x3a h3b y5d ff9 fs16 fc0 sc0 ls4 ws31">= +</div><div class="t m0 x3b h3c y5f ff6 fs16 fc0 sc0 ls4 ws30">¨<span class="_36 blank"> </span>¸ ¨<span class="_57 blank"> </span>¸</div><div class="t m0 x3b h3c y60 ff6 fs16 fc0 sc0 ls4 ws30">©<span class="_36 blank"> </span>¹ ©<span class="_57 blank"> </span>¹</div><div class="t m0 x39 h3d y61 ff2 fs16 fc0 sc0 ls4">G</div><div class="t m0 x0 h2e y62 ff1 fs12 fc0 sc0 ls4 ws6">The magnitude is </div><div class="t m0 x3c h3e y63 ff1 fs17 fc0 sc0 ls4 ws32">2 2</div><div class="t m0 xe h3e y64 ff1 fs17 fc0 sc0 ls4 ws33">avg</div><div class="t m0 x3d h3f y65 ff1 fs18 fc0 sc0 ls4 ws6">|<span class="_22 blank"> </span>|<span class="_3c blank"> </span>(<span class="_c blank"> </span>2.90<span class="_c blank"> </span> km/h<span class="_c blank"> </span>)<span class="_3a blank"> </span>(7.01 km/h)<span class="_58 blank"> </span>7.59 km/h.<span class="_59 blank"></span><span class="ff3 ls17">v<span class="ff9 ls4 ws34">=<span class="_5a blank"> </span>+ =</span></span></div><div class="t m0 x3e h40 y66 ff2 fs18 fc0 sc0 ls4">G</div><div class="t m0 x0 h2e y67 ff1 fs12 fc0 sc0 ls4 ws6">(b) The angle is given by</div><div class="t m0 x3f h41 y68 ff1 fs19 fc0 sc0 ls18">1<span class="fs12 ls4 ws6 v7">7<span class="_2a blank"> </span>.<span class="_29 blank"> </span>0<span class="_2a blank"> </span>1<span class="_f blank"></span> <span class="_2a blank"> </span>k<span class="_2a blank"> </span>m<span class="_2a blank"> </span>/<span class="_2a blank"> </span>h</span></div><div class="t m0 x25 h2e y69 ff1 fs12 fc0 sc0 ls4 ws6">t<span class="_2a blank"> </span>a<span class="_2a blank"> </span>n<span class="_5b blank"> </span>6<span class="_2a blank"> </span>7<span class="_2a blank"> </span>.<span class="_29 blank"> </span>5<span class="_1d blank"> </span> <span class="_2a blank"> </span> <span class="_2a blank"> </span>(<span class="_2a blank"> </span>n<span class="_29 blank"> </span>o<span class="_2a blank"> </span>r<span class="_2a blank"> </span>t<span class="_29 blank"> </span>h<span class="_2a blank"> </span> <span class="_29 blank"> </span>o<span class="_2a blank"> </span>f<span class="_2a blank"> </span> <span class="_29 blank"> </span>e<span class="_2a blank"> </span>a<span class="_2a blank"> </span>s<span class="_2a blank"> </span>t<span class="_29 blank"> </span>)<span class="_2a blank"> </span>,</div><div class="t m0 x40 h2e y6a ff1 fs12 fc0 sc0 ls4 ws6">2<span class="_2a blank"> </span>.<span class="_2a blank"> </span>9<span class="_29 blank"> </span>0<span class="_c blank"> </span> <span class="_2a blank"> </span>k<span class="_2a blank"> </span>m<span class="_2a blank"> </span>/<span class="_29 blank"> </span>h</div><div class="t m2 x34 h42 y6b ff9 fs1a fc0 sc0 ls4">\u03b8</div><div class="t m0 x14 h43 y68 ff9 fs19 fc0 sc0 ls19">\u2212<span class="ff6 fs10 ls4 ws35 v8">§ ·</span></div><div class="t m0 x41 h44 y6b ff9 fs12 fc0 sc0 ls4 ws36">=<span class="_5c blank"> </span>= °</div><div class="t m0 x16 h45 y6c ff6 fs10 fc0 sc0 ls4 ws35">¨ ¸</div><div class="t m0 x16 h45 y6d ff6 fs10 fc0 sc0 ls4 ws35">© ¹</div><div class="t m0 x0 h2e y6e ff1 fs12 fc0 sc0 ls4 ws37">or <span class="fs16 ws38">22.5<span class="ff9 ls1a">°</span></span><span class="ws6"> east of due north. </span></div><div class="t m0 x0 h46 y6f ff1 fs12 fc0 sc0 ls4 ws6">7. <span class="_11 blank"> </span>The <span class="_a blank"> </span>average <span class="_11 blank"> </span>velocity <span class="_11 blank"> </span>is <span class="_11 blank"> </span>given <span class="_a blank"> </span>by <span class="_11 blank"> </span>Eq. <span class="_11 blank"> </span>4-8. <span class="_11 blank"> </span>The <span class="_a blank"> </span>total <span class="_11 blank"> </span>displacement <span class="_11 blank"> </span><span class="ff9 fs1b ls1b">\u2206<span class="ff2 ls4 v1">G</span></span></div></div><div class="c x42 y70 wb h47"><div class="t m0 x4 h48 y71 ff3 fs1b fc0 sc0 ls4">r</div></div><div class="c x4 y14 w4 h14"><div class="t m0 x43 h2e y72 ff1 fs12 fc0 sc0 ls4 ws6"> is <span class="_11 blank"> </span>the <span class="_a blank"> </span>sum <span class="_11 blank"> </span>of </div><div class="t m0 x0 h2e y73 ff1 fs12 fc0 sc0 ls4 ws6">three <span class="_a blank"> </span>displacements, <span class="_29 blank"> </span>each <span class="_a blank"> </span>result <span class="_a blank"> </span>of <span class="_a blank"> </span>a <span class="_29 blank"> </span>(constant) <span class="_a blank"> </span>velocity <span class="_a blank"> </span>during <span class="_29 blank"> </span>a <span class="_a blank"> </span>given <span class="_a blank"> </span>time. <span class="_29 blank"> </span>We <span class="_a blank"> </span>use <span class="_a blank"> </span>a </div><div class="t m0 x0 h2e y74 ff1 fs12 fc0 sc0 ls4 ws6">coordinate system with +<span class="ff3 ls1c">x</span> East and +<span class="ff3 ls1d">y</span> North.</div><div class="t m0 x0 h2e y75 ff1 fs12 fc0 sc0 ls4 ws6">(a) In unit-vector notation, the first displacement is given by </div><div class="t m0 x44 h49 y76 ff1 fs19 fc0 sc0 ls4">1</div><div class="t m0 x45 h4a y77 ff1 fs1c fc0 sc0 ls4 ws6">km<span class="_38 blank"> </span>40.0 min<span class="_33 blank"> </span><span class="ws39 vc">\u2022 \u2022</span></div><div class="t m0 x29 h4a y78 ff1 fs1c fc0 sc0 ls4 ws6"> = <span class="_5d blank"> </span>60.0 <span class="_5e blank"> </span>i = (40.0<span class="_c blank"> </span> km)<span class="_c blank"> </span>i.