WJEC Further Unit 3 (Further Unit 3) 2024 June

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. 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]
      \end{tabular} & Examiner only
      \hline \end{tabular} \end{center}
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}

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 \(5 \cdot 6 \mathrm {~J}\).

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}

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.

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 \(( - \mathbf { i } + 3 \mathbf { k } )\).

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}

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