CAIE FP2 (Further Pure Mathematics 2) 2017 June

1 A bullet of mass 0.08 kg is fired horizontally into a fixed vertical barrier. It enters the barrier horizontally with speed \(300 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and emerges horizontally after 0.02 s . There is a constant horizontal resisting force of magnitude 1000 N . Find the speed with which the bullet emerges from the barrier.
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\includegraphics[max width=\textwidth, alt={}, center]{c437c752-5518-4185-b02f-74206dc4b13c-04_748_561_260_794} A uniform smooth disc with centre \(O\) and radius \(a\) is fixed at the point \(D\) on a horizontal surface. A uniform rod of length \(3 a\) and weight \(W\) rests on the disc with its end \(A\) in contact with a rough vertical wall. The rod and the disc lie in a vertical plane that is perpendicular to the wall. The wall meets the horizontal surface at the point \(E\) such that \(A E = a\) and \(E D = \frac { 5 } { 4 } a\). A particle of weight \(k W\) is hung from the rod at \(B\) (see diagram). The coefficient of friction between the rod and the wall is \(\frac { 1 } { 8 }\) and the system is in limiting equilibrium. Find the value of \(k\).

3 Two uniform small smooth spheres \(A\) and \(B\) have equal radii and masses \(3 m\) and \(m\) respectively. Sphere \(A\) is moving with speed \(u\) on a smooth horizontal surface when it collides directly with sphere \(B\) which is at rest. The coefficient of restitution between the spheres is \(e\).

1. Find, in terms of \(u\) and \(e\), expressions for the velocities of \(A\) and \(B\) after the collision.
   Sphere \(B\) continues to move until it strikes a fixed smooth vertical barrier which is perpendicular to the direction of motion of \(B\). The coefficient of restitution between \(B\) and the barrier is \(\frac { 3 } { 4 }\). When the spheres subsequently collide, \(A\) is brought to rest.
2. Find the value of \(e\).
   \includegraphics[max width=\textwidth, alt={}, center]{c437c752-5518-4185-b02f-74206dc4b13c-08_608_652_258_744} Three identical uniform discs, \(A , B\) and \(C\), each have mass \(m\) and radius \(a\). They are joined together by uniform rods, each of which has mass \(\frac { 1 } { 3 } m\) and length \(2 a\). The discs lie in the same plane and their centres form the vertices of an equilateral triangle of side \(4 a\). Each rod has one end rigidly attached to the circumference of a disc and the other end rigidly attached to the circumference of an adjacent disc, so that the rod lies along the line joining the centres of the two discs (see diagram).

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\includegraphics[max width=\textwidth, alt={}, center]{c437c752-5518-4185-b02f-74206dc4b13c-10_445_735_264_696} A particle of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The point \(A\) is such that \(O A = a\) and \(O A\) makes an angle \(\alpha\) with the upward vertical through \(O\). The particle is held at \(A\) and then projected downwards with speed \(\sqrt { } ( a g )\) so that it begins to move in a vertical circle with centre \(O\). There is a small smooth peg at the point \(B\) which is at the same horizontal level as \(O\) and at a distance \(\frac { 1 } { 3 } a\) from \(O\) on the opposite side of \(O\) to \(A\) (see diagram).

1. Show that, when the string first makes contact with the peg, the speed of the particle is \(\sqrt { } ( \operatorname { ag } ( 1 + 2 \cos \alpha ) )\).
   The particle now begins to move in a vertical circle with centre \(B\). When the particle is at the point \(C\) where angle \(C B O = 150 ^ { \circ }\), the tension in the string is the same as it was when the particle was at the point \(A\).
2. Find the value of \(\cos \alpha\).