</div><div class="t m0 x17 h4a y79 ff1 fs1c fc0 sc0 ls4 ws6">h<span class="_5f blank"> </span>60 min/h</div><div class="t m0 x46 h4b y78 ff3 fs9 fc0 sc0 ls1e">r<span class="ff6 fs1c ls4 ws3a v3">§<span class="_46 blank"> </span>· §<span class="_8 blank"> </span>·</span></div><div class="t m0 xf h4c y7a ff9 fs1c fc0 sc0 ls1f">\u2206<span class="ff6 ls4 ws3a vd">¨<span class="_46 blank"> </span>¸ ¨<span class="_8 blank"> </span>¸</span></div><div class="t m0 x1a h4d y7b ff6 fs1c fc0 sc0 ls4 ws3a">©<span class="_46 blank"> </span>¹ ©<span class="_8 blank"> </span>¹</div><div class="t m0 x46 h4e y7c ff2 fs9 fc0 sc0 ls4">G</div><div class="t m0 x0 h4f y7d ff1 fs12 fc0 sc0 ls4 ws6">The <span class="_2d blank"> </span>seco<span class="_c blank"> </span>nd <span class="_2d blank"> </span>displacement <span class="_60 blank"> </span>has <span class="_2d blank"> </span>a <span class="_60 blank"> </span>magnitude <span class="_60 blank"> </span>of <span class="_40 blank"> </span><span class="fs1d ve">20.0 m<span class="_f blank"></span>in</span></div><div class="t m0 x47 h50 y7e ff1 fs1d fc0 sc0 ls4 ws3b">km</div><div class="t m0 x48 h50 y7f ff1 fs1d fc0 sc0 ls4 ws6">h<span class="_5f blank"> </span>60 min/h</div><div class="t m0 x49 h2e y80 ff1 fs12 fc0 sc0 ls4 ws6">(60.0<span class="_61 blank"> </span>)<span class="_b blank"> </span>20.0 km,</div><div class="t m0 x4a h51 y81 ff1 fs1e fc0 sc0 ls4 ws3c">) (<span class="_1b blank"> </span><span class="ff9 fs12 ws3d">=<span class="_62 blank"></span><span class="fs1e ls20">\u22c5<span class="ff1 fs12 ls4 ws6"> and <span class="_2d blank"> </span>its </span></span></span></div><div class="t m0 x0 h2e y82 ff1 fs12 fc0 sc0 ls4 ws6">direction is 40° north of east. Therefore, </div><div class="t m0 x4b h52 y83 ff1 fs1f fc0 sc0 ls21">2<span class="fs1b ls4 ws3e v3">\u2022<span class="_63 blank"> </span>\u2022 \u2022<span class="_4e blank"> </span>\u2022</span></div><div class="t m0 x4c h53 y84 ff1 fs1b fc0 sc0 ls4 ws6">(<span class="_c blank"> </span>20.0<span class="_c blank"> </span> km) c<span class="_c blank"> </span>os(40<span class="_c blank"> </span>.0<span class="_2b blank"> </span>)<span class="_a blank"> </span>i <span class="_12 blank"> </span>(<span class="_c blank"> </span>20.<span class="_c blank"> </span>0 km) s<span class="_c blank"> </span>in(40<span class="_c blank"> </span>.0<span class="_2b blank"> </span>)<span class="_1 blank"> </span>j<span class="_17 blank"> </span>(<span class="_e blank"></span>15.<span class="_c blank"> </span>3 km)<span class="_11 blank"> </span>i <span class="_12 blank"> </span>(<span class="_e blank"></span>12.9 k<span class="_c blank"> </span>m) j.<span class="_64 blank"></span><span class="ff3 ws3f">r<span class="_35 blank"></span><span class="ff9 ws40">\u2206 =<span class="_65 blank"> </span>°<span class="_31 blank"> </span>+<span class="_66 blank"> </span>°<span class="_67 blank"> </span>=<span class="_68 blank"> </span>+</span></span></div><div class="t m0 x4d h54 y85 ff2 fs1b fc0 sc0 ls4">G</div><div class="t m0 x0 h2e y86 ff1 fs12 fc0 sc0 ls4 ws6">And the third displacement is </div><div class="t m0 xf h49 y87 ff1 fs19 fc0 sc0 ls4">3</div><div class="t m0 x15 h4a y88 ff1 fs1c fc0 sc0 ls4 ws6">km<span class="_38 blank"> </span>50.0 min<span class="_69 blank"> </span><span class="ws41 vc">\u2022 \u2022</span></div><div class="t m0 x16 h4a y89 ff1 fs1c fc0 sc0 ls4 ws6">60.0<span class="_5e blank"> </span> i =<span class="_c blank"> </span> (<span class="_3b blank"> </span>50.0<span class="_2a blank"> </span> km)<span class="_a blank"> </span>i.</div><div class="t m0 x45 h4a y8a ff1 fs1c fc0 sc0 ls4 ws6">h<span class="_5f blank"> </span>60 min/<span class="_c blank"> </span>h</div><div class="t m0 xe h55 y89 ff3 fs9 fc0 sc0 ls22">r<span class="ff6 fs1c ls4 ws42 v3">§<span class="_5 blank"> </span>· §<span class="_6a blank"> </span>·</span></div><div class="t m0 x4e h4c y8b ff9 fs1c fc0 sc0 ls4 ws43">\u2206<span class="_12 blank"> </span>= \u2212<span class="_6b blank"> </span>\u2212</div><div class="t m0 x3f h4d y8c ff6 fs1c fc0 sc0 ls4 ws42">¨<span class="_5 blank"> </span>¸ ¨<span class="_6a blank"> </span>¸</div><div class="t m0 x3f h4d y8d ff6 fs1c fc0 sc0 ls4 ws42">©<span class="_5 blank"> </span>¹ ©<span class="_6a blank"> </span>¹</div><div class="t m0 x4f h4e y8e ff2 fs9 fc0 sc0 ls4">G</div></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf6" class="pf w0 h0" data-page-no="6"><div class="pc pc6 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x50 y8f wc h56" alt="" src="https://files.passeidireto.com/ad0bf699-331c-402e-9de4-dbbf94f7e157/bg6.png"><div class="t m0 x0 h57 y90 ff8 fs20 fc0 sc0 ls4 ws6">11. We ap<span class="_f blank"></span>ply Eq. 4-10 and Eq. 4<span class="_f blank"></span>-16. </div><div class="t m0 x0 h57 y91 ff8 fs20 fc0 sc0 ls4 ws6">(a) <span class="_c blank"> </span>Taking <span class="_c blank"> </span>the <span class="_c blank"> </span>derivative <span class="_c blank"> </span>of <span class="_2a blank"> </span>the <span class="_c blank"> </span>position <span class="_c blank"> </span>vector <span class="_c blank"> </span>with <span class="_c blank"> </span>respect <span class="_c blank"> </span>to <span class="_c blank"> </span>time, <span class="_c blank"> </span>we <span class="_2a blank"> </span>have, <span class="_c blank"> </span>in <span class="_c blank"> </span>SI <span class="_c blank"> </span>units </div><div class="t m0 x0 h57 y92 ff8 fs20 fc0 sc0 ls4 ws6">(m/s), </div><div class="t m0 x51 h58 y93 ff8 fs21 fc0 sc0 ls4">2</div><div class="t m0 x11 h59 y94 ff1 fs22 fc0 sc0 ls4 ws44">\u2022<span class="_4 blank"> </span>\u2022<span class="_6c blank"> </span>\u2022<span class="_4 blank"> </span>\u2022 \u2022</div><div class="t m0 x1 h59 y95 ff1 fs22 fc0 sc0 ls4 ws6"> = <span class="_3a blank"> </span>(i<span class="_c blank"> </span> + 4<span class="_d blank"> </span>j + <span class="_6d blank"> </span>k) = 8<span class="_10 blank"> </span>j + k .