Questions Further Unit 3 (40 questions)

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WJEC Further Unit 3 2019 June Q1
8 marks Standard +0.3
  1. The diagram shows a spring of natural length 0.15 m enclosed in a smooth horizontal tube. One end of the spring \(A\) is fixed and the other end \(B\) is compressed against a ball of mass \(0 \cdot 1 \mathrm {~kg}\). \includegraphics[max width=\textwidth, alt={}, center]{b430aa50-27e3-46f7-afef-7b8e75d46e1f-2_241_714_639_632}
Initially, the ball is held in equilibrium by a force of 21 N so that the compressed length of the spring is \(\frac { 2 } { 5 }\) of its natural length.
  1. Calculate the modulus of elasticity of the spring.
  2. The ball is released by removing the force. Determine the speed of the ball when the spring reaches its natural length. Give your answer correct to two significant figures.
WJEC Further Unit 3 2019 June Q2
10 marks Standard +0.3
2. A particle of mass 0.5 kg is moving under the action of a single force \(\mathbf { F N }\) so that its velocity \(\mathrm { v } \mathrm { ms } ^ { - 1 }\) at time \(t\) seconds is given by $$\mathbf { v } = 3 t ^ { 2 } \mathbf { i } - 8 t \mathbf { j } + 2 \mathrm { e } ^ { - t } \mathbf { k }$$
  1. Find an expression for the acceleration of the particle at time \(t \mathrm {~s}\).
  2. Determine an expression for F.v at time \(t \mathrm {~s}\).
  3. Find the kinetic energy of the particle at time \(t \mathrm {~s}\).
  4. Describe the relationship between the kinetic energy of a particle and the rate of working of the force acting on the particle. Verify this relationship using your answers to part (b) and part (c).
WJEC Further Unit 3 2019 June Q3
10 marks Standard +0.8
3. The position vectors \(\mathbf { r } _ { A }\) and \(\mathbf { r } _ { B }\), in kilometres, of two small aeroplanes \(A\) and \(B\) relative to a fixed point \(O\) are given by $$\begin{aligned} & \mathbf { r } _ { A } = ( 60 \mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k } ) + ( 168 \mathbf { i } + 132 \mathbf { j } ) t \\ & \mathbf { r } _ { B } = ( 62 \mathbf { i } + 3 \mathbf { k } ) + ( 160 \mathbf { i } + p \mathbf { j } + q \mathbf { k } ) t \end{aligned}$$ where \(t\) denotes the time in hours after 9:00 a.m. and \(p , q\) are constants.
The aeroplanes \(A\) and \(B\) are on course to collide.
  1. Show that \(p = 140\) and \(q = 4\).
  2. Find an expression for the square of the distance between \(A\) and \(B\) at time \(t\) hours after 9:00 a.m.
  3. Both aeroplanes have systems that activate an alarm if they come within 600 m of each other. Determine the time when the alarms are first activated.
WJEC Further Unit 3 2019 June Q4
9 marks Standard +0.3
4. A car of mass 1200 kg has an engine that is capable of producing a maximum power of 80 kW . When in motion, the car experiences a constant resistive force of 2000 N .
  1. Calculate the maximum possible speed of the car when travelling on a straight horizontal road.
  2. The car travels up a slope inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 20 }\). If the car's engine is working at \(80 \%\) capacity, calculate the acceleration of the car at the instant when its speed is \(20 \mathrm {~ms} ^ { - 1 }\).
  3. Explain why the assumption of a constant resistive force may be unrealistic.
WJEC Further Unit 3 2019 June Q5
8 marks Moderate -0.5
5. The diagram shows a fairground ride that consists of a number of seats suspended by chains that swing out as the centre rotates. \includegraphics[max width=\textwidth, alt={}, center]{b430aa50-27e3-46f7-afef-7b8e75d46e1f-4_711_718_466_678} When the ride rotates at a constant angular speed of \(\omega = 1.4 \mathrm { rads } ^ { - 1 }\), the seats move in a horizontal circle with each chain making an angle \(\theta\) with the vertical. Each of the seats and the chains may be modelled as light. Assume that all chains have the same length and are inextensible. When a man of mass 75 kg occupies a seat, the tension in the chain is \(490 \sqrt { 3 } \mathrm {~N}\).
