Standard +0.8 This is a multi-part SHM question requiring understanding of the position-time relationship, use of inverse trigonometric functions, and careful geometric reasoning about the window positions. Part (a) requires setting up equations for positions at different times and algebraic manipulation to reach the given result. Parts (b) and (c) involve velocity calculations and time intervals in SHM. While the techniques are standard M3 content, the question requires careful setup, multiple steps, and the 'show that' proof in part (a) adds difficulty since students must arrive at a specific form.
In a game at a fair, a small target \(C\) moves horizontally with simple harmonic motion between the points \(A\) and \(B\), where \(AB = 4L\). The target moves inside a box and takes 3 s to travel from \(A\) to \(B\). A player has to shoot at \(C\), but \(C\) is only visible to the player when it passes a window \(PQ\), where \(PQ = b\). The window is initially placed with \(Q\) at the point as shown in Figure 4. The target \(C\) takes 0.75 s to pass from \(Q\) to \(P\).
Show that \(b = (2 - \sqrt{2})L\). [5]
Find the speed of \(C\) as it passes \(P\). [2]
\includegraphics{figure_5}
For advanced players, the window \(PQ\) is moved to the centre of \(AB\) so that \(AP = QB\), as shown in Figure 5.
Find the time, in seconds to 2 decimal places, taken for \(C\) to pass from \(Q\) to \(P\) in this new position. [3]
\begin{tikzpicture}[>=Stealth, thick]
% Dimensions
\def\L{10} % total length (represents 4L)
\def\H{1.4} % bar height
\def\b{3.0} % width of inner rectangle (represents b)
% Derived
\pgfmathsetmacro{\px}{\L - \b} % left edge of inner rect
\pgfmathsetmacro{\qx}{\L} % right edge of inner rect (flush with bar)
% --- Filled bar (cyan) ---
\fill[cyan!50] (0, 0) rectangle (\L, \H);
% --- White inner rectangle (cutout region of width b) ---
\fill[white] (\px, 0) rectangle (\qx, \H);
\draw (\px, 0) rectangle (\qx, \H);
% --- Outer bar outline ---
\draw (0, 0) rectangle (\L, \H);
% --- Dashed centre line A to B ---
\draw[dashed] (0, {\H/2}) -- (\L, {\H/2});
% --- Points A and B ---
\node[left] at (0, {\H/2}) {$A$};
\node[right] at (\L, {\H/2}) {$B$};
% --- Points P, Q, C ---
\node[above, yshift=1pt] at (\px, {\H/2}) {$P$};
\node[above, yshift=1pt] at (\qx, {\H/2}) {$Q$};
\pgfmathsetmacro{\cx}{0.5*(\px + \qx)}
\fill (\cx, {\H/2}) circle (2pt);
\node[below, yshift=-1pt] at (\cx, {\H/2}) {$C$};
% --- Dimension: b (above the bar) ---
\draw[<->] (\px, {\H + 0.5}) -- node[above] {$b$} (\qx, {\H + 0.5});
\draw (\px, \H) -- (\px, {\H + 0.6});
\draw (\qx, \H) -- (\qx, {\H + 0.6});
% --- Dimension: 4L (below the bar) ---
\draw[<->] (0, -0.5) -- node[below] {$4L$} (\L, -0.5);
\draw (0, 0) -- (0, -0.6);
\draw (\L, 0) -- (\L, -0.6);
\end{tikzpicture}
In a game at a fair, a small target $C$ moves horizontally with simple harmonic motion between the points $A$ and $B$, where $AB = 4L$. The target moves inside a box and takes 3 s to travel from $A$ to $B$. A player has to shoot at $C$, but $C$ is only visible to the player when it passes a window $PQ$, where $PQ = b$. The window is initially placed with $Q$ at the point as shown in Figure 4. The target $C$ takes 0.75 s to pass from $Q$ to $P$.
\begin{enumerate}[label=(\alph*)]
\item Show that $b = (2 - \sqrt{2})L$. [5]
\item Find the speed of $C$ as it passes $P$. [2]
\end{enumerate}
\includegraphics{figure_5}
For advanced players, the window $PQ$ is moved to the centre of $AB$ so that $AP = QB$, as shown in Figure 5.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Find the time, in seconds to 2 decimal places, taken for $C$ to pass from $Q$ to $P$ in this new position. [3]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 2005 Q4 [10]}}