</div><div class="t m0 x10 h5a y96 ff3 fs22 fc0 sc0 ls4">d</div><div class="t m0 x1a h5a y95 ff3 fs22 fc0 sc0 ls4 ws45">v<span class="_53 blank"> </span>t t<span class="_6e blank"> </span>t</div><div class="t m0 x19 h5a y97 ff3 fs22 fc0 sc0 ls4 ws46">dt</div><div class="t m0 x52 h5b y98 ff2 fs22 fc0 sc0 ls4">G</div><div class="t m0 x0 h5c y99 ff1 fs20 fc0 sc0 ls4 ws6">(b) Taking another<span class="_f blank"></span> derivative with res<span class="_f blank"></span>pect to tim<span class="_f blank"></span>e leads to, <span class="_f blank"></span>in SI units (m<span class="_f blank"></span>/s<span class="fs23 ls23 vf">2</span><span class="ws47">),</span></div><div class="t m0 x53 h59 y9a ff1 fs22 fc0 sc0 ls4 ws48">\u2022 \u2022<span class="_6f blank"> </span>\u2022</div><div class="t m0 x54 h59 y9b ff1 fs22 fc0 sc0 ls4 ws6">= <span class="_3a blank"> </span> (8<span class="_10 blank"> </span>j + k) = 8<span class="_1 blank"> </span>j .</div><div class="t m0 x55 h5a y9c ff3 fs22 fc0 sc0 ls4">d</div><div class="t m0 x10 h5a y9b ff3 fs22 fc0 sc0 ls4 ws49">a t</div><div class="t m0 x56 h5a y9d ff3 fs22 fc0 sc0 ls4 ws46">dt</div><div class="t m0 x10 h5d y9e ff2 fs24 fc0 sc0 ls4">G</div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf7" class="pf w0 h0" data-page-no="7"><div class="pc pc7 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x4c y9f wd h5e" alt="" src="https://files.passeidireto.com/ad0bf699-331c-402e-9de4-dbbf94f7e157/bg7.png"><div class="c x4 y14 w4 h14"><div class="t m0 x38 h5f ya0 ff8 fs25 fc0 sc0 ls4 ws4a">2.00 <span class="fs26 ws4b v3">\u2022<span class="_70 blank"> </span>\u2022<span class="_71 blank"> </span>\u2022 \u2022</span></div><div class="t m0 x2a h60 ya1 ff8 fs26 fc0 sc0 ls4 ws6"> [2.00(8)<span class="_43 blank"> </span>5.00(2)]<span class="_29 blank"> </span>i + [6.00<span class="_17 blank"> </span>7.<span class="_f blank"></span>00(16)]<span class="_42 blank"> </span>j <span class="_69 blank"> </span> (6.00<span class="_a blank"> </span>i <span class="_33 blank"> </span> 106<span class="_1 blank"> </span>j<span class="_f blank"></span>)<span class="_c blank"> </span> m</div><div class="t m0 x57 h61 ya0 ff3 fs27 fc0 sc0 ls4">t</div><div class="t m0 x58 h62 ya1 ff3 fs26 fc0 sc0 ls24">r<span class="ffa fs27 ls25 vd">=</span><span class="ffa ls4 ws4c">=<span class="_72 blank"> </span>\u2212<span class="_73 blank"> </span>\u2212<span class="_68 blank"> </span>= \u2212</span></div><div class="t m0 x58 h63 ya2 ff2 fs0 fc0 sc0 ls4">G</div><div class="t m0 x0 h60 ya3 ff8 fs26 fc0 sc0 ls4 ws6">(b) Taking the derivative of th<span class="_f blank"></span>e given expression produces </div><div class="t m0 x59 h58 ya4 ff8 fs21 fc0 sc0 ls4 ws4d">2 3</div><div class="t m0 x49 h64 ya5 ff8 fs28 fc0 sc0 ls4 ws4e">\u2022 \u2022</div><div class="t m0 x16 h62 ya6 ff8 fs28 fc0 sc0 ls4 ws6">(<span class="_74 blank"> </span>)<span class="_c blank"> </span> = (6.00<span class="_17 blank"> </span> <span class="_3b blank"> </span> 5.00) <span class="_c blank"> </span>i <span class="_12 blank"> </span> 28.0<span class="_43 blank"> </span> j<span class="_75 blank"></span><span class="ff3 fs26 ws4f">v t<span class="_57 blank"> </span>t<span class="_76 blank"> </span>t<span class="_77 blank"></span><span class="ffa fs28 ws50">\u2212 \u2212</span></span></div><div class="t m0 x3f h63 ya7 ff2 fs0 fc0 sc0 ls4">G</div><div class="t m0 x0 h62 ya8 ff8 fs28 fc0 sc0 ls4 ws6">where we have written <span class="ff3 fs26 ls26">v</span><span class="ws51">(<span class="ff3 fs26 ls27">t</span></span>) to emphasize its dependence on tim<span class="_f blank"></span>e. This becomes, at </div><div class="t m0 x0 h65 ya9 ff3 fs26 fc0 sc0 ls28">t<span class="ff8 fs28 ls4 ws6"> = 2.00 s, <span class="_78 blank"> </span><span class="fs29 ws52 v5">\u2022 \u2022</span></span></div><div class="t m0 x39 h66 yaa ff8 fs29 fc0 sc0 ls4 ws6"> = (19<span class="_c blank"> </span>.0<span class="_a blank"> </span>i <span class="_67 blank"> </span> 224 j<span class="_2a blank"> </span>)<span class="_2a blank"> </span> m/s.<span class="_79 blank"></span><span class="ff3 ls29">v<span class="ffa ls4">\u2212</span></span></div><div class="t m0 x4c h67 yab ff2 fs2a fc0 sc0 ls4">G</div><div class="t m0 x0 h68 yac ff8 fs28 fc0 sc0 ls4 ws6">(c) <span class="_1 blank"> </span>Differentiating the <span class="_2d blank"> </span><span class="ff2 fs1b v1">G</span></div><div class="t m0 x23 h69 yad ff3 fs1b fc0 sc0 ls4 ws53">v t<span class="_7a blank"></span><span class="ff8 fs1 ws54">(<span class="_2b blank"> </span>) <span class="fs28 ws6 v0"> found above, <span class="_1 blank"> </span>with <span class="_1 blank"> </span>respect <span class="_1 blank"> </span>to <span class="_1 blank"> </span><span class="ff3 fs26 ls2a">t</span> produces <span class="_7b blank"> </span><span class="fs2b v6">2</span></span></span></div><div class="t m0 x5a h66 yae ff8 fs29 fc0 sc0 ls4 ws55">\u2022 \u2022</div><div class="t m0 x5b h66 yaf ff8 fs29 fc0 sc0 ls4 ws56">12.0 i<span class="_3b blank"> </span>84.