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WJEC Further Unit 5 2023 June Q1
11 marks Standard +0.3
  1. The average time it takes for a new kettle to boil, when full of water, is 305 seconds. An old kettle will take longer, on average, to boil. Alun suspects that a particular kettle is an old kettle. He boils the full kettle on 9 occasions and the times taken, in seconds, are shown below.
    305
    295
    310
    310
    315
    307
    300
    311
    306
You may assume the times taken to boil the full kettle are normally distributed.
  1. Calculate unbiased estimates for the mean and variance of the times taken to boil the full kettle.
  2. Test, at the \(5 \%\) level of significance, whether there is evidence to suggest that this is an old kettle.
  3. State a factor that Alun should control when carrying out this investigation.
WJEC Further Unit 5 2023 June Q2
19 marks Standard +0.3
2. The random variables \(X\) and \(Y\) are independent, with \(X\) having mean \(\mu\) and variance \(\sigma ^ { 2 }\), and \(Y\) having mean \(\mu\) and variance \(k \sigma ^ { 2 }\), where \(k\) is a positive constant. Let \(\bar { X }\) denote the mean of a random sample of 20 observations of \(X\), and let \(\bar { Y }\) denote the mean of a random sample of 25 observations of \(Y\).
  1. Given that \(T _ { 1 } = \frac { 3 \bar { X } + 7 \bar { Y } } { 10 }\), show that \(T _ { 1 }\) is an unbiased estimator for \(\mu\).
  2. Given that \(T _ { 2 } = \frac { \bar { X } + a ^ { 2 } \bar { Y } } { 1 + a } , a > 0\), and \(T _ { 2 }\) is an unbiased estimator for \(\mu\), prove that \(a = 1\).
  3. Find and simplify expressions for the variances of \(T _ { 1 }\) and \(T _ { 2 }\).
  4. Show that the value of \(k\) for which \(T _ { 1 }\) and \(T _ { 2 }\) are equally good estimators is \(\frac { 5 } { 6 }\).
  5. Given that \(T _ { 3 } = ( 1 - \lambda ) \bar { X } + \lambda \bar { Y }\), find an expression for \(\lambda\), in terms of \(k\), for which \(T _ { 3 }\) has the smallest possible variance.
WJEC Further Unit 5 2023 June Q3
11 marks Standard +0.3
3. Athletes who compete in the 400 m event have resting heart rates (RHR), measured in beats per minute, which are normally distributed with known standard deviation \(4 \cdot 7\). A random sample of 90 athletes who compete in the 400 m event is taken. Their resting heart rates are summarised by $$\sum x = 4014 \quad \text { and } \quad \sum x ^ { 2 } = 182257 .$$
  1. Find a \(99 \%\) confidence interval for the mean of the RHR of athletes who compete in the 400 m event. Give the limits of your interval correct to 1 decimal place.
  2. Without doing any further calculation, explain how the width of a \(95 \%\) confidence interval would compare to the width of your interval in part (a). Athletes who compete in the discus event have RHR which are normally distributed with known standard deviation \(\sigma\). A random sample of 100 athletes who compete in the discus event is taken. A 95\% confidence interval for the mean of the RHR is calculated as [49•4, 52•6].
  3. Determine the value of \(\sigma\) that was used to calculate this confidence interval.
  4. Referring to the confidence intervals, state, with a reason, what can be said about the RHR of athletes who compete in the 400 m event compared to the RHR of athletes who compete in the discus event.
WJEC Further Unit 5 2023 June Q4
12 marks Standard +0.3
4. Llŷr believes that he will have more social media followers by appearing on a certain Welsh television show. To investigate his belief, he collects data on 9 randomly selected contestants who have appeared on the show. Llŷr records the number of social media followers one week before and one week after the contestants appeared on the show. The data he collects are shown in the table below.