  1. Show that \(\theta = 30 ^ { \circ }\).
  2. Calculate the length of each chain.
WJEC Further Unit 3 2019 June Q6
13 marks Standard +0.8
6. The diagram shows a rollercoaster at an amusement park where a car is projected from a launch point \(O\) so that it performs a loop before instantaneously coming to rest at point \(C\). The car then performs the same journey in reverse. \includegraphics[max width=\textwidth, alt={}, center]{b430aa50-27e3-46f7-afef-7b8e75d46e1f-5_677_1733_552_166} The loop section is modelled by considering the track to be a vertical circle of radius 10 m and the car as a particle of mass \(m\) kg moving on the inside surface of the circular loop. You may assume that the track is smooth. At point \(A\), which is the lowest point of the circle, the car has velocity \(u \mathrm {~ms} ^ { - 1 }\) such that \(u ^ { 2 } = 60 g\). When the car is at point \(B\) the radius makes an angle \(\theta\) with the downward vertical.
  1. Find, in terms of \(\theta\) and \(g\), an expression for \(v ^ { 2 }\), where \(v \mathrm {~ms} ^ { - 1 }\) is the speed of the car at \(B\).
  2. Show that \(R \mathrm {~N}\), the reaction of the track on the car at \(B\), is given by $$R = m g ( 4 + 3 \cos \theta ) .$$
  3. Explain why the expression for \(R\) in part (b) shows that the car will perform a complete loop.
  4. This model predicts that the car will stop at \(C\) at a vertical height of 30 m above \(A\). However, after the car has completed the loop, the track becomes rough and the car only reaches a point \(D\) at a vertical height of 28 m above \(A\). The resistance to motion of the car beyond the loop is of constant magnitude \(\frac { m g } { 32 } \mathrm {~N}\). Calculate the length of the rough track between \(A\) and \(D\).
WJEC Further Unit 3 2019 June Q7
12 marks Standard +0.3
7. Three spheres \(A , B , C\), of equal radii and each of mass \(m \mathrm {~kg}\), lie at rest on a smooth horizontal surface such that their centres are in a straight line with \(B\) between \(A\) and \(C\). The coefficient of restitution between \(A\) and \(B\) is \(e\). Sphere \(A\) is projected towards \(B\) with speed \(u \mathrm {~ms} ^ { - 1 }\) so that it collides with \(B\).
  1. Find expressions, in terms of \(e\) and \(u\), for the speed of \(A\) and the speed of \(B\) after they collide. You are now given that \(e = \frac { 1 } { 2 }\).
  2. Find, in terms of \(m\) and \(u\), the loss in kinetic energy due to the collision between \(A\) and \(B\).
  3. After the collision between \(A\) and \(B\), sphere \(B\) then collides with \(C\). The coefficient of restitution between \(B\) and \(C\) is \(e _ { 1 }\). Show that there will be no further collisions if \(e _ { 1 } \leqslant \frac { 1 } { 3 }\).
WJEC Further Unit 3 2024 June Q1
14 marks Standard +0.3
1. Two particles \(A\) and \(B\), of masses 2 kg and 5 kg respectively, are moving in the same direction along a smooth horizontal surface when they collide directly. Before the collision, \(B\) is moving with speed \(1.2 \mathrm {~ms} ^ { - 1 }\) and, immediately after the collision, its speed is \(3.8 \mathrm {~ms} ^ { - 1 }\). The coefficient of restitution between the particles \(A\) and \(B\) is 0.3 .
    1. Find the impulse exerted by \(A\) on \(B\).
    2. Given that the particles \(A\) and \(B\) were in contact for 0.08 seconds, find the average force between \(A\) and \(B\).
