Challenging +1.2 This is a multi-step mechanics problem requiring coordinate geometry and coefficient of restitution, but follows a standard pattern for Further Maths mechanics. The key insight that velocity components parallel to walls are preserved while perpendicular components are scaled by e is a standard technique. The algebra is straightforward once the setup is understood, making this slightly above average difficulty but well within typical FM2 mechanics scope.
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A rectangular pool table \(K L M N\) has \(K L = a\) and \(K N = 2 a\). A ball lies at rest on the table just outside the pocket at \(L\) and is projected along the table with speed \(u\) in a direction making an angle \(\theta\) with the edge \(L M\). The ball hits the edge \(K N\) at \(Y\), rebounds to hit the edge \(L M\) at \(X\) and then rebounds into the pocket at \(N\). Angle \(L X Y\) is denoted by \(\phi\) (see diagram). The coefficient of restitution between the ball and an edge is \(\frac { 3 } { 4 }\), and all resistances to motion may be neglected. Show that \(\tan \phi = \frac { 3 } { 4 } \tan \theta\). [3]
Show that \(X M = \left( 2 - \frac { 7 } { 3 } \cot \theta \right) a\), and find the value of \(\theta\).
Find the speed with which the ball reaches \(N\), giving the answer in the form \(k u\), where \(k\) is correct to 3 significant figures.
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A rectangular pool table $K L M N$ has $K L = a$ and $K N = 2 a$. A ball lies at rest on the table just outside the pocket at $L$ and is projected along the table with speed $u$ in a direction making an angle $\theta$ with the edge $L M$. The ball hits the edge $K N$ at $Y$, rebounds to hit the edge $L M$ at $X$ and then rebounds into the pocket at $N$. Angle $L X Y$ is denoted by $\phi$ (see diagram). The coefficient of restitution between the ball and an edge is $\frac { 3 } { 4 }$, and all resistances to motion may be neglected. Show that $\tan \phi = \frac { 3 } { 4 } \tan \theta$. [3]
Show that $X M = \left( 2 - \frac { 7 } { 3 } \cot \theta \right) a$, and find the value of $\theta$.
Find the speed with which the ball reaches $N$, giving the answer in the form $k u$, where $k$ is correct to 3 significant figures.
\hfill \mbox{\textit{CAIE FP2 2010 Q5 [12]}}