ContestantABCDEFGH1
Before48010080344351781876741457
After8419987513449545428201011644
    1. Carry out a Wilcoxon signed-rank test on this data set, at a significance level as close to 10\% as possible.
    2. Suggest a possible course of action that Llŷr might take.
  2. Give two reasons why the Wilcoxon signed-rank test is appropriate in this case.
WJEC Further Unit 5 2023 June Q5
13 marks Standard +0.3
5. The masses, \(X\), in kg, of men who work for a large company are normally distributed with mean 75 and standard deviation 10.
  1. Find the probability that the mean mass of a random sample of 5 men is less than 70 kg .
  2. The mean mass, in kg , of a random sample of \(n\) men drawn from this distribution is \(\bar { X }\). Given that \(\mathrm { P } ( \bar { X } > 80 )\) is approximately \(0 \cdot 007\), find \(n\). The masses, in kg, of women who work for the company are normally distributed with mean 68 and standard deviation 6 . A lift in the company building will not move if the total mass in the lift is more than 500 kg .
  3. A random sample of 3 men and 4 women get in the lift. Find the probability that the lift will not move.
  4. State a modelling assumption you have made in calculating your answer for part (c).
WJEC Further Unit 5 2023 June Q6
7 marks Standard +0.3
6. A triathlon race organiser wishes to know whether competitors who are members of a triathlon club race more frequently than competitors who are not members of a triathlon club. Six competitors from a triathlon club and six competitors who are not members of a triathlon club are selected at random. The table below shows the number of triathlon races they each entered last year.
Club
members
11412537
Not club
members
294086
  1. Use a Mann-Whitney U test at a significance level as close to \(5 \%\) as possible to carry out the race organiser's investigation.
  2. Briefly explain why a Wilcoxon signed-rank test is not appropriate in this case.
WJEC Further Unit 5 2023 June Q7
7 marks Challenging +1.2
7. Branwen intends to buy a new bike, either a Cannotrek or a Bianchondale. If there is evidence that the difference in the mean times on the two bikes over a 10 km time trial is more than 1.25 minutes, she will buy the faster bike. Otherwise, she will base her decision on other factors. She negotiates a test period to try both bikes. The times, in minutes, taken by Branwen to complete a 10 km time trial on the Cannotrek may be modelled by a normal distribution with mean \(\mu _ { C }\) and standard deviation \(0 \cdot 75\). The times, in minutes, taken by Branwen to complete a 10 km time trial on the Bianchondale may be modelled by a normal distribution with mean \(\mu _ { B }\) and standard deviation \(0 \cdot 6\). During the test period, she completes 6 time trials with a mean time of 19.5 minutes on the Cannotrek, and 5 time trials with a mean time of 17.3 minutes on the Bianchondale. She calculates a \(p \%\) confidence interval for \(\mu _ { C } - \mu _ { B }\).
  1. What would be the largest value of \(p\) that would lead Branwen to base her purchasing decision on the time trials, without considering other factors?
  2. State an assumption you have made in part (a).
WJEC Further Unit 6 2019 June Q1
15 marks Standard +0.8
A large aeroplane, of mass 360 tonnes, starts from rest at the beginning of a straight horizontal runway. The aeroplane produces a constant thrust of 980 kN and experiences a variable resistance to motion of magnitude \(\left( 80 + 0 \cdot 1 v ^ { 2 } \right) \mathrm { kN }\), where \(v \mathrm {~ms} ^ { - 1 }\) is the speed of the aeroplane after it has travelled \(x\) metres.
    1. Find the maximum speed that the aeroplane can attain.
    2. Show that \(v\) satisfies the differential equation $$3600 v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 9000 - v ^ { 2 } .$$
    (b) Find an expression for \(v ^ { 2 }\) in terms of \(x\).
  2. Given that the aeroplane must achieve a speed of at least \(85 \mathrm {~ms} ^ { - 1 }\) to take off, determine the minimum length of the runway.