<span class="_c blank"> </span>0<span class="_d blank"> </span>j,<span class="_7c blank"></span><span class="ff3 ws57">t t<span class="_7d blank"></span><span class="ffa">\u2212</span></span></div><div class="t m0 x0 h6a yb0 ff8 fs28 fc0 sc0 ls4 ws6">which yields <span class="_7e blank"> </span><span class="fs2c v6">2</span></div><div class="t m0 x5c h66 yb1 ff8 fs29 fc0 sc0 ls4 ws58">\u2022 \u2022</div><div class="t m0 x5d h66 yb2 ff8 fs29 fc0 sc0 ls4 ws6"> =(24.<span class="_c blank"> </span>0<span class="_a blank"> </span>i<span class="_17 blank"> </span>33<span class="_c blank"> </span>6<span class="_1 blank"> </span>j<span class="_2a blank"> </span>)<span class="_2a blank"> </span> m/s<span class="_34 blank"></span><span class="ff3 ls2b">a<span class="ffa ls4">\u2212</span></span></div><div class="t m0 x9 h67 yb3 ff2 fs2a fc0 sc0 ls2c">G<span class="ff8 fs28 ls4 ws6 v4"> at <span class="ff3 fs26 ls27">t</span> = 2.00 s. </span></div><div class="t m0 x0 h6b yb4 ff8 fs28 fc0 sc0 ls4 ws6">(d) The angle of <span class="_1 blank"> </span><span class="ff2 fs1b v1">G</span></div></div><div class="c x5e yb5 we h47"><div class="t m0 x4 h48 yb6 ff3 fs1b fc0 sc0 ls4">v</div></div><div class="t m0 x34 h62 yb7 ff8 fs28 fc0 sc0 ls4 ws6">, measured from +<span class="ff3 fs26 ws59">x</span>, is either </div><div class="t m0 x40 h6c yb8 ff8 fs2c fc0 sc0 ls2d">1<span class="fs22 ls4 ws5a v7">224<span class="_1 blank"> </span>m/s</span></div><div class="t m0 xb h59 yb9 ff8 fs22 fc0 sc0 ls4 ws5b">tan<span class="_7f blank"> </span>85.2<span class="_12 blank"> </span>or<span class="_80 blank"> </span>94.8</div><div class="t m0 x10 h59 yba ff8 fs22 fc0 sc0 ls4 ws5c">19.0 m/s</div><div class="t m0 x5f h6d yb8 ffa fs2d fc0 sc0 ls2e">\u2212<span class="ff6 fs16 ls4 ws5d v10">§ ·</span></div><div class="t m0 x19 h6e ybb ffa fs16 fc0 sc0 ls2f">\u2212<span class="ffb ls4 ws5e va">= \u2212<span class="_81 blank"> </span>°<span class="_3f blank"> </span>°</span></div><div class="t m0 x60 h3c ybc ff6 fs16 fc0 sc0 ls4 ws5d">¨ ¸</div><div class="t m0 x60 h3c ybd ff6 fs16 fc0 sc0 ls4 ws5d">© ¹</div><div class="t m0 x0 h6f ybe ffc fs28 fc0 sc0 ls4 ws6">where <span class="_2d blank"> </span>we <span class="_2d blank"> </span>settle <span class="_2d blank"> </span>on <span class="_2d blank"> </span>the <span class="_2d blank"> </span>first <span class="_2d blank"> </span>choice <span class="_2d blank"> </span>(!85.2°, <span class="_2d blank"> </span>which <span class="_2d blank"> </span>is <span class="_2d blank"> </span>equivalent <span class="_2d blank"> </span>to <span class="_2d blank"> </span>275° <span class="_2d blank"> </span>measured </div><div class="t m0 x0 h62 ybf ffc fs28 fc0 sc0 ls4 ws6">counterclockwise <span class="_29 blank"> </span>from <span class="_29 blank"> </span>the <span class="_a blank"> </span>+<span class="_f blank"></span><span class="ff3 fs26 ls30">x<span class="ffc fs28 ls4"> <span class="_29 blank"> </span>axis) <span class="_29 blank"> </span>since <span class="_29 blank"> </span>the <span class="_a blank"> </span>signs <span class="_2a blank"> </span>of <span class="_a blank"> </span>its <span class="_2a blank"> </span>components <span class="_29 blank"> </span>imply <span class="_a blank"> </span>that <span class="_2a blank"> </span>it <span class="_29 blank"> </span>is <span class="_a blank"> </span>in </span></span></div><div class="t m0 x0 h6f yc0 ffc fs28 fc0 sc0 ls4 ws6">the fourth quadrant. </div><div class="t m0 x0 h62 yc1 ffc fs26 fc0 sc0 ls4 ws6">13. <span class="_c blank"> </span>In <span class="_c blank"> </span>parts <span class="_c blank"> </span>(b) <span class="_c blank"> </span>and <span class="_2a blank"> </span>(c), <span class="_c blank"> </span>we <span class="_c blank"> </span>use <span class="_c blank"> </span>Eq. <span class="_c blank"> </span>4-10 <span class="_c blank"> </span>and <span class="_2a blank"> </span>Eq. <span class="_c blank"> </span>4-16. <span class="_c blank"> </span>For <span class="_c blank"> </span>part <span class="_c blank"> </span>(d), <span class="_c blank"> </span>we <span class="_2a blank"> </span>find <span class="_c blank"> </span>the <span class="_c blank"> </span>direction </div><div class="t m0 x0 h62 yc2 ffc fs26 fc0 sc0 ls4 ws6">of the velocity computed in <span class="_f blank"></span>part (b), since that represents the asked-for tangent <span class="_f blank"></span>line. </div><div class="t m0 x0 h62 yc3 ffc fs26 fc0 sc0 ls4 ws6">(a) Plugging into the g<span class="_f blank"></span>iven expression, we obtai<span class="_f blank"></span>n </div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf8" class="pf w0 h0" data-page-no="8"><div class="pc pc8 w0 h0"><div class="t m0 x0 h70 yc4 ffc fs28 fc0 sc0 ls4 ws6">15. <span class="_c blank"> </span>We <span class="_c blank"> </span>find <span class="_c blank"> </span><span class="ff3 fs26 ls31 v0">t</span><span class="v0"> <span class="_c blank"> </span>by <span class="_c blank"> </span>applying <span class="_c blank"> </span>Eq. <span class="_c blank"> </span>2-11 <span class="_c blank"> </span>to <span class="_c blank"> </span>motion along <span class="_c blank"> </span>the <span class="_c blank"> </span><span class="ff3 fs26 ls32">y</span>axis <span class="_c blank"> </span>(with <span class="_c blank"> </span><span class="ff3 fs26 ls26">v<span class="fs23 ls4 