Edit Problem

Year

Paper

Question Number

Course

Section

Worksheet Citation (for copying)

Click the copy button or select the text to copy this citation for use in worksheets.

Problem Text

Solution (Optional)

\begin{center}
    \begin{tikzpicture}[scale=2]
        \draw (2, 0) -- (2,2);
        \draw (2,2) -- (0,2);

\filldraw (1,0) circle (1pt); 
        \draw[dashed] (1,0) -- (2,1) -- (1.5,2) -- (0, 2-1.5);
    \end{tikzpicture}
\end{center}

The puck sets off at some velocity $\displaystyle \binom{u_x}{u_y}$, after the first bounce off the wall parallel to the $y$-axis, it has velocity $\displaystyle  \binom{-eu_x}{u_y}$. After it bounces off the wall parallel to the $x$-axis, it has velocity $\displaystyle \binom{-eu_x}{-eu_y}$ which is clearly parallel to the original velocity.

\begin{center}
    \begin{tikzpicture}[scale=2]
        \draw (0,0) -- (2,0) -- (2, 3) -- (0, 3) -- cycle;

\def\t{0.5};
        \def\s{1.2};
        \def\r{0.8};

\filldraw ({\r},0) circle (1pt);

\draw[dashed, -latex] (0.8,0) -- ({0.5*(0.8+2)},{0.5*(2-\r)*\t});
        \draw[dashed] ({0.5*(0.8+2)},{0.5*(2-\r)*\t}) -- (2,{(2-\r)*\t}); % gradient \t
        \draw[dashed] (2,{(2-\r)*\t}) -- ({\s},3); % gradient (3-\t*(2-\r))/(\s-2)
        \draw[dashed] ({\s},3)-- (0, {3-\s*\t});
        \draw[dashed] (0, {3-\s*\t}) -- ({-(\s-2)/(3-\t*(2-\r))*(3-\s*\t)} , 0);
    \end{tikzpicture}
\end{center}

If the puck bounces off 3 walls and returns to the same point the shape formed must be a parallelogram.  We need to hit the point on the opposite side which is in a ratio of $1:z$, but this must be possible if we aim towards the side further away.

\begin{center}
    \begin{tikzpicture}[scale=2]
        \draw (0,0) -- (2,0) -- (2, 3) -- (0, 3) -- cycle;

\def\t{0.5};
        \def\e{.3};
        % \def\r{0.8}; % Original fixed value for comparison if needed
        % Calculated \r
        \pgfmathsetmacro{\rval}{(3*\e*(-2 + \t))/(\e*(-2 + \t) - \t)};

\filldraw ({\t},0) circle (1pt);

% --- Force calculation of gOne beforehand ---
        \pgfmathsetmacro{\gOneValue}{\rval/(2-\t)};

% Use the calculated value \gOneValue
        \draw[dashed, -latex] (\t,0) -- (2,{0+(2-\t)*\gOneValue});

\pgfmathsetmacro{\xTwoValue}{(2-(3-\rval)*\e/\gOneValue)};
        \pgfmathsetmacro{\yThreeValue}{(3 - \xTwoValue*\gOneValue)}; 
        \pgfmathsetmacro{\zFourValue}{(\yThreeValue * \e / \gOneValue)};

\draw[dashed] (2, {\rval}) -- ({\xTwoValue} , 3);
        \draw[dashed] ({\xTwoValue}, 3) -- (0, {\yThreeValue});
        \draw[dashed] (0, {\yThreeValue}) -- ({\zFourValue},0);

\def\t{1.5};
        % \def\r{0.8}; % Original fixed value for comparison if needed
        % Calculated \r
        \pgfmathsetmacro{\rval}{(3*\e*(-2 + \t))/(\e*(-2 + \t) - \t)};

\filldraw ({\t},0) circle (1pt);

% --- Force calculation of gOne beforehand ---
        \pgfmathsetmacro{\gOneValue}{\rval/(2-\t)};

% Use the calculated value \gOneValue
        \draw[dashed, -latex, blue] (\t,0) -- (2,{0+(2-\t)*\gOneValue});

\draw[dashed, blue] (2, {\rval}) -- ({\xTwoValue} , 3);
        \draw[dashed, blue] ({\xTwoValue}, 3) -- (0, {\yThreeValue});
        \draw[dashed, blue] (0, {\yThreeValue}) -- ({\zFourValue},0);

\end{tikzpicture}
\end{center}

For a fixed path, as $z$ increases we generate more parallelograms which cross ours (on two of the legs) twice. As they move the full length it will cover the full leg of the parallogram. Similarly going the other way will cover the other leg of the parallelogram. Therefore from every point there are two circuits round the table

Preview

Problem

Solution

Cancel

Current Ratings

Difficulty Rating: 1500.0

Difficulty Comparisons: 0

Banger Rating: 1500.0

Banger Comparisons: 0