  3. Explain why, according to this model, the aeroplane will not reach the speed found in (a)(i).
WJEC Further Unit 6 2019 June Q2
14 marks Challenging +1.8
2. A metal sign is formed by removing triangle \(B C D\) from a rectangular lamina \(A C E F\) made of uniform material, and adding a quarter circle XYZ, made of the same uniform material, with a particle attached to its vertex at \(Y\). The sign is supported by two light chains fixed at \(E\) and \(F\). The quarter circle has radius 24 cm and the particle at \(Y\) has a mass equal to half of that of the removed triangle. \(X D\) is parallel to \(A C\) and \(B Z\) is parallel to \(A F\). The dimensions, in cm , are as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{3578a810-46da-4d9e-a98f-248be72a517a-3_885_636_712_715}
  1. Calculate the distance of the centre of mass of the sign from
    1. \(A F\),
    2. \(A C\).
  2. The support at \(F\) comes loose so that the sign is freely suspended at \(E\) by one chain alone. Given that it then hangs in equilibrium, calculate the angle that \(E F\) makes with the vertical.
WJEC Further Unit 6 2019 June Q3
14 marks Standard +0.8
3. A light elastic string, of natural length \(l \mathrm {~m}\) and modulus of elasticity 14 N , is hanging vertically with its upper end fixed and a particle of mass \(m \mathrm {~kg}\) attached to the lower end. The particle is initially in equilibrium and air resistance is to be neglected.
  1. Find, in terms of \(m , g\) and \(l\), the extension, \(e\), of the string when the particle is in equilibrium. The particle is pulled vertically downwards a further distance from its equilibrium position and released. In its subsequent motion, the string remains taut. Let \(x \mathrm {~m}\) denote the extension of the string from the equilibrium position at time \(t \mathrm {~s}\).
    1. Write down, in terms of \(x , m , g\) and \(l\), an expression for the tension in the string.
    2. Hence, show that the particle is moving with Simple Harmonic Motion which satisfies the differential equation, $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - \frac { 14 } { m l } x$$
    3. State the maximum distance that the particle could be pulled vertically downwards from its equilibrium position and still move with Simple Harmonic Motion. Give a reason for your answer.
  3. Given that \(m = 0.5 , l = 0.7\) and that the particle is pulled to the position where \(x = 0.2\) before being released,
    1. find the maximum speed of the particle,
    2. determine the time taken for the particle to reach \(x = 0.15\) for the first time.
WJEC Further Unit 6 2019 June Q4
15 marks Standard +0.3
4. Ryan is playing a game of snooker. The horizontal table is modelled as the horizontal \(x - y\) plane with the point \(O\) as the origin and unit vectors parallel to the \(x\)-axis and the \(y\)-axis denoted by \(\mathbf { i }\) and \(\mathbf { j }\) respectively. All balls on the table have a common mass \(m \mathrm {~kg}\). The table and the four sides, called cushions, are modelled as smooth surfaces. The dimensions of the table, in metres, are as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{3578a810-46da-4d9e-a98f-248be72a517a-5_663_1138_667_482} Initially, all balls are stationary. Ryan strikes ball \(A\) so that it collides with ball \(B\). Before the collision, \(A\) has velocity \(( - \mathbf { i } + 8 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) and, after the collision, it has velocity \(( 2 \mathbf { i } + \mathbf { j } ) \mathrm { ms } ^ { - 1 }\).
  1. Show that the velocity of ball \(B\) after the collision is \(( - 3 \mathbf { i } + 7 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). After the collision with ball \(A\), ball \(B\) hits the cushion at point \(C\) before rebounding and moving towards the pocket at \(P\). The cushion is parallel to the vector \(\mathbf { i }\) and the coefficient of restitution between the cushion and ball \(B\) is \(\frac { 5 } { 7 }\).
  2. Calculate the velocity of ball \(B\) after impact with the cushion.
  3. Find, in terms of \(m\), the magnitude of the impulse exerted on ball \(B\) by the cushion at \(C\), stating your units clearly.
