CAIE M1 (Mechanics 1) 2022 June

Small smooth spheres \(A\) and \(B\), of equal radii and of masses \(5\text{kg}\) and \(3\text{kg}\) respectively, lie on a smooth horizontal plane. Initially \(B\) is at rest and \(A\) is moving towards \(B\) with speed \(8.5\text{ms}^{-1}\). The spheres collide and after the collision \(A\) continues to move in the same direction but with a quarter of the speed of \(B\).

1. Find the speed of \(B\) after the collision. [3]
2. Find the loss of kinetic energy of the system due to the collision. [2]

\includegraphics{figure_2} Coplanar forces of magnitudes \(60\text{N}\), \(20\text{N}\), \(16\text{N}\) and \(14\text{N}\) act at a point in the directions shown in the diagram. Find the magnitude and direction of the resultant force. [6]

Two particles \(A\) and \(B\), of masses \(2.4\text{kg}\) and \(1.2\text{kg}\) respectively, are connected by a light inextensible string which passes over a fixed smooth pulley. \(A\) is held at a distance of \(2.1\text{m}\) above a horizontal plane and \(B\) is \(1.5\text{m}\) above the plane. The particles hang vertically and are released from rest. In the subsequent motion \(A\) reaches the plane and does not rebound and \(B\) does not reach the pulley.

1. Show that the tension in the string before \(A\) reaches the plane is \(16\text{N}\) and find the magnitude of the acceleration of the particles before \(A\) reaches the plane. [4]
2. Find the greatest height of \(B\) above the plane. [3]

A particle \(A\), moving along a straight horizontal track with constant speed \(8\text{ms}^{-1}\), passes a fixed point \(O\). Four seconds later, another particle \(B\) passes \(O\), moving along a parallel track in the same direction as \(A\). Particle \(B\) has speed \(20\text{ms}^{-1}\) when it passes \(O\) and has a constant deceleration of \(2\text{ms}^{-2}\). \(B\) comes to rest when it returns to \(O\).

1. Find expressions, in terms of \(t\), for the displacement from \(O\) of each particle \(t\) seconds after \(B\) passes \(O\). [3]
2. Find the values of \(t\) when the particles are the same distance from \(O\). [3]
3. On the given axes, sketch the displacement-time graphs for both particles, for values of \(t\) from \(0\) to \(20\). [3] $$s \text{ (m)}$$ $$200$$ $$100$$ $$0 \quad 0 \quad 10 \quad 20 \quad t \text{ (s)}$$

\includegraphics{figure_5} A block of mass \(12\text{kg}\) is placed on a plane which is inclined at an angle of \(24°\) to the horizontal. A light string, making an angle of \(36°\) above a line of greatest slope, is attached to the block. The tension in the string is \(65\text{N}\) (see diagram). The coefficient of friction between the block and plane is \(\mu\). The block is in limiting equilibrium and is on the point of sliding up the plane. Find \(\mu\). [6]

A car of mass \(900\text{kg}\) is moving up a hill inclined at \(\sin^{-1} 0.12\) to the horizontal. The initial speed of the car is \(11\text{ms}^{-1}\). After \(12\text{s}\), the car has travelled \(150\text{m}\) up the hill and has speed \(16\text{ms}^{-1}\). The engine of the car is working at a constant rate of \(24\text{kW}\).

1. Find the work done against the resistive forces during the \(12\text{s}\). [5]
2. The car then travels along a straight horizontal road. There is a resistance to the motion of the car of \((1520 + 4v)\text{N}\) when the speed of the car is \(v\text{ms}^{-1}\). The car travels at a constant speed with the engine working at a constant rate of \(32\text{kW}\). Find this speed. [3]

A particle \(P\) moves in a straight line. The velocity \(v\text{ms}^{-1}\) at time \(t\) seconds is given by $$v = 0.5t \quad \text{for } 0 \leqslant t \leqslant 10,$$ $$v = 0.25t^2 - 8t + 60 \quad \text{for } 10 < t \leqslant 20.$$

1. Show that there is an instantaneous change in the acceleration of the particle at \(t = 10\). [3]
2. Find the total distance covered by \(P\) in the interval \(0 \leqslant t \leqslant 20\). [6]