WJEC Further Unit 3 (Further Unit 3) Specimen

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 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. [3]
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. [5]
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. [4]

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 \(2\sqrt{3}\) m horizontally from \(O\). It is then released and allowed to fall under gravity.

1. Show that the speed of \(P\) when it first begins to move in a circle is \(\sqrt{3g}\). [4]
2. In the subsequent motion, when the string first makes an angle of 45° with the downwards vertical,
   1. calculate the speed \(v\) of \(P\),
   2. determine the tension in the string. [8]

At time \(t = 0\) s, the position vector of an object \(A\) is \(\mathbf{i}\) m and the position vector of another object \(B\) is \(3\mathbf{i}\) m. The constant velocity vector of \(A\) is \(2\mathbf{i} + 5\mathbf{j} - 4k\) ms\(^{-1}\) and the constant velocity vector of \(B\) is \(\mathbf{i} + 3\mathbf{j} - 5k\) ms\(^{-1}\). Determine the value of \(t\) when \(A\) and \(B\) are closest together and find the least distance between \(A\) and \(B\). [9]

Relative to a fixed origin \(O\), the position vector \(\mathbf{r}\) m at time \(t\) s of a particle \(P\), of mass 0.4 kg, is given by $$\mathbf{r} = e^{2t}\mathbf{i} + \sin(2t)\mathbf{j} + \cos(2t)\mathbf{k}.$$

1. Show that the velocity vector \(\mathbf{v}\) and the position vector \(\mathbf{r}\) are never perpendicular to each other. [6]
2. Given that the speed of \(P\) at time \(t\) is \(v\) ms\(^{-1}\), show that $$v^2 = 4e^{4t} + 4.$$ [2]
3. Find the kinetic energy of \(P\) at time \(t\). [1]
4. Calculate the work done by the force acting on \(P\) in the interval \(0 < t < 1\). [2]
5. Determine an expression for the rate at which the force acting on \(P\) is working at time \(t\). [2]

A particle of mass \(m\) 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 ms\(^{-1}\). Calculate the angle the string makes with the vertical. [6]

\includegraphics{figure_6} A particle of mass 5 kg is attached to a string \(AB\) and a rod \(BC\) at the point \(B\). The string \(AB\) is light and elastic with modulus \(\lambda\) N and natural length 2 m. The rod \(BC\) 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 \(AC = 2\) m. When the particle rests in equilibrium, \(AB\) makes an angle of 50° with the downward vertical.

1. Determine, in terms of \(\lambda\), the tension in the string \(AB\). [3]
2. Calculate, in terms of \(\lambda\), the energy stored in the string \(AB\). [2]
3. Find, in terms of \(\lambda\), the thrust in the rod \(BC\). [4]

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 \(P\) W. The constant resistance to motion of the vehicle is \(R\) N. At the instant the vehicle is moving with velocity \(\frac{16}{5}\) ms\(^{-1}\), its acceleration is 2 ms\(^{-2}\). The maximum velocity of the vehicle is \(\frac{16}{3}\) ms\(^{-1}\). Determine the value of \(P\) and the value of \(R\). [9]