  4. Given that \(C\) has position vector \(( x \mathbf { i } + 1 \cdot 75 \mathbf { j } ) \mathrm { m }\),
    1. determine the time taken between the ball hitting the cushion at \(C\) and entering the pocket at \(P\),
    2. find the value of \(x\).
  5. Describe one way in which the model used could be refined. Explain how your refinement would affect your answer to (d)(i).
WJEC Further Unit 6 2019 June Q5
10 marks Standard +0.8
5.
  1. Show, by integration, that the centre of mass of a uniform solid hemisphere of radius \(r\) is at a distance of \(\frac { 3 r } { 8 }\) from the plane face.
  2. The diagram shows a composite solid body which consists of a uniform right circular cylinder capped by a uniform hemisphere. The total height of the solid is \(3 r \mathrm {~cm}\), where \(r\) represents the common radius of the hemisphere and the cylinder. \includegraphics[max width=\textwidth, alt={}, center]{3578a810-46da-4d9e-a98f-248be72a517a-6_397_340_762_858} Given that the density of the hemisphere is \(50 \%\) more than that of the cylinder, find the distance of the centre of mass of the solid from its base along the axis of symmetry.
WJEC Further Unit 6 2019 June Q6
12 marks Challenging +1.2
6. \includegraphics[max width=\textwidth, alt={}, center]{3578a810-46da-4d9e-a98f-248be72a517a-7_606_506_365_781} A uniform ladder \(A B\), of mass 10 kg and length 5 m , rests with one end \(A\) against a smooth vertical wall and the other end \(B\) on rough horizontal ground. The ladder is inclined at an angle \(\theta\) to the horizontal. A woman of mass 75 kg stands on the ladder so that her weight acts at a distance \(x \mathrm {~m}\) from \(B\).
  1. Show that the frictional force, \(F \mathrm {~N}\), between the ladder and the horizontal ground is given by $$F = 5 g \cot \theta ( 1 + 3 x ) .$$ For safety reasons, it is recommended that \(\theta\) is chosen such that the ratio \(C B : C A\) is \(1 : 4\).
  2. Determine the least value of the coefficient of friction such that the ladder will not slip however high the woman climbs.
  3. State one modelling assumption that you have made in your solution.
WJEC Further Unit 6 2022 June Q1
12 marks Standard +0.8
  1. A particle is moving along the \(x\)-axis. At time \(t\) seconds the particle is \(x\) metres from the origin, \(O\), and its velocity \(v \mathrm {~ms} ^ { - 1 }\) is given by
$$v = \frac { 24 } { 4 x + 9 }$$
  1. Find, in terms of \(x\), an expression for the acceleration of the particle at time \(t \mathrm {~s}\).
  2. At \(t = T\) the acceleration of the particle is \(- \frac { 4 } { 3 } \mathrm {~ms} ^ { - 2 }\).
    1. Determine the value of \(x\) when \(t = T\).
    2. Given that \(x = - 2\) when \(t = 0\), find an expression for \(t\) in terms of \(x\) and hence find the value of \(T\).
WJEC Further Unit 6 2022 June Q2
15 marks Standard +0.8
2. A particle \(P\) moves along the \(x\)-axis such that its position \(x\) metres, after \(t\) seconds, is given by $$x = \sin ( \pi t ) + \sqrt { 3 } \cos ( \pi t )$$
    1. Show that the motion of the particle \(P\) is Simple Harmonic. State the value of \(x\) at the centre of motion.
    2. Show that the period of the motion of \(P\) is 2 s and determine the amplitude. Suppose that another particle \(Q\) is introduced so that it also moves along the \(x\)-axis with Simple Harmonic Motion with centre of motion, \(O\), and period equal to that of particle \(P\). When \(t = 0\), the particle \(Q\) is at \(O\) and when it is \(2 \sqrt { 3 } \mathrm {~m}\) from \(O\) its speed is \(2 \pi \mathrm {~ms} ^ { - 1 }\).