ws5f v11">y</span></span> <span class="_c blank"> </span>= <span class="_c blank"> </span>0 <span class="_c blank"> </span>characterizing </span></div><div class="t m0 x0 h62 yc5 ff3 fs26 fc0 sc0 ls26">y<span class="ffc fs28 ls4 ws6"> = </span><span class="ls33">y<span class="ffc fs23 ls4 ws6 v11">max </span><span class="ffc fs28 ls4 ws51">):</span></span></div><div class="t m0 x23 h71 yc6 ffc fs28 fc0 sc0 ls4 ws6">0 = (12 m/s) + (<span class="ffd ls34">\u2212</span>2.0 m/s<span class="fs23 ls35 vf">2</span><span class="ls36">)<span class="ff3 fs26 ls37">t<span class="ff6 fs0 ls38 v0">\ue09f</span><span class="ls31 v0">t</span></span></span><span class="v0"> = 6.0 s. </span></div><div class="t m0 x0 h70 yc7 ffc fs28 fc0 sc0 ls4 ws6">Then, Eq. 2-11 applies to motion along the <span class="ff3 fs26 ls39 v0">x</span><span class="v0">axis to determ<span class="_f blank"></span>ine the answer: </span></div><div class="t m0 x46 h72 yc8 ff3 fs26 fc0 sc0 ls4 ws59">v<span class="fs23 ws5f v11">x</span><span class="ffc fs28 ws6"> = (8.0 m/s) + (4.0 m/s<span class="fs23 ls23 vf">2</span>)(6.0 s) = 32 m<span class="_f blank"></span>/s. </span></div><div class="t m0 x0 h70 yc9 ffc fs28 fc0 sc0 ls4 ws6">Therefore, the velocity of the cart, when it reaches <span class="ff3 fs26 ls26 v0">y</span><span class="v0"> <span class="_f blank"></span>= <span class="ff3 fs26 ls3a">y</span><span class="fs23 v11">max </span>, is (32 m/s)i</span></div><div class="t m0 x61 h73 yca ffc fs23 fc0 sc0 ls4 ws5f">^<span class="fs28 vb">.</span></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf9" class="pf w0 h0" data-page-no="9"><div class="pc pc9 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x58 y9f wf h74" alt="" src="https://files.passeidireto.com/ad0bf699-331c-402e-9de4-dbbf94f7e157/bg9.png"><div class="c x4 y14 w4 h14"><div class="t m0 x0 h75 ycb ffc fs2e fc0 sc0 ls4 ws6">17. <span class="_2a blank"> </span>Constant <span class="_29 blank"> </span>acceleration <span class="_2a blank"> </span>in <span class="_2a blank"> </span>both <span class="_29 blank"> </span>directions <span class="_29 blank"> </span>(<span class="ff3 ls3b">x</span> <span class="_29 blank"> </span>and <span class="_2a blank"> </span><span class="ff3 ls3c">y</span>) <span class="_29 blank"> </span>allows <span class="_29 blank"> </span>us <span class="_2a blank"> </span>to <span class="_29 blank"> </span>use <span class="_29 blank"> </span>Table <span class="_2a blank"> </span>2-1 <span class="_29 blank"> </span>for <span class="_29 blank"> </span>the </div><div class="t m0 x0 h75 ycc ffc fs2e fc0 sc0 ls4 ws6">motion along each <span class="_c blank"> </span>direction. This can be <span class="_c blank"> </span>handled individually (for <span class="ff9 ws60">\u2206<span class="ff3 ws61">x</span></span> <span class="_c blank"> </span>and <span class="ff9 ws60">\u2206<span class="ff3 ws61">y</span></span>) or <span class="_c blank"> </span>together </div><div class="t m0 x0 h75 ycd ffc fs2e fc0 sc0 ls4 ws6">with <span class="_42 blank"> </span>the <span class="_42 blank"> </span>unit-vector <span class="_42 blank"> </span>notation <span class="_42 blank"> </span>(for <span class="_42 blank"> </span><span class="ff9 ws60">\u2206<span class="ff3 ws61">r</span></span>). <span class="_42 blank"> </span>Where <span class="_42 blank"> </span>units <span class="_42 blank"> </span>are <span class="_42 blank"> </span>not <span class="_42 blank"> </span>shown, <span class="_1 blank"> </span>S<span class="_f blank"></span>I <span class="_42 blank"> </span>units <span class="_42 blank"> </span>are <span class="_42 blank"> </span>to <span class="_1 blank"> </span>be </div><div class="t m0 x0 h75 yce ffc fs2e fc0 sc0 ls4 ws61">understood.</div><div class="t m0 x0 h76 ycf ffc fs2e fc0 sc0 ls4 ws6">(a) <span class="_11 blank"> </span>The <span class="_42 blank"> </span>velocity <span class="_42 blank"> </span>of <span class="_11 blank"> </span>the <span class="_42 blank"> </span>particle <span class="_42 blank"> </span>at <span class="_11 blank"> </span>any <span class="_42 blank"> </span>time <span class="_11 blank"> </span><span class="ff3 ls3d">t</span> <span class="_42 blank"> </span>is <span class="_42 blank"> </span>given <span class="_11 blank"> </span>by <span class="ff2 fs1b ws62 v1">G G<span class="_38 blank"> </span>G</span></div><div class="t m0 x62 h48 yd0 ff3 fs1b fc0 sc0 ls4 ws63">v<span class="_30 blank"> </span>v<span class="_3c blank"> </span>a t<span class="_82 blank"></span><span class="ff9 ws64">= +</span></div><div class="t m0 x63 h77 yd1 ffc fs2f fc0 sc0 ls3e">0<span class="fs2e ls4 ws6 v2">, <span class="_11 blank"> </span>where <span class="ff2 fs1b v1">G</span></span></div><div class="t m0 x64 h78 yd0 ff3 fs1b fc0 sc0 ls3f">v<span class="ffc fs2f ls40 v12">0<span class="fs2e ls4 ws6 v2"> is <span class="_11 blank"> </span>the </span></span></div><div class="t m0 x0 h79 yd2 ffc fs2e fc0 sc0 ls4 ws6">initial <span class="_2a blank"> </span>velocity <span class="_29 blank"> </span>and <span class="ff2 fs1b v1">G</span></div><div class="t m0 x24 h7a yd3 ff3 fs1b fc0 sc0 ls41">a<span class="ffc fs2e ls4 ws6 v0"> is <span class="_2a blank"> </span>the <span class="_29 blank"> </span>(constant) <span class="_2a blank"> </span>acceleration. <span class="_2a blank"> </span>The <span class="_29 blank"> </span><span class="ff3 ws61">x</span> <span class="_2a blank"> </span>component <span class="_29 blank"> </span>is <span class="_2a blank"> </span><span class="ff3 ws61">v<span class="fs23 ws5f v11">x</span></span> <span class="_29 blank"> </span>= <span class="_29 blank"> </span><span class="ff3 ws61">v</span><span class="fs23 ls23 v11">0</span></span><span class="fs23 ls4 ws5f v11">x</span><span class="ffc fs2e ls4 ws6 v0"> <span class="_2a blank"> </span>+ <span class="_29 blank"> </span><span class="ff3 ws61">a<span class="fs23 ws5f v11">x</span><span class="ls42">t</span></span> <span class="_29 blank"> </span>= </span></div><div class="t m0 x0 h75 yd4 ffc fs2e fc0 sc0 ls4 ws6">3.00 ! 