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<p>A particle moves back and forth in a horizontal line between two end points. A fixed origin \(O\) is the midpoint between these end points. Its displacement \(x\) metres from \(O\) at time \(t\) seconds is \(x=4\sin(2t+\frac\pi3)\), where&nbsp;\(0\leq t\leq\pi\).</p>
<ol class="abc-list"><li data-list-item-id="p3-tzc-q02-item01" data-list-item-path="p3-tzc-q02-item01"><div class="list-item-paragraph">Determine</div><ol class="i-list"><li data-list-item-id="p3-tzc-q02-sub01-item01" data-list-item-path="p3-tzc-q02-item01/p3-tzc-q02-sub01-item01"><div class="list-item-paragraph">the amplitude;<span class="inline-marks">[1]</span></div></li><li data-list-item-id="p3-tzc-q02-sub01-item02" data-list-item-path="p3-tzc-q02-item01/p3-tzc-q02-sub01-item02"><div class="list-item-paragraph">the initial displacement;<span class="inline-marks">[2]</span></div></li><li data-list-item-id="p3-tzc-q02-sub01-item03" data-list-item-path="p3-tzc-q02-item01/p3-tzc-q02-sub01-item03"><div class="list-item-paragraph">the value of \(t\) when the particle first passes through \(O\).<span class="inline-marks">[2]</span></div></li></ol></li><li data-list-item-id="p3-tzc-q02-item02" data-list-item-path="p3-tzc-q02-item02"><div class="list-item-paragraph">The acceleration \(a\) at displacement \(s\) satisfies \(a=-n^2s\), where&nbsp;\(n&gt;0\).</div><div class="list-item-paragraph">The displacement is \(s=A\sin(nt+b)\), where&nbsp;\(t\geq0\), \(A,n&gt;0\) and \(-\pi\leq b\leq\pi\).</div><div class="list-item-paragraph">By finding expressions for&nbsp;\(\frac{\mathrm{d}s}{\mathrm{d}t}\) and \(\frac{\mathrm{d}^2s}{\mathrm{d}t^2}\), verify that&nbsp;\(s=A\sin(nt+b)\) satisfies \(a=-n^2s\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="p3-tzc-q02-item03" data-list-item-path="p3-tzc-q02-item03"><ol class="i-list"><li data-list-item-id="p3-tzc-q02-sub03-item01" data-list-item-path="p3-tzc-q02-item03/p3-tzc-q02-sub03-item01"><div class="list-item-paragraph">Use the chain rule to show that&nbsp;\(a=v\frac{\mathrm{d}v}{\mathrm{d}s}\), where&nbsp;\(v\) is velocity.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="p3-tzc-q02-sub03-item02" data-list-item-path="p3-tzc-q02-item03/p3-tzc-q02-sub03-item02"><div class="list-item-paragraph">By solving \(v\frac{\mathrm{d}v}{\mathrm{d}s}=-n^2s\), show that&nbsp;\(v^2=n^2(A^2-s^2)\).<span class="inline-marks">[5]</span></div></li><li data-list-item-id="p3-tzc-q02-sub03-item03" data-list-item-path="p3-tzc-q02-item03/p3-tzc-q02-sub03-item03"><div class="list-item-paragraph">Hence, or otherwise, find the particle’s maximum speed.<span class="inline-marks">[2]</span></div></li></ol></li><li data-list-item-id="p3-tzc-q02-item04" data-list-item-path="p3-tzc-q02-item04"><div class="list-item-paragraph">The continuous random variable \(S\) denotes the particle’s displacement \(s\) from \(O\) at time \(t\).</div><div class="list-item-paragraph">\[f(s)=\begin{cases}\frac1{\pi\sqrt{A^2-s^2}},&-A&lt;s&lt;A,\\0,&\text{otherwise}.\end{cases}\]</div><div class="list-item-paragraph">Show that&nbsp;\(P(-\frac A2\leq S\leq\frac A2)=\frac13\).<span class="inline-marks">[4]</span></div></li><li data-list-item-id="p3-tzc-q02-item05" data-list-item-path="p3-tzc-q02-item05"><div class="list-item-paragraph">For \(-A&lt;s&lt;A\), \(f(s)\) can be expressed as \(\frac{k}{|v(s)|}\), where \(k&gt;0\) and \(v(s)\) is the particle’s velocity at displacement \(s\) from \(O\).</div><div class="list-item-paragraph">Find the value of \(k\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="p3-tzc-q02-item06" data-list-item-path="p3-tzc-q02-item06"><ol class="i-list"><li data-list-item-id="p3-tzc-q02-sub06-item01" data-list-item-path="p3-tzc-q02-item06/p3-tzc-q02-sub06-item01"><div class="list-item-paragraph">Determine \(E(S)\), justifying your answer.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="p3-tzc-q02-sub06-item02" data-list-item-path="p3-tzc-q02-item06/p3-tzc-q02-sub06-item02"><div class="list-item-paragraph">Interpret the result in the context of the particle’s motion.<span class="inline-marks">[1]</span></div></li></ol></li></ol>

A particle moves back and forth in a horizontal line between two end points. A fixed origin \(O\) is the midpoint between these end points. Its displacement \(x\) metres from \(O\) at time \(t\) seconds is \(x=4\sin(2t+\frac\pi3)\), where \(0\leq t\leq\pi\).

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