      [0pt] [4]
  2. Calculate the speed of \(A\) before and after the collision.
  3. After the collision between \(A\) and \(B\), particle \(B\) continues to move with speed \(3.8 \mathrm {~ms} ^ { - 1 }\) until it collides directly with a stationary particle \(C\) of mass 4 kg . When \(B\) and \(C\) collide, they coalesce to form a single particle.
    1. Write down the coefficient of restitution between \(B\) and \(C\).
    2. Determine the speed of the combined particle after the collision.
      \section*{PLEASE DO NOT WRITE ON THIS PAGE}
WJEC Further Unit 3 2024 June Q2
10 marks Standard +0.3
  1. The diagram below shows a light spring of natural length 1.2 m and modulus of elasticity 84 N . One end of the spring \(A\) is fixed and the other end is attached to an object \(P\) of mass 4 kg . \includegraphics[max width=\textwidth, alt={}, center]{ae23a093-1419-4be4-8285-951650dc5a35-06_542_451_466_808}
Initially, \(P\) is held at rest with the spring stretched to a total length of 2.2 m and \(A P\) vertical.
  1. Show that the elastic energy stored in the spring is 35 J .
  2. The object \(P\) is then released. Find the speed of \(P\) at the instant when the elastic energy in the spring is reduced to \(5 \cdot 6 \mathrm {~J}\).
WJEC Further Unit 3 2024 June Q3
5 marks Moderate -0.8
3. Three forces \(( 4 \mathbf { i } - 7 \mathbf { j } + 9 \mathbf { k } ) \mathrm { N } , ( 5 \mathbf { i } + 3 \mathbf { j } - 8 \mathbf { k } ) \mathrm { N }\) and \(( - 2 \mathbf { i } + 6 \mathbf { j } - 11 \mathbf { k } ) \mathrm { N }\) act on a particle.
  1. Find the resultant \(\mathbf { R }\) of the three forces.
  2. The points \(A\) and \(B\) have position vectors \(( 3 \mathbf { i } + 4 \mathbf { j } - 12 \mathbf { k } ) \mathrm { m }\) and \(( a \mathbf { i } + 7 \mathbf { j } - 10 \mathbf { k } ) \mathrm { m }\) respectively, where \(a\) is a constant. The work done by \(\mathbf { R }\) in moving the particle from \(A\) to \(B\) is 21 J . Calculate the value of \(a\).
    \section*{PLEASE DO NOT WRITE ON THIS PAGE}
WJEC Further Unit 3 2024 June Q4
7 marks Moderate -0.3
  1. The diagram below shows a particle \(P\), of mass 5 kg , attached to one end of a light inextensible string of length 3 m . The other end is fixed at a point \(A\). The particle \(P\) is moving in a horizonal circle with centre \(C\), where the point \(C\) is vertically below \(A\). The string is inclined at an angle \(\theta\) to the downward vertical, where \(\tan \theta = \frac { 20 } { 21 }\). \includegraphics[max width=\textwidth, alt={}, center]{ae23a093-1419-4be4-8285-951650dc5a35-10_725_796_639_628}
Find the speed of the particle.
WJEC Further Unit 3 2024 June Q5
9 marks Standard +0.8
5. A particle of mass 2 kg is moving under the action of a force \(\mathbf { F N }\) which, at time \(t\) seconds, is given by $$\mathbf { F } = 4 t \mathbf { i } - \sqrt { t } \mathbf { j } + 6 \mathbf { k }$$ When \(t = 1\), the velocity of the particle is \(\left( 3 \mathbf { i } - \frac { 1 } { 3 } \mathbf { j } - \mathbf { k } \right) \mathrm { ms } ^ { - 1 }\).
  1. Find an expression for the velocity vector of the particle at time \(t \mathrm {~s}\).
  2. Determine the values of \(t\) when the particle is moving in a direction perpendicular to the vector \(( - \mathbf { i } + 3 \mathbf { k } )\).
