CAIE M1 (Mechanics 1) 2012 November

1 An object is released from rest at a height of 125 m above horizontal ground and falls freely under gravity, hitting a moving target \(P\). The target \(P\) is moving on the ground in a straight line, with constant acceleration \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). At the instant the object is released \(P\) passes through a point \(O\) with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the distance from \(O\) to the point where \(P\) is hit by the object.

2
\includegraphics[max width=\textwidth, alt={}, center]{2bb3c9bb-60f0-440d-a148-b4db3478ca31-2_212_625_528_761} Particles \(A\) and \(B\), of masses 0.3 kg and 0.2 kg respectively, are attached to the ends of a light inextensible string. \(A\) is held at rest on a rough horizontal table with the string passing over a small smooth pulley at the edge of the table. \(B\) hangs vertically below the pulley (see diagram). The system is released and \(B\) starts to move downwards with acceleration \(1.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find

1. the tension in the string after the system is released,
2. the frictional force acting on \(A\).

3
\includegraphics[max width=\textwidth, alt={}, center]{2bb3c9bb-60f0-440d-a148-b4db3478ca31-2_241_535_1247_806} A particle \(P\) of mass 0.5 kg rests on a rough plane inclined at angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.28\). A force of magnitude 0.6 N , acting upwards on \(P\) at angle \(\alpha\) from a line of greatest slope of the plane, is just sufficient to prevent \(P\) sliding down the plane (see diagram). Find

1. the normal component of the contact force on \(P\),
2. the frictional component of the contact force on \(P\),
3. the coefficient of friction between \(P\) and the plane.

4
\includegraphics[max width=\textwidth, alt={}, center]{2bb3c9bb-60f0-440d-a148-b4db3478ca31-2_387_1091_2019_525} Three coplanar forces of magnitudes \(8 \mathrm {~N} , 12 \mathrm {~N}\) and 2 N act at a point. The resultant of the forces has magnitude \(R \mathrm {~N}\). The directions of the three forces and the resultant are shown in the diagram. Find \(R\) and \(\theta\).

5 Particle \(P\) travels along a straight line from \(A\) to \(B\) with constant acceleration \(0.05 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Its speed at \(A\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its speed at \(B\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the time taken for \(P\) to travel from \(A\) to \(B\), and find also the distance \(A B\). Particle \(Q\) also travels along the same straight line from \(A\) to \(B\), starting from rest at \(A\). At time \(t \mathrm {~s}\) after leaving \(A\), the speed of \(Q\) is \(k t ^ { 3 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(k\) is a constant. \(Q\) takes the same time to travel from \(A\) to \(B\) as \(P\) does.
2. Find the value of \(k\) and find \(Q\) 's speed at \(B\).

6
\includegraphics[max width=\textwidth, alt={}, center]{2bb3c9bb-60f0-440d-a148-b4db3478ca31-3_382_1451_797_347} The diagram shows the vertical cross-section \(A B C D\) of a surface. \(B C\) is a circular arc, and \(A B\) and \(C D\) are tangents to \(B C\) at \(B\) and \(C\) respectively. \(A\) and \(D\) are at the same horizontal level, and \(B\) and \(C\) are at heights 2.7 m and 3.0 m respectively above the level of \(A\) and \(D\). A particle \(P\) of mass 0.2 kg is given a velocity of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(A\), in the direction of \(A B\) (see diagram). The parts of the surface containing \(A B\) and \(B C\) are smooth.

1. Find the decrease in the speed of \(P\) as \(P\) moves along the surface from \(B\) to \(C\). The part of the surface containing \(C D\) exerts a constant frictional force on \(P\), as it moves from \(C\) to \(D\), and \(P\) comes to rest as it reaches \(D\).
2. Find the speed of \(P\) when it is at the mid-point of \(C D\).

7 A car of mass 1200 kg moves in a straight line along horizontal ground. The resistance to motion of the car is constant and has magnitude 960 N . The car's engine works at a rate of 17280 W .

1. Calculate the acceleration of the car at an instant when its speed is \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car passes through the points \(A\) and \(B\). While the car is moving between \(A\) and \(B\) it has constant speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Show that \(V = 18\). At the instant that the car reaches \(B\) the engine is switched off and subsequently provides no energy. The car continues along the straight line until it comes to rest at the point \(C\). The time taken for the car to travel from \(A\) to \(C\) is 52.5 s .
3. Find the distance \(A C\).