Edexcel M1 (Mechanics 1)

A particle, \(P\), of mass 5 kg moves with speed 3 m s\(^{-1}\) along a smooth horizontal track. It strikes a particle \(Q\) of mass 2 kg which is at rest on the track. Immediately after the collision, \(P\) and \(Q\) move in the same direction with speeds \(v\) and 2v m s\(^{-1}\) respectively.

1. Calculate the value of \(v\). [3 marks]
2. Calculate the magnitude of the impulse received by \(Q\) on impact. [2 marks]

A particle \(P\) moves with a constant velocity \((3\mathbf{i} + 2\mathbf{j})\) m s\(^{-1}\) with respect to a fixed origin \(O\). It passes through the point \(A\) whose position vector is \((2\mathbf{i} + 11\mathbf{j})\) m at \(t = 0\).

1. Find the angle in degrees that the velocity vector of \(P\) makes with the vector \(\mathbf{i}\). [2 marks]
2. Calculate the distance of \(P\) from \(O\) when \(t = 2\). [4 marks]

A car of mass 1250 kg is moving at constant speed up a hill, inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac{1}{10}\). The driving force produced by the engine is 1800 N.

1. Calculate the resistance to motion which the car experiences. [4 marks]

At the top of the hill, the road becomes horizontal.

1. Find the initial acceleration of the car. [3 marks]

A non-uniform plank \(AB\) of mass 20 kg and length 6 m is supported at both ends so that it is horizontal. When a woman of mass 60 kg stands on the plank at a distance of 2 m from \(B\), the magnitude of the reaction at \(A\) is 35g N.

1. Suggest a suitable model for
   1. the plank, [2 marks]
   2. the woman.
2. Calculate the magnitude of the reaction at \(B\), giving your answer in terms of \(g\). [2 marks]
3. Explain briefly, in the context of the problem, the term 'non-uniform'. [2 marks]
4. Find the distance of the centre of mass of the plank from \(A\). [4 marks]

\includegraphics{figure_1} The points \(A\), \(O\) and \(B\) lie on a straight horizontal track as shown in Figure 1. \(A\) is 20 m from \(O\) and \(B\) is on the other side of \(O\) at a distance \(x\) m from \(O\). At time \(t = 0\), a particle \(P\) starts from rest at \(O\) and moves towards \(B\) with uniform acceleration of 3 m s\(^{-2}\). At the same instant, another particle \(Q\), which is at the point \(A\), is moving with a velocity of 3 m s\(^{-1}\) in the direction of \(O\) with uniform acceleration of 4 m s\(^{-2}\) in the same direction. Given that the \(Q\) collides with \(P\) at \(B\), find the value of \(x\). [10 marks]

A sledge of mass 4 kg rests in limiting equilibrium on a rough slope inclined at an angle 10° to the horizontal. By modelling the sledge as a particle,

1. show that the coefficient of friction, \(\mu\), between the sledge and the ground is 0.176 correct to 3 significant figures. [6 marks]

The sledge is placed on a steeper part of the slope which is inclined at an angle 30° to the horizontal. The value of \(\mu\) remains unchanged.

1. Find the minimum extra force required along the line of greatest slope to prevent the sledge from slipping down the hill. [5 marks]

Whilst looking over the edge of a vertical cliff, 122.5 metres in height, Jim dislodges a stone. The stone falls freely from rest towards the sea below. Ignoring the effect of air resistance,

1. calculate the time it would take for the stone to reach the sea, [3 marks]
2. find the speed with which the stone would hit the water. [2 marks]

Two seconds after the stone begins to fall, Jim throws a tennis ball downwards at the stone. The tennis ball's initial speed is \(u\) m s\(^{-1}\) and it hits the stone before they both reach the water.

1. Find the minimum value of \(u\). [5 marks]
2. If you had taken air resistance into account in your calculations, what effect would this have had on your answer to part (c)? Explain your answer. [2 marks]

\includegraphics{figure_2} Figure 2 shows two particles \(P\) and \(Q\), of mass 3 kg and 2 kg respectively, attached to the ends of a light, inextensible string which passes over a smooth, fixed pulley. The system is released from rest with \(P\) and \(Q\) at the same level 1.5 metres above the ground and 2 metres below the pulley.

1. Show that the initial acceleration of the system is \(\frac{g}{5}\) m s\(^{-2}\). [4 marks]
2. Find the tension in the string. [2 marks]
3. Find the speed with which \(P\) hits the ground. [3 marks]

When \(P\) hits the ground, it does not rebound.

1. What is the closest that \(Q\) gets to the pulley. [5 marks]