WJEC Further Unit 3 2024 June Q6
10 marks Challenging +1.2
6. A slope is inclined at an angle of \(5 ^ { \circ }\) to the horizontal. A car, of mass 1500 kg , has an engine that is working at a constant rate of \(P \mathrm {~W}\). The resistance to motion of the car is constant at 4500 N . When the car is moving up the slope, its acceleration is \(a \mathrm {~ms} ^ { - 2 }\) at the instant when its speed is \(10 \mathrm {~ms} ^ { - 1 }\). When the car is moving down the slope, its deceleration is \(a \mathrm {~ms} ^ { - 2 }\) at the instant when its speed is \(20 \mathrm {~ms} ^ { - 1 }\). Determine the value of \(P\) and the value of \(a\). \includegraphics[max width=\textwidth, alt={}, center]{ae23a093-1419-4be4-8285-951650dc5a35-14_87_1609_635_267}
WJEC Further Unit 3 2024 June Q7
15 marks Standard +0.3
7. One end of a light rod of length \(\frac { 5 } { 7 } \mathrm {~m}\) is attached to a fixed point \(O\) and the other end is attached to a particle \(P\), of mass \(m \mathrm {~kg}\). The particle \(P\) is projected from the point \(A\), which is vertically below \(O\), with a horizontal speed of \(u \mathrm {~ms} ^ { - 1 }\) so that it moves in a vertical circle with centre \(O\). When the rod \(O P\) is inclined at an angle \(\theta\) to the downward vertical, the speed of \(P\) is \(v \mathrm {~ms} ^ { - 1 }\) and the tension in the rod is \(T \mathrm {~N}\). \includegraphics[max width=\textwidth, alt={}, center]{ae23a093-1419-4be4-8285-951650dc5a35-16_629_593_646_735}
  1. Show that $$v ^ { 2 } = u ^ { 2 } - 14 + 14 \cos \theta$$
  2. Hence determine the least possible value of \(u ^ { 2 }\) for the particle to reach the highest point of the circle.
  3. Given that \(u ^ { 2 } = 32 \cdot 2\),
    1. find, in terms of \(m\) and \(\theta\), an expression for \(T\),
    2. calculate the range of values of \(\theta\) such that the rod is exerting a thrust.
      State whether your answer to (c)(ii) would be different if the mass of the particle was reduced. Give a reason for your answer. Additional page, if required. Write the question number(s) in the left-hand margin. only
WJEC Further Unit 3 2018 June Q1
13 marks Standard +0.8
Two objects, \(A\) of mass 18 kg and \(B\) of mass 7 kg, are moving in the same straight line on a smooth horizontal surface. Initially, they are moving with the same speed of \(4\text{ ms}^{-1}\) and in the same direction. Object \(B\) collides with a vertical wall which is perpendicular to its direction of motion and rebounds with a speed of \(3\text{ ms}^{-1}\). Subsequently, the two objects \(A\) and \(B\) collide directly. The coefficient of restitution between the two objects is \(\frac{5}{7}\).
  1. Find the coefficient of restitution between \(B\) and the wall. [1]
  2. Determine the speed of \(A\) and the speed of \(B\) immediately after the two objects collide. [7]
  3. Calculate the impulse exerted by \(A\) on \(B\) due to the collision and clearly state its units. [2]
  4. Find the loss in energy due to the collision between \(A\) and \(B\). [2]
  5. State the direction of motion of \(A\) relative to the wall after the collision with \(B\). [1]
WJEC Further Unit 3 2018 June Q2
10 marks Challenging +1.2
A car of mass 750 kg is moving on a slope inclined at an angle \(\theta\) to the horizontal, where \(\sin\theta = 0.1\). When the car's engine is working at a constant power \(PW\), the car can travel at maximum speeds of \(14\text{ ms}^{-1}\) up the slope and \(28\text{ ms}^{-1}\) down the slope. In each case, the resistance to motion experienced by the car is proportional to the square of its speed. Find the value of \(P\) and determine the resistance to the motion of the car when its speed is \(10.5\text{ ms}^{-1}\). [10]
WJEC Further Unit 3 2018 June Q3
10 marks Challenging +1.8
A light elastic string of natural length \(1.5\) m and modulus of elasticity \(490\) N has one end attached to a fixed point \(A\) and the other end attached to a particle \(P\) of mass \(30\) kg. Initially, \(P\) is held at rest vertically below \(A\) such that the distance \(AP\) is \(0.6\) m. It is then allowed to fall vertically.