1.00<span class="ff3 ls43">t</span>, and the <span class="ff3 ls44">y</span> co<span class="_f blank"></span>mponent is</div><div class="t m0 x10 h7b yd5 ff3 fs2e fc0 sc0 ls4 ws61">v<span class="fs23 ws5f v11">y</span><span class="ffc ws6 v0"> = </span><span class="v0">v<span class="ffc fs23 ws5f v11">0y</span><span class="ffc ws6"> + </span>a<span class="fs23 ws5f v11">y</span>t<span class="ffc ws6"> = !0.500</span>t</span></div><div class="t m0 x0 h75 yd6 ffc fs2e fc0 sc0 ls4 ws65">since <span class="ff3 ws61">v</span><span class="fs23 ws5f v11">0y</span><span class="ws6 v0"> = 0. <span class="_c blank"> </span>Wh<span class="_f blank"></span>en the particle <span class="_c blank"> </span>reaches its maximum <span class="ff3 ws61">x</span> coordinate at <span class="ff3 ls45">t</span> = <span class="_c blank"> </span><span class="ff3 ls46">t<span class="fs23 ls4 ws5f v11">m</span></span>, we must have </span></div><div class="t m0 x0 h7c yd7 ff3 fs2e fc0 sc0 ls4 ws61">v<span class="fs23 ws5f v11">x</span><span class="ffc ws6 v0"> = 0. <span class="_c blank"> </span>Therefore, 3.00 ! 1.00<span class="ff3 ls47">t</span></span><span class="fs23 ws5f v11">m</span><span class="ffc ws6 v0"> = <span class="_c blank"> </span>0 or <span class="ff3 ls47">t</span></span><span class="fs23 ws5f v11">m</span><span class="ffc ws6 v0"> <span class="_c blank"> </span>= 3.00 s. The </span><span class="v0">y<span class="ffc ws6"> <span class="_c blank"> </span>component of the <span class="_c blank"> </span>velocity at this </span></span></div><div class="t m0 x0 h75 yd8 ffc fs2e fc0 sc0 ls4 ws6">time is </div><div class="t m0 x1a h75 yd9 ff3 fs2e fc0 sc0 ls4 ws61">v<span class="fs23 ws5f v11">y</span><span class="ffc ws6 v0"> = 0 ! 0.500(3.00) = !1.50 m<span class="_f blank"></span>/s; </span></div><div class="t m0 x0 h75 yda ffc fs2e fc0 sc0 ls4 ws6">this is the only nonzero com<span class="_f blank"></span>ponent of </div></div><div class="c x13 ydb w10 h7d"><div class="t m0 x4 h26 ydc ff2 fse fc0 sc0 ls4">G</div></div><div class="c x65 ydd w11 h27"><div class="t m0 x4 h28 yde ff3 fse fc0 sc0 ls4">v</div></div><div class="t m0 x59 h75 ydf ffc fs2e fc0 sc0 ls4 ws6"> at <span class="ff3 ls48">t<span class="fs23 ls4 ws5f v11">m</span><span class="ls4">.</span></span></div><div class="t m0 x0 h75 ye0 ffc fs2e fc0 sc0 ls4 ws6">(b) <span class="_c blank"> </span>Since <span class="_c blank"> </span>it <span class="_c blank"> </span>started <span class="_2a blank"> </span>at <span class="_c blank"> </span>the <span class="_c blank"> </span>origin, <span class="_c blank"> </span>the <span class="_c blank"> </span>coordinates <span class="_c blank"> </span>of <span class="_2a blank"> </span>the <span class="_c blank"> </span>particle <span class="_c blank"> </span>at <span class="_c blank"> </span>any <span class="_c blank"> </span>time<span class="ff3"> <span class="_c blank"> </span>t</span> <span class="_2a blank"> </span>are <span class="_c blank"> </span>given <span class="_c blank"> </span>by </div><div class="t m0 x1f h7e ye1 ff2 fs30 fc0 sc0 ls4 ws66">G G<span class="_83 blank"> </span>G</div><div class="t m0 x1f h7f ye2 ff3 fs30 fc0 sc0 ls4 ws67">r<span class="_49 blank"> </span>v<span class="_2b blank"> </span>t<span class="_6c blank"> </span>a t<span class="_84 blank"></span><span class="ff9 ws68">= +</span></div><div class="t m0 x66 h80 ye3 ffc fs2f fc0 sc0 ls4">0</div><div class="t m0 x67 h80 ye4 ffc fs2f fc0 sc0 ls4">1</div><div class="t m0 x67 h80 ye5 ffc fs2f fc0 sc0 ls4">2</div><div class="t m0 x68 h80 ye6 ffc fs2f fc0 sc0 ls49">2<span class="fs2e ls4 ws6 vb">. At <span class="ff3 ls4a">t</span> = <span class="ff3 ls4b">t<span class="fs23 ls4 ws5f v11">m</span></span> this becom<span class="_f blank"></span>es </span></div><div class="t m3 x5d h81 ye7 ff7 fs31 fc0 sc0 ls4 ws69">( )</div><div class="t m4 xf h82 ye8 ff7 fs32 fc0 sc0 ls4 ws6a">( )</div><div class="t m3 x19 h81 ye7 ff7 fs31 fc0 sc0 ls4 ws6b">( )</div><div class="t m4 x49 h82 ye8 ff7 fs32 fc0 sc0 ls4 ws6a">( )</div><div class="t m0 x47 h83 ye9 ffc fs33 fc0 sc0 ls4">2</div><div class="t m0 x69 h84 yea ffc fs34 fc0 sc0 ls4">1</div><div class="t m0 x4e h85 yeb ffe fs34 fc0 sc0 ls4 ws6c">"<span class="_66 blank"> </span>" "<span class="_85 blank"> </span>" "</div><div class="t m0 x3a h85 yec ffe fs34 fc0 sc0 ls4 ws6">3.00<span class="_c blank"> </span>i<span class="_33 blank"> </span>3.00<span class="_86 blank"> </span>1.00<span class="_a blank"> </span>i<span class="_69 blank"> </span>0.50<span class="_1 blank"> </span>j<span class="_17 blank"> </span>3.00<span class="_87 blank"> </span>(<span class="_c blank"> </span>4.50<span class="_a blank"> </span>i<span class="_69 blank"> </span>2.25<span class="_1 blank"> </span>j) m.