  2. Find the amplitude of particle \(Q\).
  3. Determine the time when particles \(P\) and \(Q\) first meet.
WJEC Further Unit 6 2022 June Q3
14 marks Challenging +1.2
3. The diagram below shows a lamina \(A B C D E\) which is made of a uniform material. It consists of a rectangle \(A B D E\) with \(A B = 6 a\) and \(A E = 8 a\), together with an isosceles triangle \(B C D\) with \(B C = D C = 5 a\). A semicircle, with its centre at the midpoint of \(A E\) and radius \(3 a\), is removed from \(A B D E\). \includegraphics[max width=\textwidth, alt={}, center]{b9c63cb4-d446-4548-be42-e30b10cb4b99-3_606_703_603_680}
  1. Write down the distance of the centre of mass of the lamina \(A B C D E\) from \(A B\).
  2. Show that the distance of the centre of mass of the lamina \(A B C D E\) from \(A E\) is \(\frac { 140 } { 40 - 3 \pi } a\).
  3. The lamina \(A B C D E\) is freely suspended from the point \(D\) and hangs in equilibrium.
    1. Calculate the angle that \(B D\) makes with the vertical.
    2. The mass of the lamina is \(M\). When a particle of mass \(k M\) is attached at the point \(C\), the lamina hangs in equilibrium with \(A B\) horizontal. Determine the value of \(k\).
WJEC Further Unit 6 2022 June Q4
12 marks Standard +0.3
4. The diagram below shows a uniform rod \(A B\), of weight 10 N , hinged to a vertical wall at \(A\). The rod is held in a horizontal position by means of a light inextensible string. One end of the string is attached to a point \(C\) on the rod and the other end is attached to a point \(D\) on the wall. The point \(D\) is 0.6 m vertically above \(A\) and the length of \(A C\) is 0.8 m . A particle of weight 25 N is attached to the rod at \(B\) and the tension in the string is 75 N . \includegraphics[max width=\textwidth, alt={}, center]{b9c63cb4-d446-4548-be42-e30b10cb4b99-4_572_808_612_625}
  1. Find the length of the rod \(A B\).
  2. Calculate the magnitude and direction of the reaction at the hinge at \(A\).
WJEC Further Unit 6 2022 June Q5
13 marks Standard +0.3
5. Two smooth spheres \(A\) and \(B\), of equal radii, are moving on a smooth horizontal plane when they collide. Immediately after the collision sphere \(A\) has velocity ( \(- 2 \mathbf { i } - 5 \mathbf { j }\) ) \(\mathrm { ms } ^ { - 1 }\) and sphere \(B\) has velocity \(( \mathbf { i } + 3 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). When the spheres collide, their line of centres is parallel to the vector \(\mathbf { i }\) and the coefficient of restitution between the spheres is \(\frac { 2 } { 5 }\). Sphere \(A\) has mass 4 kg and sphere \(B\) has mass 2 kg .
  1. Find the velocity of \(A\) and the velocity of \(B\) immediately before the collision. After the collision, sphere \(A\) continues to move with velocity ( \(- 2 \mathbf { i } - 5 \mathbf { j }\) ) \(\mathrm { ms } ^ { - 1 }\) until it collides with a smooth vertical wall. The impulse exerted by the wall on \(A\) is \(32 \mathbf { j }\) Ns.
  2. State whether the wall is parallel to the vector \(\mathbf { i }\) or to the vector \(\mathbf { j }\). Give a reason for your answer.
  3. Find the speed of \(A\) after the collision with the wall.
  4. Calculate the loss of kinetic energy caused by the collision of sphere \(A\) with the wall.
WJEC Further Unit 6 2022 June Q6
14 marks Standard +0.8
6. The diagram shows a particle \(P\), of mass 4 kg , lying on a smooth horizontal surface. It is attached by two light springs to fixed points \(A\) and \(B\), where \(A B = 2.8 \mathrm {~m}\).
