CAIE M1 (Mechanics 1) 2011 June

1 A block is pulled for a distance of 50 m along a horizontal floor, by a rope that is inclined at an angle of \(\alpha ^ { \circ }\) to the floor. The tension in the rope is 180 N and the work done by the tension is 8200 J . Find the value of \(\alpha\).

2 A car of mass 1250 kg is travelling along a straight horizontal road with its engine working at a constant rate of \(P \mathrm {~W}\). The resistance to the car's motion is constant and equal to \(R \mathrm {~N}\). When the speed of the car is \(19 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) its acceleration is \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), and when the speed of the car is \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) its acceleration is \(0.16 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the values of \(P\) and \(R\).

3
\includegraphics[max width=\textwidth, alt={}, center]{8d64372d-0b9a-4b93-8c41-7096c813f714-2_443_825_755_661} A particle \(P\) is projected from the top of a smooth ramp with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and travels down a line of greatest slope. The ramp has length 6.4 m and is inclined at \(30 ^ { \circ }\) to the horizontal. Another particle \(Q\) is released from rest at a point 3.2 m vertically above the bottom of the ramp, at the same instant that \(P\) is projected (see diagram). Given that \(P\) and \(Q\) reach the bottom of the ramp simultaneously, find

1. the value of \(u\),
2. the speed with which \(P\) reaches the bottom of the ramp.
   \includegraphics[max width=\textwidth, alt={}, center]{8d64372d-0b9a-4b93-8c41-7096c813f714-3_609_1539_255_303} The diagram shows the velocity-time graphs for the motion of two particles \(P\) and \(Q\), which travel in the same direction along a straight line. \(P\) and \(Q\) both start at the same point \(X\) on the line, but \(Q\) starts to move \(T\) s later than \(P\). Each particle moves with speed \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for the first 20 s of its motion. The speed of each particle changes instantaneously to \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) after it has been moving for 20 s and the particle continues at this speed.

1. Make a rough copy of the diagram and shade the region whose area represents the displacement of \(P\) from \(X\) at the instant when \(Q\) starts. It is given that \(P\) has travelled 70 m at the instant when \(Q\) starts.
2. Find the value of \(T\).
3. Find the distance between \(P\) and \(Q\) when \(Q\) 's speed reaches \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
4. Sketch a single diagram showing the displacement-time graphs for both \(P\) and \(Q\), with values shown on the \(t\)-axis at which the speed of either particle changes.

5
\includegraphics[max width=\textwidth, alt={}, center]{8d64372d-0b9a-4b93-8c41-7096c813f714-4_620_623_255_760} A small block of mass 1.25 kg is on a horizontal surface. Three horizontal forces, with magnitudes and directions as shown in the diagram, are applied to the block. The angle \(\theta\) is such that \(\cos \theta = 0.28\) and \(\sin \theta = 0.96\). A horizontal frictional force also acts on the block, and the block is in equilibrium.

1. Show that the magnitude of the frictional force is 7.5 N and state the direction of this force.
2. Given that the block is in limiting equilibrium, find the coefficient of friction between the block and the surface. The force of magnitude 6.1 N is now replaced by a force of magnitude 8.6 N acting in the same direction, and the block begins to move.
3. Find the magnitude and direction of the acceleration of the block.

6 A lorry of mass 15000 kg climbs a hill of length 500 m at a constant speed. The hill is inclined at \(2.5 ^ { \circ }\) to the horizontal. The resistance to the lorry's motion is constant and equal to 800 N .

1. Find the work done by the lorry's driving force. On its return journey the lorry reaches the top of the hill with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and continues down the hill with a constant driving force of 2000 N . The resistance to the lorry's motion is again constant and equal to 800 N .
2. Find the speed of the lorry when it reaches the bottom of the hill.

7 A particle travels in a straight line from \(A\) to \(B\) in 20 s . Its acceleration \(t\) seconds after leaving \(A\) is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where \(a = \frac { 3 } { 160 } t ^ { 2 } - \frac { 1 } { 800 } t ^ { 3 }\). It is given that the particle comes to rest at \(B\).

1. Show that the initial speed of the particle is zero.
2. Find the maximum speed of the particle.
3. Find the distance \(A B\).