CAIE Further Paper 3 (Further Paper 3) 2023 June

1 A particle \(P\) 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 particle \(P\) is held at the point \(A\), where \(O A\) makes an angle \(\alpha\) with the downward vertical through \(O\), and with the string taut. The particle \(P\) is projected perpendicular to \(O A\) in an upwards direction with speed \(\sqrt { 3 a g }\). It then starts to move along a circular path in a vertical plane. The string goes slack when \(P\) is at \(B\), where \(O B\) makes an angle \(\theta\) with the upward vertical. Given that \(\cos \alpha = \frac { 4 } { 5 }\), find the value of \(\cos \theta\).

2 One end of a light elastic string, of natural length \(a\) and modulus of elasticity \(\lambda \mathrm { mg }\), is attached to a fixed point \(O\). The string lies on a smooth horizontal surface. A particle \(P\) of mass \(m\) is attached to the other end of the string. The particle \(P\) is projected in the direction \(O P\). When the length of the string is \(\frac { 4 } { 3 } a\), the speed of \(P\) is \(\sqrt { 2 \mathrm { ag } }\). When the length of the string is \(\frac { 5 } { 3 } a\), the speed of \(P\) is \(\frac { 1 } { 2 } \sqrt { 2 \mathrm { ag } }\). Find the value of \(\lambda\).
\includegraphics[max width=\textwidth, alt={}, center]{8a62b72b-cffc-4d6d-b09a-8b0cb8f8eea3-04_654_502_267_817} A uniform lamina is in the form of a triangle \(A B C\), with \(A C = 8 a , B C = 6 a\) and angle \(A C B = 90 ^ { \circ }\). The point \(D\) on \(A C\) is such that \(A D = 3 a\). The point \(E\) on \(C B\) is such that \(C E = x\) (see diagram). The triangle \(C D E\) is removed from the lamina.

1. Find, in terms of \(a\) and \(x\), the distance of the centre of mass of the remaining object \(A D E B\) from \(A C\).
   The object \(A D E B\) is on the point of toppling about the point \(E\) when the object is in the vertical plane with its edge \(E B\) on a smooth horizontal surface.
2. Find \(x\) in terms of \(a\).
   \includegraphics[max width=\textwidth, alt={}, center]{8a62b72b-cffc-4d6d-b09a-8b0cb8f8eea3-06_419_1160_274_488} Two identical smooth uniform spheres \(A\) and \(B\) each have mass \(m\). The two spheres are moving on a smooth horizontal surface when they collide with speeds \(u\) and \(2 u\) respectively. Immediately before the collision, \(A\) 's direction of motion makes an angle of \(30 ^ { \circ }\) with the line of centres, and \(B\) 's direction of motion is perpendicular to the line of centres (see diagram). After the collision, \(A\) and \(B\) are moving in the same direction. The coefficient of restitution between the spheres is \(e\).

1. Find the value of \(e\).
2. Find the loss in the total kinetic energy of the spheres as a result of the collision.

5 One end of a light elastic string, of natural length \(12 a\) and modulus of elasticity \(k m g\), is attached to a fixed point \(O\). The other end of the string is attached to a particle of mass \(m\). The particle moves with constant speed \(\frac { 3 } { 2 } \sqrt { 3 a g }\) in a horizontal circle with centre at a distance \(12 a\) below \(O\). The string is inclined at an angle \(\theta\) to the downward vertical through \(O\).

1. Find, in terms of \(a\), the extension of the string.
2. Find the value of \(k\).

6 A particle of mass \(m \mathrm {~kg}\) falls vertically under gravity, from rest. At time \(t \mathrm {~s} , P\) has fallen \(x \mathrm {~m}\) and has velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The only forces acting on \(P\) are its weight and a resistance of magnitude \(k m g v \mathrm {~N}\), where \(k\) is a constant.

1. Find an expression for \(v\) in terms of \(t , g\) and \(k\).
2. Given that \(k = 0.05\), find, in metres, how far \(P\) has fallen when its speed is \(12 \mathrm {~ms} ^ { - 1 }\).

7 The points \(O\) and \(P\) are on a horizontal plane, a distance 8 m apart. A ball is thrown from \(O\) with speed \(u \mathrm {~ms} ^ { - 1 }\) at an angle \(\theta\) above the horizontal, where \(\tan \theta = \frac { 4 } { 3 }\). At the same instant, a model aircraft is launched with speed \(5 \mathrm {~ms} ^ { - 1 }\) parallel to the horizontal plane from a point 4 m vertically above \(P\). The model aircraft moves in the same vertical plane as the ball and in the same horizontal direction as the ball. The model aircraft moves horizontally with a constant speed of \(5 \mathrm {~ms} ^ { - 1 }\). After \(T \mathrm {~s}\), the ball and the model aircraft collide.

1. Find the value of \(T\).
2. Find the direction in which the ball is moving immediately before the collision.
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