Spring \(A P\) has natural length 0.8 m and modulus of elasticity 60 N .
Spring \(P B\) has natural length 1.2 m and modulus of elasticity 30 N . \includegraphics[max width=\textwidth, alt={}, center]{b9c63cb4-d446-4548-be42-e30b10cb4b99-5_231_1253_612_404} When \(P\) is in equilibrium, it is at the point \(C\).
  1. Show that \(A C = 1 \mathrm {~m}\).
  2. The particle \(P\) is pulled horizontally and is initially held at rest at the midpoint of \(A B\). The system is then released.
    1. Show that \(P\) performs Simple Harmonic Motion about centre \(C\) and find the period of its motion.
    2. Determine the shortest time taken for \(P\) to reach a position where there is no tension in the spring \(A P\). \section*{END OF PAPER}
WJEC Further Unit 6 2023 June Q1
13 marks Challenging +1.2
  1. The diagram shows a uniform rod \(A B\), of length 8 m and mass 23 kg , in limiting equilibrium with its end \(A\) on rough horizontal ground and point \(C\) resting against a smooth fixed cylinder. The rod is inclined at an angle of \(30 ^ { \circ }\) to the ground. \includegraphics[max width=\textwidth, alt={}, center]{d7f600c5-af4a-4708-bfd9-92b37a95c634-3_240_869_603_598}
The coefficient of friction between the ground and the rod is \(\frac { 2 } { 3 }\).
  1. Calculate the magnitude of the normal reaction at \(C\) and the magnitude of the normal reaction to the ground at \(A\).
  2. Find the length \(A C\).
  3. Suppose instead that the rod is non-uniform with its centre of mass closer to \(A\) than to \(B\). Without carrying out any further calculations, state whether or not this will affect your answers in part (a). Give a reason for your answer.
WJEC Further Unit 6 2023 June Q2
7 marks Challenging +1.2
2. You are given that the centre of mass of a uniform solid cone of height \(h\) and base radius \(r\) is at a height of \(\frac { 1 } { 4 } h\) above its base. The diagram shows a solid conical frustum. It is formed by taking a uniform right circular cone, of base radius \(3 x\) and height \(6 y\), and removing a smaller cone, of base radius \(x\), with the same vertex. \includegraphics[max width=\textwidth, alt={}, center]{d7f600c5-af4a-4708-bfd9-92b37a95c634-3_490_903_1937_575} Show that the distance of the centre of mass of the frustum from its base along the axis of symmetry is \(\frac { 18 } { 13 } y\).
WJEC Further Unit 6 2023 June Q3
13 marks Standard +0.3
3. The vertical motion of a point on the surface of the water in a certain harbour may be modelled as Simple Harmonic Motion about a mean level. The diagram shows that, on a particular day, the depth of water in the harbour at low tide is 2 m and the depth of the water in the harbour at high tide is 10 m . The table below shows the times of high and low tides on this day. \includegraphics[max width=\textwidth, alt={}, center]{d7f600c5-af4a-4708-bfd9-92b37a95c634-4_405_912_621_233}
Tidal Times
High/LowTime
Depth
(metres)
Low Tide5 a.m.2
High Tide11 a.m.10
Low Tide5 p.m.2
High Tide11 p.m.10
  1. Write down the period and amplitude of the motion.
  2. Let \(x \mathrm {~m}\) denote the height of water above mean level \(t\) hours after 5a.m. Find an expression for \(x\) in terms of \(t\).
  3. The depth of water must be at least 4 m for boats to safely use the harbour. Determine the earliest time, after low tide at 5 a.m., at which boats can safely leave the harbour and hence find the latest possible time of return before the next low tide.
