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\includegraphics[max width=\textwidth, alt={}, center]{f6887893-66c5-40df-ba8d-9439a5c268eb-3_531_908_1674_616} 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.