Questions — WJEC Further Unit 3 (34 questions)

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WJEC Further Unit 3 2024 June Q5
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
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 \(a\).
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WJEC Further Unit 3 2024 June Q7
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 \(T \mathrm {~N}\).
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  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 Specimen Q1
\begin{enumerate} \item By burning a charge, a cannon fires a cannon ball of mass 12 kg horizontally. As the cannon ball leaves the cannon, its speed is \(600 \mathrm {~ms} ^ { - 1 }\). The recoiling part of the cannon has a mass of 1600 kg .
  1. Determine the speed of the recoiling part immediately after the cannon ball leaves the cannon.
  2. Find the energy created by the burning of the charge. State any assumption you have made in your solution and briefly explain how the assumption affects your answer.
  3. Calculate the constant force needed to bring the recoiling part to rest in 1.2 m . State, with a reason, whether your answer is an overestimate or an underestimate of the actual force required. \item A particle \(P\), of mass 3 kg , is attached to a fixed point \(O\) by a light inextensible string of length 4 m . Initially, particle \(P\) is held at rest at a point which is \(2 \sqrt { 3 } \mathrm {~m}\) horizontally from \(O\). It is then released and allowed to fall under gravity.
WJEC Further Unit 3 Specimen Q5
5. A particle of mass \(m \mathrm {~kg}\) is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point \(O\). The particle is set in motion such that it moves in a horizontal circle of radius 2 m with constant speed \(4.8 \mathrm {~ms} ^ { - 1 }\). Calculate the angle the string makes with the vertical.
WJEC Further Unit 3 Specimen Q6
6. A particle of mass 5 kg is attached to a string \(A B\) and a rod \(B C\) at the point \(B\). The string \(A B\) is light and elastic with modulus \(\lambda \mathrm { N }\) and natural length 2 m . The rod \(B C\) is light and of length 2 m . The end \(A\) of the string is attached to a fixed point and the end \(C\) of the rod is attached to another fixed point such that \(A\) is vertically above \(C\) with \(A C = 2 \mathrm {~m}\). When the particle rests in equilibrium, \(A B\) makes an angle of \(50 ^ { \circ }\) with the downward vertical.
  1. Determine, in terms of \(\lambda\), the tension in the string \(A B\).
  2. Calculate, in terms of \(\lambda\), the energy stored in the string \(A B\).
  3. Find, in terms of \(\lambda\), the thrust in the rod \(B C\).
WJEC Further Unit 3 Specimen Q7
7. A vehicle of mass 6000 kg is moving up a slope inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 6 } { 49 }\). The vehicle's engine exerts a constant power of \(P \mathrm {~W}\). The constant resistance to motion of the vehicle is \(R \mathrm {~N}\). At the instant the vehicle is moving with velocity \(\frac { 16 } { 5 } \mathrm {~ms} ^ { - 1 }\), its acceleration is \(2 \mathrm {~ms} ^ { - 2 }\). The maximum velocity of the vehicle is \(\frac { 16 } { 3 } \mathrm {~ms} ^ { - 1 }\). Determine the value of \(P\) and the value of \(R\).
WJEC Further Unit 3 2018 June Q2
  1. Calculate the distance \(A P\) when \(P\) is instantaneously at rest for the first time, giving your answer correct to 2 decimal places.
  2. Estimate the distance \(A P\) when \(P\) is instantaneously at rest for the second time and clearly state one assumption that you have made in making your estimate. \item The position vector \(\mathbf { x }\) metres at time \(t\) seconds of an object of mass 3 kg may be modelled by \end{enumerate} $$\mathbf { x } = 3 \sin t \mathbf { i } - 4 \cos 2 t \mathbf { j } + 5 \sin t \mathbf { k }$$
  3. Find an expression for the velocity vector \(\mathbf { v } \mathrm { ms } ^ { - 1 }\) at time \(t\) seconds and determine the least value of \(t\) when the object is instantaneously at rest.
  4. Write down the momentum vector at time \(t\) seconds.
  5. Find, in vector form, an expression for the force acting on the object at time \(t\) seconds.
WJEC Further Unit 3 Specimen Q2
  1. Show that the speed of \(P\) when it first begins to move in a circle is \(\sqrt { 3 g }\).
  2. In the subsequent motion, when the string first makes an angle of \(45 ^ { \circ }\) with the downwards vertical,
    1. calculate the speed \(v\) of \(P\),
    2. determine the tension in the string. \item At time \(t = 0 \mathrm {~s}\), the position vector of an object \(A\) is \(\mathbf { i } \mathrm { m }\) and the position vector of another object \(B\) is \(3 \mathbf { i } \mathrm {~m}\). The constant velocity vector of \(A\) is \(2 \mathbf { i } + 5 \mathbf { j } - 4 \mathbf { k } \mathrm {~ms} ^ { - 1 }\) and the constant velocity vector of \(B\) is \(\mathbf { i } + 3 \mathbf { j } - 5 \mathbf { k } \mathrm {~ms} ^ { - 1 }\). Determine the value of \(t\) when \(A\) and \(B\) are closest together and find the least distance between \(A\) and \(B\). \item Relative to a fixed origin \(O\), the position vector \(\mathbf { r } \mathrm { m }\) at time \(t \mathrm {~s}\) of a particle \(P\), of mass 0.4 kg , is given by \end{enumerate} $$\mathbf { r } = \mathrm { e } ^ { 2 t } \mathbf { i } + \sin ( 2 t ) \mathbf { j } + \cos ( 2 t ) \mathbf { k }$$
  3. Show that the velocity vector \(\mathbf { v }\) and the position vector \(\mathbf { r }\) are never perpendicular to each other.
  4. Given that the speed of \(P\) at time \(t\) is \(v _ { \mathrm { ms } } ^ { - 1 }\), show that $$v ^ { 2 } = 4 \mathrm { e } ^ { 4 t } + 4$$
  5. Find the kinetic energy of \(P\) at time \(t\).
  6. Calculate the work done by the force acting on \(P\) in the interval \(0 < t < 1\).
  7. Determine an expression for the rate at which the force acting on \(P\) is working at time \(t\).