Edexcel M2 (Mechanics 2)

A particle \(P\) moves in a straight line so that its velocity \(v\) ms\(^{-1}\) at time \(t\) seconds is given, for \(t > 1\), by the formula \(v = 2t + \frac{8}{t^2}\). Find the time when the acceleration of \(P\) is zero. [5 marks]

\includegraphics{figure_2} A key is modelled as a lamina which consists of a circle of radius 3 cm, with a circle of radius 1 cm removed from its centre, attached to a rectangle of length 8 cm and width 1 cm, with a rectangle measuring 3 cm by 1 cm fixed to its end as shown. Calculate the distance of the centre of mass of the key from the line marked \(AB\). [7 marks]

A van of mass 1600 kg is moving with constant speed down a straight road inclined at 7° to the horizontal. The non-gravitational resistance to the van's motion has a constant magnitude of 2000 N and the engine of the van is working at a rate of 1.5 kW. Find

1. the constant speed of the van, [5 marks]
2. the acceleration of the van if the resistance is suddenly reduced to 1900 N. [2 marks]

\(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular unit vectors in a horizontal plane. A body of mass 1 kg moves under the action of a constant force \((4\mathbf{i} + 5\mathbf{j})\) N. The body moves from the point \(P\) with position vector \((-3\mathbf{i} - 15\mathbf{j})\) m to the point \(Q\) with position vector \(9\mathbf{i}\) m.

1. Find the work done by the force in moving the body from \(P\) to \(Q\). [5 marks]
2. Given that the body started from rest at \(P\), find its speed when it is at \(Q\). [5 marks]

Two railway trucks \(A\) and \(B\), whose masses are \(6m\) and \(5m\) respectively, are moving in the same direction along a straight track with speeds \(5u\) and \(3u\) respectively, and collide directly. Immediately after this impact the speeds of \(A\) and \(B\) are \(v\) and \(kv\) respectively, in the same direction as before. The coefficient of restitution between \(A\) and \(B\) is \(e\). Modelling the trucks as particles,

1. show that
   1. \(v = \frac{45u}{5k + 6}\),
   2. \(v = \frac{2eu}{k - 1}\).
   [8 marks]
2. Use the fact that \(0 \leq e \leq 1\) to deduce the range of possible values of \(k\). [5 marks]

A piece of lead and a table tennis ball are dropped together from a point \(P\) near the top of the Leaning Tower of Pisa. The lead hits the ground after 3.3 seconds.

1. Calculate the height above ground from which the lead was dropped. [2 marks]

According to a simple model, the ball hits the ground at the same time as the lead.

1. State why this may not be true in practice and describe a refinement to the model which could lead to a more realistic solution. [2 marks]

The piece of lead is now thrown again from \(P\), with speed 7 ms\(^{-1}\) at an angle of 30° to the horizontal, as shown. \includegraphics{figure_6}

1. Find expressions in terms of \(t\) for \(x\) and \(y\), the horizontal and vertical displacements respectively of the piece of lead from \(P\) at time \(t\) seconds after it is thrown. [4 marks]
2. Deduce that \(y = \frac{\sqrt{3}}{3}x - \frac{2}{15}x^2\). [3 marks]
3. Find the speed of the piece of lead when it has travelled 10 m horizontally from \(P\). [5 marks]

\includegraphics{figure_7} A uniform ladder \(AB\), of mass \(m\) kg and length \(2a\) m, rests with its upper end \(A\) in contact with a smooth vertical wall and its lower end \(B\) in contact with a fixed peg on horizontal ground. The ladder makes an angle \(\alpha\) with the ground, where \(\tan \alpha = \frac{3}{4}\).

1. Show that the magnitude of the resultant force acting on the ladder at \(B\) is \(\frac{\sqrt{13}}{3}mg\). [7 marks]
2. Find, to the nearest degree, the direction of this resultant force at \(B\). [3 marks]

The peg will break when the horizontal force acting on it exceeds \(2mg\) N. A painter of mass \(6m\) kg starts to climb the ladder from \(B\).

1. Find, in terms of \(a\), the greatest distance up the ladder that the painter can safely climb. [7 marks]