CAIE M1 (Mechanics 1) 2024 November

Two particles, of masses \(1.8\) kg and \(1.2\) kg, are connected by a light inextensible string that passes over a fixed smooth pulley. The particles hang vertically. The system is released from rest. Find the magnitude of the acceleration of the particles and find the tension in the string. [4]

\includegraphics{figure_2} A particle of mass \(7.5\) kg, starting from rest at \(A\), slides down an inclined plane \(AB\). The point \(B\) is \(12.5\) metres vertically below the level of \(A\), as shown in the diagram.

1. Given that the plane is smooth, use an energy method to find the speed of the particle at \(B\). [2]
2. It is given instead that the plane is rough and the particle reaches \(B\) with a speed of \(8 \text{ ms}^{-1}\). The plane is \(25\) m long and the constant frictional force has magnitude \(F\) N. Find the value of \(F\). [3]

\includegraphics{figure_3} Coplanar forces of magnitudes \(52\) N, \(39\) N and \(P\) N act at a point in the directions shown in the diagram. The system is in equilibrium. Find the values of \(P\) and \(\theta\). [4]

A bus travels between two stops, \(A\) and \(B\). The bus starts from rest at \(A\) and accelerates at a constant rate of \(a \text{ ms}^{-2}\) until it reaches a speed of \(16 \text{ ms}^{-1}\). It then travels at this constant speed before decelerating at a constant rate of \(0.75 \text{ ms}^{-2}\), coming to rest at \(B\). The total time for the journey is \(240\) s.

1. Sketch the velocity-time graph for the bus's journey from \(A\) to \(B\). [1]
2. Find an expression, in terms of \(a\), for the length of time that the bus is travelling with constant speed. [2]
3. Given that the distance from \(A\) to \(B\) is \(3000\) m, find the value of \(a\). [3]

A particle, \(A\), is projected vertically upwards from a point \(O\) with a speed of \(80 \text{ ms}^{-1}\). One second later a second particle, \(B\), with the same mass as \(A\), is projected vertically upwards from \(O\) with a speed of \(100 \text{ ms}^{-1}\). At time \(T\) s after the first particle is projected, the two particles collide and coalesce to form a particle \(C\).

1. Show that \(T = 3.5\). [4]
2. Find the height above \(O\) at which the particles collide. [1]
3. Find the time from \(A\) being projected until \(C\) returns to \(O\). [5]

\includegraphics{figure_6} A particle of mass \(1.2\) kg is placed on a rough plane which is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac{3}{5}\). The particle is kept in equilibrium by a horizontal force of magnitude \(P\) N acting in a vertical plane containing a line of greatest slope (see diagram). The coefficient of friction between the particle and the plane is \(0.15\). Find the least possible value of \(P\). [6]

A car has mass \(1200\) kg. When the car is travelling at a speed of \(v \text{ ms}^{-1}\), there is a resistive force of magnitude \(kv\) N. The maximum power of the car's engine is \(92.16\) kW.

1. The car travels along a straight level road.
   1. The car has a greatest possible constant speed of \(48 \text{ ms}^{-1}\). Show that \(k = 40\). [1]
   2. At an instant when its speed is \(45 \text{ ms}^{-1}\), find the greatest possible acceleration of the car. [3]
2. The car now travels at a constant speed up a hill inclined at an angle of \(\sin^{-1} 0.15\) to the horizontal. Find the greatest possible speed of the car going up the hill. [4]

A particle \(P\) moves in a straight line, passing through a point \(O\) with velocity \(4.2 \text{ ms}^{-1}\). At time \(t\) s after \(P\) passes \(O\), the acceleration, \(a \text{ ms}^{-2}\), of \(P\) is given by \(a = 0.6t - 2.7\). Find the distance \(P\) travels between the times at which it is at instantaneous rest. [7]