WJEC Further Unit 6 (Further Unit 6) 2024 June

1. Two smooth spheres \(A\) and \(B\) are moving on a smooth horizontal plane when they collide obliquely. When the spheres collide, the line joining their centres is parallel to the vector \(\mathbf { j }\), as shown in the diagram below.

Immediately before the collision, sphere \(A\) has velocity ( \(6 \mathbf { i } - 3 \mathbf { j }\) ) \(\mathrm { ms } ^ { - 1 }\) and sphere \(B\) has velocity \(( - 4 \mathbf { i } + 7 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). Sphere \(A\) has mass 6 kg and sphere \(B\) has mass 2 kg . \includegraphics[max width=\textwidth, alt={}, center]{36112cfa-20c4-4ba8-b972-6b7b44e5182f-02_595_972_753_534} Immediately after the collision, sphere \(B\) has velocity \(( - 4 \mathbf { i } - 5 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\).

1. Find the velocity of \(A\) immediately after the collision.
2. Calculate the coefficient of restitution between \(A\) and \(B\).
3. Find the angle through which the direction of motion of \(B\) is deflected as a result of the collision. Give your answer correct to the nearest degree.
4. After the collision, sphere \(B\) continues to move with velocity \(( - 4 \mathbf { i } - 5 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) until it collides with another sphere \(C\), which exerts an impulse of \(( - 20 \mathbf { i } + 18 \mathbf { j } )\) Ns on \(B\). Find the velocity of \(B\) after the collision with \(C\).
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1. An object, of mass 1.8 kg , is falling vertically downwards under gravity. During the motion, it experiences a variable resistance of \(0 \cdot 2 v ^ { 2 } \mathrm {~N}\), where \(v \mathrm {~ms} ^ { - 1 }\) is the speed of the object at time \(t\) seconds.
   1. Show that \(v\) satisfies the differential equation
   $$\frac { \mathrm { d } v } { \mathrm {~d} t } = \frac { 9 g - v ^ { 2 } } { 9 }$$ At time \(t = 0\), the object passes a point \(A\) with a speed of \(\sqrt { g } \mathrm {~ms} ^ { - 1 }\). The object then hits the ground with a speed of \(8 \mathrm {~ms} ^ { - 1 }\).
2. Calculate the time taken for the object to hit the ground.
   …………………………………………………………………………………………………………………………………………………... \includegraphics[max width=\textwidth, alt={}, center]{36112cfa-20c4-4ba8-b972-6b7b44e5182f-07_67_1614_639_264} \includegraphics[max width=\textwidth, alt={}, center]{36112cfa-20c4-4ba8-b972-6b7b44e5182f-07_76_1614_717_264} \includegraphics[max width=\textwidth, alt={}, center]{36112cfa-20c4-4ba8-b972-6b7b44e5182f-07_79_1614_801_264} \includegraphics[max width=\textwidth, alt={}, center]{36112cfa-20c4-4ba8-b972-6b7b44e5182f-07_72_1609_895_267} \includegraphics[max width=\textwidth, alt={}, center]{36112cfa-20c4-4ba8-b972-6b7b44e5182f-07_75_1614_979_264}
3. Given that the distance of the object from \(A\) at time \(t\) is \(x\) metres, form another differential equation to find an expression for \(x\) in terms of \(v\). Hence, find the height of \(A\) above the ground.
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1. \(A C B\) is the diameter of a semi-circular lamina of radius \(2 a\) and centre \(C\). Another semi-circular lamina, having \(A C\) as its diameter, is added to form a uniform lamina, as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{36112cfa-20c4-4ba8-b972-6b7b44e5182f-10_755_521_520_772}
   1. (i) Show that the distance of the centre of mass of the lamina from \(A B\) is \(\frac { 28 } { 15 \pi } a\).
      (ii) Calculate the distance of the centre of mass of the lamina from a line drawn through \(A\) that is perpendicular to \(A B\).
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   2. Suppose that the lamina is suspended in equilibrium by means of two vertical wires attached at \(A\) and \(B\) so that \(A B\) is horizontal. Find the fraction of the lamina's weight that is supported by the wire attached at \(B\).
      