</div><div class="t m0 x69 h85 yed ffe fs34 fc0 sc0 ls4">2</div><div class="t m0 x6a h84 yec ff3 fs34 fc0 sc0 ls4c">r<span class="fff ls4 ws6d">=<span class="_7 blank"> </span>+ \u2212<span class="_4 blank"> </span>\u2212<span class="_88 blank"> </span>=<span class="_89 blank"> </span>\u2212</span></div><div class="t m0 x6a h86 yee ff2 fs34 fc0 sc0 ls4">G</div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pfa" class="pf w0 h0" data-page-no="a"><div class="pc pca w0 h0"><img fetchpriority="low" loading="lazy" class="bi x6b yef w12 h87" alt="" src="https://files.passeidireto.com/ad0bf699-331c-402e-9de4-dbbf94f7e157/bga.png"><div class="c x4 y14 w4 h14"><div class="t m0 x0 h88 yf0 ffc fs35 fc0 sc0 ls4 ws6">20. <span class="_c blank"> </span>The <span class="_c blank"> </span>acceleration <span class="_c blank"> </span>is <span class="_c blank"> </span>constant <span class="_c blank"> </span>so <span class="_c blank"> </span>that <span class="_c blank"> </span>use <span class="_2a blank"> </span>of <span class="_c blank"> </span>Table <span class="_c blank"> </span>2-1 <span class="_c blank"> </span>(for <span class="_c blank"> </span>both <span class="_c blank"> </span>the <span class="_c blank"> </span><span class="ff3 ls4d">x</span> <span class="_c blank"> </span>and <span class="_c blank"> </span><span class="ff3 ls4d">y</span> <span class="_c blank"> </span>motions) <span class="_c blank"> </span>is </div><div class="t m0 x0 h88 yf1 ffc fs35 fc0 sc0 ls4 ws6">permitted. <span class="_29 blank"> </span>Where <span class="_a blank"> </span>units<span class="_f blank"></span> <span class="_29 blank"> </span>are <span class="_a blank"> </span>not <span class="_2a blank"> </span>shown, <span class="_a blank"> </span>SI <span class="_2a blank"> </span>unit<span class="_c blank"> </span>s <span class="_2a blank"> </span>ar<span class="_c blank"> </span>e <span class="_29 blank"> </span>to <span class="_29 blank"> </span>be <span class="_a blank"> </span>u<span class="_f blank"></span>nderstood. <span class="_29 blank"> </span>Coll<span class="_c blank"> </span>ision <span class="_2a blank"> </span>b<span class="_c blank"> </span>etween </div><div class="t m0 x0 h88 yf2 ffc fs35 fc0 sc0 ls4 ws6e">particles <span class="ff3 ws6f">A</span><span class="ws6"> and <span class="ff3 ws6f">B</span> requires two things. First, the <span class="ff3 ls4e">y</span> motion of <span class="ff3 ws6f">B</span> must satisfy (using Eq. 2-15</span></div><div class="t m0 x0 h88 yf3 ffc fs35 fc0 sc0 ls4 ws6">and noting that </div><div class="t m5 x3a h89 yf4 fff fs36 fc0 sc0 ls4 ws70">\u03b8</div><div class="t m0 x5e h88 yf4 ffc fs35 fc0 sc0 ls4 ws6"> is measured from the <span class="ff3 ls4f">y</span> axis) </div><div class="t m0 x5f h80 yf5 ffc fs2f fc0 sc0 ls4 ws71">2<span class="_8a blank"> </span>2 2</div><div class="t m0 x6c h84 yf6 ffc fs34 fc0 sc0 ls4 ws72">1 1</div><div class="t m0 x69 h84 yf7 ffc fs34 fc0 sc0 ls4 ws6"> <span class="_38 blank"> </span> <span class="_1d blank"> </span>30<span class="_c blank"> </span> m<span class="_19 blank"> </span>(<span class="_c blank"> </span>0.40<span class="_1 blank"> </span>m/s<span class="_41 blank"> </span>) cos<span class="_6f blank"> </span>.</div><div class="t m0 x6c h84 yf8 ffc fs34 fc0 sc0 ls4 ws73">2 2</div><div class="t m0 x6d h8a yf9 ff3 fs37 fc0 sc0 ls4">y</div><div class="t m0 x41 h84 yfa ff3 fs34 fc0 sc0 ls4 ws74">y<span class="_8b blank"> </span>a t<span class="_8c blank"> </span>t</div><div class="t m6 x6e h8b yfb fff fs38 fc0 sc0 ls4 ws75">\u03b8</div><div class="t m0 x6f h8c yfc ff6 fs24 fc0 sc0 ls4 ws76">ª º</div><div class="t m0 x5c h8c yfb fff fs34 fc0 sc0 ls50">=<span class="ff6 fs24 ls51">\ue09f</span><span class="ls52">=<span class="ff6 fs24 ls4 ws76 v13">¬ ¼</span></span></div><div class="t m0 x0 h88 yfd ffc fs35 fc0 sc0 ls4 ws6">Second, the <span class="ff3 ls4d">x</span> motions of <span class="ff3 ws6f">A</span> and <span class="ff3 ws6f">B</span> must coincide: </div><div class="t m0 x1a h8d yfe ffc fs39 fc0 sc0 ls4 ws77">2<span class="_8d blank"> </span>2 2</div><div class="t m0 x46 h84 yff ffc fs34 fc0 sc0 ls4 ws78">1 1</div><div class="t m0 x12 h84 y100 ffc fs34 fc0 sc0 ls4 ws79">(3.0<span class="_1 blank"> </span>m/s )<span class="_8e blank"> </span>(0.4 0<span class="_1 blank"> </span>m/s<span class="_41 blank"> </span>)<span class="_1 blank"> </span>sin<span class="_8f blank"> </span>.</div><div class="t m0 x46 h84 y101 ffc fs34 fc0 sc0 ls4 ws78">2 2</div><div class="t m0 x14 h8d y102 ff3 fs39 fc0 sc0 ls4">x</div><div class="t m0 x34 h84 y103 ff3 fs34 fc0 sc0 ls4 ws7a">vt<span class="_8b blank"> </span>a t<span class="_5a blank"> </span>t<span class="_90 blank"> </span>t</div><div class="t m7 x70 h8e y103 fff fs3a fc0 sc0 ls4 ws7b">\u03b8</div><div class="t m0 x71 h8f y104 ff6 fs22 fc0 sc0 ls4 ws7c">ª º</div><div class="t m0 x6b h8f y105 fff fs22 fc0 sc0 ls53">=<span class="ff6 ls54">\ue09f</span><span class="ls55">=<span class="ff6 ls4 ws7c v13">¬ ¼</span></span></div><div class="t m0 x0 h88 y106 ffc fs35 fc0 sc0 ls4 ws6">We eliminate a factor of <span class="ff3 ls56">t</span> in the last relationship and formally solve for time: </div><div class="t m0 x72 h90 y107 ffc fs3b fc0 sc0 ls4">2</div><div class="t m0 x54 h84 y108 ffc fs34 fc0 sc0 ls4 ws7d">2<span class="_91 blank"> </span>2(3.0<span class="_1 blank"> </span>m/s )<span class="_58 blank"> </span><span class="va">.