Edexcel M2 (Mechanics 2)

A small ball \(A\) is moving with velocity \((7\mathbf{i} + 12\mathbf{j})\) ms\(^{-1}\). It collides in mid-air with another ball \(B\), of mass \(0.4\) kg, moving with velocity \((-\mathbf{i} + 7\mathbf{j})\) ms\(^{-1}\). Immediately after the collision, \(A\) has velocity \((-3\mathbf{i} + 4\mathbf{j})\) ms\(^{-1}\) and \(B\) has velocity \((6.5\mathbf{i} + 13\mathbf{j})\) ms\(^{-1}\). Calculate the mass of \(A\). [4 marks]

A stick of mass \(0.75\) kg is at rest with one end \(X\) on a rough horizontal floor and the other end \(Y\) leaning against a smooth vertical wall. The coefficient of friction between the stick and the floor is \(0.6\). Modelling the stick as a uniform rod, find the smallest angle that the stick can make with the floor before it starts to slip. \includegraphics{figure_2} [6 marks]

An engine of mass \(20\,000\) kg climbs a hill inclined at \(10°\) to the horizontal. The total non-gravitational resistance to its motion has magnitude \(35\,000\) N and the maximum speed of the engine on the hill is \(15\) ms\(^{-1}\).

1. Find, in kW, the maximum rate at which the engine can work. [4 marks]
2. Find the maximum speed of the engine when it is travelling on a horizontal track against the same non-gravitational resistance as before. [3 marks]

Relative to a fixed origin \(O\), the points \(X\) and \(Y\) have position vectors \((4\mathbf{i} - 5\mathbf{j})\) m and \((12\mathbf{i} + \mathbf{j})\) m respectively, where \(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular unit vectors in the directions due east and due north respectively. A particle \(P\) starts from \(X\), and \(t\) seconds later its position vector relative to \(O\) is \((2t + 4)\mathbf{i} + (kt^2 - 5)\mathbf{j}\).

1. Find the value of \(k\) if \(P\) takes \(4\) seconds to reach \(Y\). [3 marks]
2. Show that \(P\) has constant acceleration and find the magnitude and direction of this acceleration. [4 marks]

Three particles \(A\), \(B\) and \(C\), of equal size and each of mass \(m\), are at rest on the same straight line on a smooth horizontal surface. The coefficient of restitution between \(A\) and \(B\), and between \(B\) and \(C\), is \(e\). \(A\) is projected with speed \(7\) ms\(^{-1}\) and strikes \(B\) directly. \(B\) then collides with \(C\), which starts to move with speed \(4\) ms\(^{-1}\). Calculate the value of \(e\). [10 marks]

A rectangular piece of cardboard \(ABCD\), measuring \(30\) cm by \(12\) cm, has a semicircle of radius \(5\) cm removed from it as shown. \includegraphics{figure_6}

1. Calculate the distances of the centre of mass of the remaining piece of cardboard from \(AB\) and from \(BC\). [7 marks]

The remaining cardboard is suspended from \(A\) and hangs in equilibrium.

1. Find the angle made by \(AB\) with the vertical. [4 marks]

A rocket is fired from a fixed point \(O\). During the first phase of its motion its velocity, \(v\) ms\(^{-1}\), is given at time \(t\) seconds after firing by the formula $$v = pt^2 + qt.$$ \(5\) seconds after firing, the rocket is travelling at \(500\) ms\(^{-1}\). \(30\) seconds after firing, the rocket is travelling at \(12\,000\) ms\(^{-1}\).

1. Find the constants \(p\) and \(q\). [4 marks]
2. Sketch a velocity-time graph for the rocket for \(0 \leq t \leq 30\). [2 marks]
3. Find the initial acceleration of the rocket. [2 marks]
4. Find the distance of the rocket from \(O\) \(30\) seconds after firing. [4 marks]

From time \(t = 30\) onwards, the rocket maintains a constant speed of \(12\,000\) ms\(^{-1}\).

1. Find the average speed of the rocket during its first \(50\) seconds of motion. [3 marks]

A golf ball is hit with initial velocity \(u\) ms\(^{-1}\) at an angle of \(45°\) above the horizontal. The ball passes over a building which is \(15\) m tall at a distance of \(30\) m horizontally from the point where the ball was hit.

1. Find the smallest possible value of \(u\). [7 marks]

When \(u\) has this minimum value,

1. show that the ball does not rise higher than the top of the building. [4 marks]
2. Deduce the total horizontal distance travelled by the ball before it hits the ground. [2 marks]
3. Briefly describe two modelling assumptions that you have made. [2 marks]