  1. Calculate the distance \(AP\) when \(P\) is instantaneously at rest for the first time, giving your answer correct to 2 decimal places. [8]
  2. Estimate the distance \(AP\) when \(P\) is instantaneously at rest for the second time and clearly state one assumption that you have made in making your estimate. [2]
WJEC Further Unit 3 2018 June Q4
11 marks Standard +0.3
The position vector \(\mathbf{x}\) metres at time \(t\) seconds of an object of mass 3 kg may be modelled by $$\mathbf{x} = 3\sin t \mathbf{i} - 4\cos 2t \mathbf{j} + 5\sin t \mathbf{k}.$$
  1. Find an expression for the velocity vector \(\mathbf{v}\text{ ms}^{-1}\) at time \(t\) seconds and determine the least value of \(t\) when the object is instantaneously at rest. [7]
  2. Write down the momentum vector at time \(t\) seconds. [1]
  3. Find, in vector form, an expression for the force acting on the object at time \(t\) seconds. [3]
WJEC Further Unit 3 2018 June Q5
15 marks Challenging +1.8
A particle \(P\), of mass \(m\) kg, is attached to one end of a light inextensible string of length \(l\) m. The other end of the string is attached to a fixed point \(O\). Initially, \(P\) is held at rest with the string just taut and making an angle of 60° with the downward vertical. It is then given a velocity \(u\text{ ms}^{-1}\) perpendicular to the string in a downward direction.
    1. When the string makes an angle \(\theta\) with the downward vertical, the velocity of the particle is \(v\) and the tension in the string is \(T\). Find an expression for \(T\) in terms of \(m\), \(l\), \(v^2\) and \(\theta\).
    2. Given that \(P\) describes complete circles in the subsequent motion, show that \(u^2 > 4lg\). [10]
  2. Given that now \(u^2 = 3lg\), find the position of the string when circular motion ceases. Briefly describe the motion of \(P\) after circular motion has ceased. [3]
  3. The string is replaced by a light rigid rod. Given that \(P\) describes complete circles in the subsequent motion, show that \(u^2 > klg\), where \(k\) is to be determined. [2]
WJEC Further Unit 3 2018 June Q6
11 marks Challenging +1.2
A vehicle of mass 1200 kg is moving with a constant speed of \(40\text{ ms}^{-1}\) around a horizontal circular path which is on a test track banked at an angle of 60° to the horizontal. There is no tendency to sideslip at this speed. The vehicle is modelled as a particle.
  1. Calculate the normal reaction of the track on the vehicle. [3]
  2. Determine
    1. the radius of the circular path,
    2. the angular speed of the vehicle and clearly state its units. [6]
  3. What further assumption have you made in your solution to (b)? Briefly explain what effect this assumption has on the radius of the circular path. [2]
WJEC Further Unit 3 2022 June Q1
4 marks Moderate -0.8
A particle of mass 1.2 kg is attached to one end of a light inextensible string of length 2 m. The other end of the string is fixed to a point O on a smooth horizontal surface. With the string taut, the particle moves on the surface with constant speed \(8\text{ ms}^{-1}\) in a horizontal circle with centre O.