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1. The diagram below shows part of a game at a funfair that consists of a target moving along a straight horizontal line \(A B\). The centre of the target may be modelled as a particle moving with Simple Harmonic Motion about centre \(O\), where \(O\) is the midpoint of \(A B\). \includegraphics[max width=\textwidth, alt={}, center]{36112cfa-20c4-4ba8-b972-6b7b44e5182f-14_245_1145_525_452}

When the target is at a distance of 84 cm from \(O\), its speed is \(52 \mathrm { cms } ^ { - 1 }\) and the magnitude of its acceleration is \(1344 \mathrm { cms } ^ { - 2 }\).

1. Show that the period of the motion is \(\frac { \pi } { 2 } \mathrm {~s}\).
   

   (b) Determine the maximum speed of the target.

   Examiner

2. During a game, players fire a ball at the target. A timer is started when the target is at \(A\). Players must wait for the target to complete at least one full cycle before firing. Given that the target is hit when it is at a distance of 67 cm from \(O\), calculate the two earliest possible times taken to hit the target.
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1. The diagram below shows a uniform rod \(A B\) of weight \(W N\) and length \(2 l\), with its lower end \(A\) resting on a rough horizontal floor. A light cable is attached to the other end \(B\). The rod is in equilibrium when it is inclined at an angle of \(\theta\) to the floor, where \(0 ^ { \circ } < \theta \leqslant 45 ^ { \circ }\). The tension in the cable is \(T \mathrm {~N}\) acting at an angle of \(2 \theta\) to the rod, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{36112cfa-20c4-4ba8-b972-6b7b44e5182f-18_508_1105_559_479}
   1. (i) Show that \(T = \frac { W } { 4 } \operatorname { cosec } \theta\).
      (ii) Hence determine the normal reaction of the floor on the rod at \(A\), giving your answer in terms of \(W\).
   2. Given that the coefficient of friction between the floor and the rod is \(\frac { \sqrt { 3 } } { 3 }\), calculate the minimum possible value for \(\theta\).
      
   3. The region \(R\), shown in the diagram below, is bounded by the coordinate axes and the curve
   $$y = \frac { a } { b } \sqrt { b ^ { 2 } - x ^ { 2 } }$$ where \(a , b\) are constants. \includegraphics[max width=\textwidth, alt={}, center]{36112cfa-20c4-4ba8-b972-6b7b44e5182f-21_451_1116_644_468} The region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis to form a uniform solid \(S\). The volume of \(S\) is \(\frac { 2 } { 3 } \pi a ^ { 2 } b\).
2. Use integration to show that the distance of the centre of mass of \(S\) from the \(y\)-axis is \(\frac { 3 b } { 8 }\).
   The diagram below shows a small tree growing in a pot. The uniform solid \(S\) described on the previous page may be used to model the part of the tree above the pot. This part of the tree has height \(h \mathrm {~cm}\) and base radius \(\frac { h } { 4 } \mathrm {~cm}\). The pot, including its contents, may be modelled as a solid cylinder of height 50 cm and radius 25 cm . \includegraphics[max width=\textwidth, alt={}, center]{36112cfa-20c4-4ba8-b972-6b7b44e5182f-22_846_839_1596_612} You may assume that the density of the pot, including its contents, is equal to 20 times the density of the part of the tree above the pot.
3. A gardener suggests that a tree is said to have outgrown its pot if the centre of mass, of both the tree and its pot, lies above the height of the pot. Determine the maximum value of \(h\) before the tree outgrows its pot.
4. Identify one possible limitation of the model used that could affect your answer to part (b). \section*{END OF PAPER} Additional page, if required. Write the question number(s) in the left-hand margin. \section*{PLEASE DO NOT WRITE ON THIS PAGE} \section*{PLEASE DO NOT WRITE ON THIS PAGE}