CAIE M1 (Mechanics 1) 2014 June

A particle moves in a straight line. At time \(t\) seconds, its displacement from a fixed point is \(s\) metres, where $$s = t^3 - 6t^2 + 9t$$

1. Find expressions for the velocity and acceleration of the particle at time \(t\). [4]
2. Find the times when the particle is at rest. [2]

A block of mass \(2\) kg is placed on a rough horizontal surface. The coefficient of friction between the block and the surface is \(0.3\).

1. Calculate the maximum frictional force that can act on the block. [2]
2. A horizontal force of \(5\) N is applied to the block. Calculate the acceleration of the block. [3]

\includegraphics{figure_3} A particle is moving under the action of three forces as shown in the diagram. The particle is in equilibrium. Find the magnitudes of forces \(P\) and \(Q\). [6]

A particle of mass \(0.5\) kg moves in a straight line under the action of a variable force. At time \(t\) seconds, the force is \((3t - 2)\) N in the direction of motion. Given that the particle starts from rest, find the velocity of the particle when \(t = 4\). [5]

\includegraphics{figure_5} A uniform rod AB has length \(2\) m and weight \(20\) N. The rod rests horizontally in equilibrium on two supports at points C and D, where AC = \(0.4\) m and BD = \(0.6\) m.

1. Find the reaction at each support. [4]
2. State what happens if the support at D is removed. [2]

\includegraphics{figure_6} A particle is projected vertically upward from ground level with speed \(u\) m s\(^{-1}\). The particle moves under gravity alone.

1. Find an expression for the maximum height reached by the particle. [3]

\includegraphics{figure_6b} The diagram shows a velocity-time graph for the motion of the particle.

1. Use the graph to find the value of \(u\). [2]
2. Find the time taken for the particle to return to ground level. [3]