  1. Find the angular velocity of the particle about O. [2]
  2. Calculate the tension in the string. [2]
WJEC Further Unit 3 2022 June Q2
11 marks Moderate -0.3
The diagram below shows a woman standing at the end of a diving platform. She is about to dive into the water below. The woman has mass 60 kg and she may be modelled as a particle positioned at the end of the platform which is 10 m above the water. \includegraphics{figure_2} When the woman dives, she projects herself from the platform with a speed of \(7.8\text{ ms}^{-1}\).
  1. Find the kinetic energy of the woman when she leaves the platform. [2]
  2. Initially, the situation is modelled ignoring air resistance. By using conservation of energy, show that the model predicts that the woman enters the water with an approximate speed of \(16\text{ ms}^{-1}\). [6]
  3. Suppose that this model is refined to include air resistance so that the speed with which the woman enters the water is now predicted to be \(13\text{ ms}^{-1}\). Determine the amount of energy lost to air resistance according to the refined model. [3]
WJEC Further Unit 3 2022 June Q3
10 marks Standard +0.3
Two spheres \(A\) and \(B\), of equal radii, are moving towards each other on a smooth horizontal surface and collide directly. Sphere \(A\) has mass \(4m\) kg and sphere \(B\) has mass \(3m\) kg. Just before the collision, \(A\) has speed \(9\text{ ms}^{-1}\) and \(B\) has speed \(3.5\text{ ms}^{-1}\). Immediately after the collision, \(A\) has speed \(1.5\text{ ms}^{-1}\) in the direction of its original motion.
  1. Show that the speed of \(B\) immediately after the collision is \(6.5\text{ ms}^{-1}\). [3]
  2. Calculate the coefficient of restitution between \(A\) and \(B\). [3]
  3. Given that the magnitude of the impulse exerted by \(B\) on \(A\) is 36 Ns, find the value of \(m\). [3]
  4. Give a reason why it is not necessary to model the spheres as particles in this question. [1]
WJEC Further Unit 3 2022 June Q4
9 marks Standard +0.3
A particle \(P\) of mass 0.5 kg is in equilibrium under the action of three forces \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\). $$\mathbf{F}_1 = (9\mathbf{i} + 6\mathbf{j} - 12\mathbf{k})\text{N} \quad \text{and} \quad \mathbf{F}_2 = (6\mathbf{i} - 7\mathbf{j} + 3\mathbf{k})\text{N}.$$
  1. Find the force \(\mathbf{F}_3\). [2]
  2. Forces \(\mathbf{F}_2\) and \(\mathbf{F}_3\) are removed so that \(P\) moves in a straight line \(AB\) under the action of the single force \(\mathbf{F}_1\). The points \(A\) and \(B\) have position vectors \((2\mathbf{i} - 9\mathbf{j} + 7\mathbf{k})\) m and \((8\mathbf{i} - 5\mathbf{j} - \mathbf{k})\) m respectively. The particle \(P\) is initially at rest at \(A\).
    1. Verify that \(\mathbf{F}_1\) acts parallel to the vector \(\overrightarrow{AB}\).
    2. Find the work done by the force \(\mathbf{F}_1\) as the particle moves from \(A\) to \(B\).
    3. By using the work-energy principle, find the speed of \(P\) as it reaches \(B\). [7]
WJEC Further Unit 3 2022 June Q5
14 marks Challenging +1.2
One end of a light elastic string, of natural length 2.5 m and modulus of elasticity \(30g\) N, is fixed to a point O. A particle \(P\), of mass 2 kg, is attached to the other end of the string. Initially, \(P\) is held at rest at the point O. It is then released and allowed to fall under gravity.
  1. Show that, while the string is taut, $$v^2 = g(5 + 2x - 6x^2),$$ where \(v\text{ ms}^{-1}\) denotes the velocity of the particle when the extension in the string is \(x\) m. [6]
  2. Calculate the maximum extension of the string. [3]
    1. Find the extension of the string when \(P\) attains its maximum speed.
    2. Hence determine the maximum speed of \(P\). [5]