WJEC Further Unit 6 (Further Unit 6) Specimen

1. A ball of mass 0.4 kg is thrown vertically upwards from a point \(O\) with initial speed \(17 \mathrm {~ms} ^ { - 1 }\). When the ball is at a height of \(x \mathrm {~m}\) above \(O\) and its speed is \(v \mathrm {~ms} ^ { - 1 }\), the air resistance acting on the ball has magnitude \(0.01 v ^ { 2 } \mathrm {~N}\).
   1. Show that, as the ball is ascending, \(v\) satisfies the differential equation
   $$40 v \frac { \mathrm {~d} v } { \mathrm {~d} x } = - \left( 392 + v ^ { 2 } \right)$$
2. Find an expression for \(v\) in terms of \(x\).
3. Calculate, correct to two decimal places, the greatest height of the ball.
4. State, with a reason, whether the speed of the ball when it returns to \(O\) is greater than \(17 \mathrm {~ms} ^ { - 1 }\), less than \(17 \mathrm {~ms} ^ { - 1 }\) or equal to \(17 \mathrm {~ms} ^ { - 1 }\).

2. (a) Prove that the centre of mass of a uniform solid cone of height \(h\) and base radius \(b\) is at a height of \(\frac { 1 } { 4 } h\) above its base.
(b) A uniform solid cone \(C _ { 1 }\) has height 3 m and base radius 2 m . A smaller cone \(C _ { 2 }\) of height 2 m and base radius 1 m is contained symmetrically inside \(C _ { 1 }\). The bases of \(C _ { 1 }\) and \(C _ { 2 }\) have a common centre and the axis of \(C _ { 2 }\) is part of the axis of \(C _ { 1 }\). If \(C _ { 2 }\) is removed from \(C _ { 1 }\), show that the centre of mass of the remaining solid is at a distance of \(\frac { 11 } { 5 } \mathrm {~m}\) from the vertex of \(C _ { 1 }\).
(c) The remaining solid is suspended from a string which is attached to a point on the outer curved surface at a distance of \(\frac { 1 } { 3 } \sqrt { 13 } \mathrm {~m}\) from the vertex of \(C _ { 1 }\). Given that the axis of symmetry is inclined at an angle of \(\alpha\) to the vertical, find \(\tan \alpha\).

3. A body, of mass 9 kg , is projected along a straight horizontal track with an initial speed of \(20 \mathrm {~ms} ^ { - 1 }\). At time \(t \mathrm {~s}\) the body experiences a resistance of magnitude \(( 0.2 + 0.03 v ) \mathrm { N }\) where \(v \mathrm {~ms} ^ { - 1 }\) is its speed.

1. Show that \(v\) satisfies the differential equation $$900 \frac { \mathrm {~d} v } { \mathrm {~d} t } = - ( 20 + 3 v )$$
2. Find an expression for \(t\) in terms of \(v\).
3. Calculate, to the nearest second, the time taken for the body to come to rest.

4. The diagram shows a uniform lamina consisting of a rectangular section \(G P Q E\) with a semi-circular section EFG of radius 4 cm . Quadrants \(A P B\) and \(C Q D\) each with radius 2 cm are removed. Dimensions in cm are as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{3efc4ef6-8a80-4267-8e95-733200e875c5-3_758_604_497_651}

1. Write down the distance of the centre of mass of the lamina \(A B C D E F G\) from \(A G\).
2. Determine the distance of the centre of mass of the lamina \(A B C D E F G\) from \(B C\).
3. The lamina \(A B C D E F G\) is suspended freely from the point \(E\) and hangs in equilibrium. Calculate the angle EG makes with the vertical.

5. A particle \(A\), of mass \(m \mathrm {~kg}\), has position vector \(11 \mathbf { i } + 6 \mathbf { j }\) and a velocity \(2 \mathbf { i } + 7 \mathbf { j }\). At the same moment, second particle \(B\), of mass \(2 m \mathrm {~kg}\), has position vector \(7 \mathbf { i } + 10 \mathbf { j }\) and a velocity \(5 \mathbf { i } + 4 \mathbf { j }\).

1. If the particles continue to move with these velocities, prove that the particles will collide. Given that the particles coalesce after collision, find the common velocity of the particles after collision.
2. Determine the impulse exerted by \(A\) on \(B\).
3. Calculate the loss of kinetic energy caused by the collision.

6. The diagram shows a playground ride consisting of a seat \(P\), of mass 12 kg , attached to a vertical spring, which is fixed to a horizontal board. When the ride is at rest with nobody on it, the compression of the spring is 0.05 m .
\includegraphics[max width=\textwidth, alt={}, center]{3efc4ef6-8a80-4267-8e95-733200e875c5-4_305_654_1032_667} The spring is of natural length 0.75 m and modulus of elasticity \(\lambda\).

1. Find the value of \(\lambda\). The seat \(P\) is now pushed vertically downwards a further 0.05 m and is then released from rest.
2. Show that \(P\) makes Simple Harmonic oscillations of period \(\frac { \pi } { 7 }\) and write down the amplitude of the motion.
3. Find the maximum speed of \(P\).
4. Calculate the speed of \(P\) when it is at a distance 0.03 m from the equilibrium position.
5. Find the distance of \(P\) from the equilibrium position 1.6 s after it is released.[3]
6. State one modelling assumption you have made about the seat and one modelling assumption you have made about the spring.