Edexcel M1 (Mechanics 1)

A tennis ball, moving horizontally, hits a wall at \(25 \text{ ms}^{-1}\) and rebounds along the same straight line at \(15 \text{ ms}^{-1}\). The impulse exerted by the wall on the ball has magnitude \(12\) Ns.

1. Calculate the mass of the ball. [4 marks]
2. State any modelling assumptions that you have made. [2 marks]

Forces of magnitude \(4\) N, \(5\) N and \(8\) N act on a particle in directions whose bearings are \(000°\), \(090°\) and \(210°\) respectively. Find the magnitude of the resultant force and the bearing of the direction in which it acts. [7 marks] \includegraphics{figure_2}

A packing-case, of mass \(60\) kg, is standing on the floor of a lift. The mass of the lift-cage is \(200\) kg. The lift-cage is raised and lowered by means of a cable attached to its roof. In each of the following cases, find the magnitude of the force exerted by the floor of the lift-cage on the packing-case and the tension in the cable supporting the lift:

1. The lift is descending with constant speed. [3 marks]
2. The lift is ascending and accelerating at \(1.2 \text{ ms}^{-2}\). [4 marks]
3. State any modelling assumptions you have made. [2 marks]

\(AB\) is a light rod. Forces \(\mathbf{F}\), \(\mathbf{G}\) and \(\mathbf{H}\), of magnitudes \(3\) N, \(2\) N and \(6\) N respectively, act upwards at right angles to the rod in a vertical plane at points dividing \(AB\) in the ratio \(1:4:2:4\), as shown. \includegraphics{figure_4} A single force \(\mathbf{P}\) is applied downwards at the point \(C\) to keep the rod horizontal in equilibrium.

1. State the magnitude of \(\mathbf{P}\). [1 mark]
2. Show that \(AC:CB = 5:6\). [5 marks]

Two particles, of weights \(3\) N and \(k\) N, are now placed on the rod at \(A\) and \(B\) respectively, while the same upward forces \(\mathbf{F}\), \(\mathbf{G}\) and \(\mathbf{H}\) act as before. It is found that a single downward force at the same point \(C\) as before keeps \(AB\) horizontal under gravity.

1. Find the value of \(k\). [6 marks]

Two smooth spheres \(A\) and \(B\), of masses \(2m\) and \(m\) respectively, are connected by a light inextensible string which passes over a smooth fixed pulley as shown. \(A\) is initially at rest on the rough horizontal surface of a table, the coefficient of friction between \(A\) and the table being \(\frac{2}{7}\). \(B\) hangs freely on the end of the vertical portion of the string. \includegraphics{figure_5} \(A\) is now given an impulse, directed away from the pulley, of magnitude \(5m\) Ns.

1. Show that the system starts to move with speed \(2.5 \text{ ms}^{-1}\). [1 mark]
2. State which modelling assumption ensures that the tensions in the two sections of the string can be taken to be equal. [1 mark]

Given that \(A\) comes to rest before it reaches the edge of the table and before \(B\) hits the pulley,

1. find the time taken for the system to come to rest. [7 marks]
2. Find the distance travelled by \(A\) before it first comes to rest. [4 marks]

The diagram shows the velocity-time graph for a cyclist's journey. Each section has constant acceleration or deceleration and the three sections are of equal duration \(x\) seconds each. \includegraphics{figure_6} Given that the total distance travelled is \(792\) m,

1. find the value of \(x\) and the acceleration for the first section of the journey. [6 marks]

Another cyclist covers the same journey in three sections of equal duration, accelerating at \(\frac{1}{11} \text{ ms}^{-2}\) for the first section, travelling at constant speed for the second section and decelerating at \(\frac{1}{11} \text{ ms}^{-2}\) for the third section.

1. Find the time taken by this cyclist to complete the journey. [6 marks]
2. Show that the maximum speeds of both cyclists are the same. [2 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.

1. Find the distance \(XY\). [2 marks]

A particle \(P\) of mass \(2\) kg moves from \(X\) to \(Y\) in \(4\) seconds, in a straight line at a constant speed.

1. Show that the velocity vector of \(P\) is \((2\mathbf{i} + 1.5\mathbf{j}) \text{ ms}^{-1}\). [3 marks]

The particle continues beyond \(Y\) with the same constant velocity.

1. Write down an expression for the position vector of \(P\) \(t\) seconds after leaving \(X\). [2 marks]
2. Find the value of \(t\) when \(P\) is at the point with position vector \((16\mathbf{i} + 4\mathbf{j})\) m. [2 marks]

When it is moving with the same constant speed, \(P\) collides directly with another particle \(Q\), of mass \(4\) kg, which is at rest. \(P\) and \(Q\) coalesce and move together as a single particle.

1. Find the velocity vector of the combined particle after the collision. [5 marks]