</span></div><div class="t m0 x3c h84 y109 ffc fs34 fc0 sc0 ls4 ws7e">(0.4 0<span class="_1 blank"> </span>m/ s<span class="_92 blank"> </span>)<span class="_1 blank"> </span>sin</div><div class="t m0 x73 h90 y10a ff3 fs3b fc0 sc0 ls4">x</div><div class="t m0 x73 h84 y10b ff3 fs34 fc0 sc0 ls4">v</div><div class="t m0 x69 h84 y10c ff3 fs34 fc0 sc0 ls57">t<span class="ls4 v14">a</span></div><div class="t m7 x74 h8e y10d fff fs3a fc0 sc0 ls4 ws7b">\u03b8</div><div class="t m0 x50 h91 y10e fff fs22 fc0 sc0 ls4 ws7f">= =</div><div class="t m0 x0 h88 y10f ffc fs35 fc0 sc0 ls4 ws6">This is then plugged into the previous equation to produce </div><div class="t m0 x75 h92 y110 ffc fs3c fc0 sc0 ls4">2</div><div class="t m0 x59 h92 y111 ffc fs3c fc0 sc0 ls4">2</div><div class="t m0 x62 h92 y112 ffc fs3c fc0 sc0 ls4">2</div><div class="t m0 x21 h93 y113 ffc fs3d fc0 sc0 ls4 ws80">1<span class="_93 blank"> </span>2(3.0<span class="_1 blank"> </span>m/s )</div><div class="t m0 x76 h93 y114 ffc fs3d fc0 sc0 ls4 ws6">30<span class="_c blank"> </span> m<span class="_6e blank"> </span>(<span class="_c blank"> </span>0.40 m<span class="_f blank"></span>/s<span class="_41 blank"> </span>) cos</div><div class="t m0 x21 h93 y115 ffc fs3d fc0 sc0 ls4 ws81">2<span class="_94 blank"> </span>(0.4 0<span class="_52 blank"> </span>m/s<span class="_41 blank"> </span>)<span class="_1 blank"> </span>si n</div><div class="t m5 x77 h94 y116 fff fs3e fc0 sc0 ls4 ws82">\u03b8</div><div class="t m5 x78 h94 y117 fff fs3e fc0 sc0 ls4 ws82">\u03b8</div><div class="t m0 x72 h95 y118 ff6 fs3d fc0 sc0 ls4 ws83">§ ·</div><div class="t m0 x1 h95 y119 ff6 fs3d fc0 sc0 ls4 ws84">ª º</div><div class="t m0 x6c h96 y11a fff fs3d fc0 sc0 ls58">=<span class="ff6 ls4 ws83 v12">¨ ¸</span></div><div class="t m0 x1 h95 y11b ff6 fs3d fc0 sc0 ls4 ws85">¬<span class="_73 blank"> </span>¼ <span class="ws83 va">© ¹</span></div><div class="t m0 x0 h97 y11c ffc fs35 fc0 sc0 ls4 ws6">which, with the use of sin<span class="fs23 vf">2</span></div><div class="t m5 x27 h89 y11d fff fs36 fc0 sc0 ls4 ws70">\u03b8</div><div class="t m0 x52 h72 y11d ffc fs35 fc0 sc0 ls4 ws6"> = 1 ! cos<span class="fs23 vf">2</span></div><div class="t m5 x79 h89 y11d fff fs36 fc0 sc0 ls4 ws70">\u03b8</div><div class="t m0 x7a h88 y11d ffc fs35 fc0 sc0 ls4 ws6">, simplifies to </div><div class="t m8 x7b h98 y11e ff10 fs3f fc0 sc0 ls4 ws86">(<span class="_95 blank"> </span>) (<span class="_38 blank"> </span>)</div><div class="t m0 x49 h99 y11f ffc fs40 fc0 sc0 ls4">2</div><div class="t m0 x15 h99 y120 ffc fs40 fc0 sc0 ls4">2</div><div class="t m0 xf h9a y121 ffc fs41 fc0 sc0 ls4 ws87">9.0 cos<span class="_96 blank"> </span>9.0</div><div class="t m0 x3a h9a y122 ffc fs41 fc0 sc0 ls4 ws88">30<span class="_97 blank"> </span>1<span class="_98 blank"> </span>cos<span class="_99 blank"> </span>cos .</div><div class="t m0 xe h9a y123 ffc fs41 fc0 sc0 ls4 ws89">0.20 1<span class="_98 blank"> </span>co<span class="_f blank"></span>s<span class="_9a blank"> </span>0.20<span class="_3b blank"> </span>30</div><div class="t m2 x10 h9b y121 ff10 fs42 fc0 sc0 ls59">\u03b8<span class="ls4 ws8a v15">\u03b8 \u03b8</span></div><div class="t m2 x7c h9b y124 ff10 fs42 fc0 sc0 ls4 ws8b">\u03b8</div><div class="t m0 x76 h9c y125 ff10 fs41 fc0 sc0 ls5a">=<span class="ff6 ls5b">\ue09f</span><span class="ls4 ws8c">\u2212 =</span></div><div class="t m0 x52 h9d y126 ff10 fs41 fc0 sc0 ls4">\u2212</div><div class="t m0 x0 h88 y127 ffc fs35 fc0 sc0 ls4 ws6">We use the quadratic formula (choosing the positive root) to solve for cos </div><div class="t m5 x7d h89 y127 ff10 fs36 fc0 sc0 ls4 ws6">\u03b8 </div><div class="t m0 x7e h88 y127 ffc fs35 fc0 sc0 ls4">:</div><div class="t m9 x7f h9e y128 ff10 fs43 fc0 sc0 ls4 ws8d">(<span class="_6c blank"> </span>) (<span class="_9b blank"> </span>)</div><div class="t m0 x22 h9f y129 ffc fs44 fc0 sc0 ls4">2</div><div class="t m0 x80 ha0 y12a ffc fs45 fc0 sc0 ls4 ws8e">1.5<span class="_50 blank"> </span>1.5<span class="_3c blank"> </span>4 1<span class="_c blank"> </span>.0<span class="_39 blank"> </span>1.0<span class="_9c blank"> </span><span class="v16">1</span></div><div class="t m0 x23 ha0 y12b ffc fs45 fc0 sc0 ls4 ws8f">cos <span class="ws90 v14">2 2</span></div><div class="t m2 x6d ha1 y12c ff10 fs46 fc0 sc0 ls4 ws91">\u03b8</div><div class="t m0 x81 ha2 y12a ff10 fs45 fc0 sc0 ls4 ws92">\u2212 +<span class="_9d blank"> </span>\u2212<span class="_1a blank"> </span>\u2212</div><div class="t m0 x1 ha2 y12d ff10 fs45 fc0 sc0 ls4 ws93">= =</div><div class="t m0 x82 h88 y12e ffc fs35 fc0 sc0 ls4 ws6">which yields</div><div class="t m2 x1d ha1 y12f ff10 fs46 fc0 sc0 ls4 ws91">\u03b8</div><div class="t m0 x36 ha3 y12f ff10 fs45 fc0 sc0 ls5c">=<span class="ff11 fs47 ls4 v3">F</span></div><div class="t m0 x4f ha4 y130 ff11 fs47 fc0 sc0 ls4">H</div><div class="t m0 x4f ha4 y131 ff11 fs47 fc0 sc0 ls4">G</div></div><div class="c x6c y132 w13 ha5"><div class="t m0 x4 ha4 y133 ff11 fs47 fc0 sc0 ls4">I</div></div><div class="t m0 x6c ha4 y134 ff11 fs47 fc0 sc0 ls4">K</div><div class="t m0 x6c ha4 y135 ff11 fs47 fc0 sc0 ls5d">J<span class="ff10 fs45 ls4 ws94 v8">= °</span></div><div class="t m0 x24 ha6 y136 ff10 fs48 fc0 sc0 ls4">\u2212</div><div class="t m0 x35 ha0 y137 ffc fs45 fc0 sc0 ls4 ws95">cos .</div><div class="t m0 x3e ha7 y136 ffc fs49 fc0 sc0 ls5e">1<span class="fs45 ls4 v7">1</span></div><div class="t m0 x46 ha8 y138 ffc fs45 fc0 sc0 ls5f">2<span class="ls4 ws96 v17">60</span></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div>
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