  4. Calculate the rate at which the level of water is falling at 2 p.m.
WJEC Further Unit 6 2023 June Q4
15 marks Challenging +1.2
4. The diagram shows three light rods \(A B , B C\) and \(C A\) rigidly joined together so that \(A B C\) is a right-angled triangle with \(A B = 45 \mathrm {~cm} , A C = 28 \mathrm {~cm}\) and \(\widehat { A B } = 90 ^ { \circ }\). The rods support a uniform lamina, of density \(2 m \mathrm {~kg} / \mathrm { cm } ^ { 2 }\), in the shape of a quarter circle \(A D E\) with radius 12 cm and centre at the vertex \(A\). Three particles are attached to \(B C\) : one at \(B\), one at \(C\) and one at \(F\), the midpoint of \(B C\). The masses at \(C , F\) and \(B\) are \(50 m \mathrm {~kg} , 30 m \mathrm {~kg}\) and \(20 m \mathrm {~kg}\) respectively. \includegraphics[max width=\textwidth, alt={}, center]{d7f600c5-af4a-4708-bfd9-92b37a95c634-5_604_908_756_575}
  1. Calculate the distance of the centre of mass of the system from
    1. \(A C\),
    2. \(A B\).
  2. When the system is freely suspended from a point \(P\) on \(A C\), it hangs in equilibrium with \(A B\) vertical. Write down the length \(A P\).
  3. When the system is freely suspended from a point \(Q\) on \(A D\), it hangs in equilibrium with \(Q B\) making an angle of \(60 ^ { \circ }\) with the vertical. Find the distance \(A Q\).
WJEC Further Unit 6 2023 June Q5
16 marks Challenging +1.2
5. In this question, \(\mathbf { i }\) and \(\mathbf { j }\) represent unit vectors due east and due north respectively. Two smooth spheres \(P\) and \(Q\), of equal radii, are moving on a smooth horizontal surface. The mass of \(P\) is 2 kg and the mass of \(Q\) is 6 kg . The velocity of \(P\) is \(( 8 \mathbf { i } - 6 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) and the velocity of \(Q\) is \(( 4 \mathbf { i } + 10 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). At a particular instant, \(Q\) is positioned 12 m east and 48 m south of \(P\).
  1. Prove that \(P\) and \(Q\) will collide. At the instant the spheres collide, the line joining their centres is parallel to the vector \(\mathbf { j }\). Immediately after the collision, sphere \(Q\) has speed \(5 \mathrm {~ms} ^ { - 1 }\).
  2. Determine the coefficient of restitution between the spheres and hence calculate the velocity of \(P\) immediately after the collision.
  3. Find the magnitude of the impulse required to stop sphere \(P\) after the collision.
WJEC Further Unit 6 2023 June Q6
16 marks Challenging +1.8
6. The diagram on the left shows a train of mass 50 tonnes approaching a buffer at the end of a straight horizontal railway track. The buffer is designed to prevent the train from running off the end of the track. The buffer may be modelled as a light horizontal spring \(A B\), as shown in the diagram on the right, which is fixed at the end \(A\). The train strikes the buffer so that \(P\) makes contact with \(B\) at \(t = 0\) seconds. While \(P\) is in contact with \(B\), an additional resistive force of \(250000 v \mathrm {~N}\) will oppose the motion of the train, where \(v \mathrm {~ms} ^ { - 1 }\) is the speed of the train at time \(t\) seconds. The spring has natural length 1 m and modulus of elasticity 312500 N . At time \(t\) seconds, the compression of the spring is \(x\) metres. \includegraphics[max width=\textwidth, alt={}, center]{d7f600c5-af4a-4708-bfd9-92b37a95c634-7_358_1506_824_283}
  1. Show that, while \(P\) is in contact with \(B\), \(x\) satisfies the differential equation $$4 \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 20 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 25 x = 0$$
  2. Given that, when \(P\) first makes contact with \(B\), the speed of the train is \(U \mathrm {~ms} ^ { - 1 }\), find an expression for \(x\) in terms of \(U\) and \(t\).
  3. When the train comes to rest, the compression of the buffer is 0.3 m . Determine the speed of the train when it strikes the buffer.
  4. State which type of damping is described by the motion of \(P\). Give a reason for your answer.