CAIE M1 (Mechanics 1) 2011 November

1 A racing cyclist, whose mass with his cycle is 75 kg , works at a rate of 720 W while moving on a straight horizontal road. The resistance to the cyclist's motion is constant and equal to \(R \mathrm {~N}\).

1. Given that the cyclist is accelerating at \(0.16 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) at an instant when his speed is \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the value of \(R\).
2. Given that the cyclist's acceleration is positive, show that his speed is less than \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

2 A block of mass 6 kg is sliding down a line of greatest slope of a plane inclined at \(8 ^ { \circ }\) to the horizontal. The coefficient of friction between the block and the plane is 0.2 .

1. Find the deceleration of the block.
2. Given that the initial speed of the block is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find how far the block travels.

3 A particle \(P\) moves in a straight line. It starts from a point \(O\) on the line with velocity \(1.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The acceleration of \(P\) at time \(t \mathrm {~s}\) after leaving \(O\) is \(0.8 t ^ { - 0.75 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the displacement of \(P\) from \(O\) when \(t = 16\).

4
\includegraphics[max width=\textwidth, alt={}, center]{2bd9f770-65b1-48c2-bf58-24e732bb6988-2_608_723_1247_712} A particle \(P\) has weight 10 N and is in limiting equilibrium on a rough horizontal table. The forces shown in the diagram represent the weight of \(P\), an applied force of magnitude 4 N acting on \(P\) in a direction at \(30 ^ { \circ }\) above the horizontal, and the contact force exerted on \(P\) by the table (the resultant of the frictional and normal components) of magnitude \(C \mathrm {~N}\).

1. Find the value of \(C\).
2. Find the coefficient of friction between \(P\) and the table.

5 Particles \(A\) and \(B\), of masses 0.9 kg and 0.6 kg respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley. The system is released from rest with the string taut, with its straight parts vertical and with the particles at the same height above the horizontal floor. In the subsequent motion, \(B\) does not reach the pulley.

1. Find the acceleration of \(A\) and the tension in the string during the motion before \(A\) hits the floor. After \(A\) hits the floor, \(B\) continues to move vertically upwards for a further 0.3 s .
2. Find the height of the particles above the floor at the instant that they started to move.

6 A lorry of mass 16000 kg climbs a straight hill \(A B C D\) which makes an angle \(\theta\) with the horizontal, where \(\sin \theta = \frac { 1 } { 20 }\). For the motion from \(A\) to \(B\), the work done by the driving force of the lorry is 1200 kJ and the resistance to motion is constant and equal to 1240 N . The speed of the lorry is \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(A\) and \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(B\).

1. Find the distance \(A B\). For the motion from \(B\) to \(D\) the gain in potential energy of the lorry is 2400 kJ .
2. Find the distance \(B D\). For the motion from \(B\) to \(D\) the driving force of the lorry is constant and equal to 7200 N . From \(B\) to \(C\) the resistance to motion is constant and equal to 1240 N and from \(C\) to \(D\) the resistance to motion is constant and equal to 1860 N .
3. Given that the speed of the lorry at \(D\) is \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the distance \(B C\).

7 A tractor travels in a straight line from a point \(A\) to a point \(B\). The velocity of the tractor is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\) after leaving \(A\).

1. 
   \includegraphics[max width=\textwidth, alt={}, center]{2bd9f770-65b1-48c2-bf58-24e732bb6988-4_668_1091_397_568} The diagram shows an approximate velocity-time graph for the motion of the tractor. The graph consists of two straight line segments. Use the graph to find an approximation for
   (a) the distance \(A B\),
   (b) the acceleration of the tractor for \(0 < t < 400\) and for \(400 < t < 800\).
2. The actual velocity of the tractor is given by \(v = 0.04 t - 0.00005 t ^ { 2 }\) for \(0 \leqslant t \leqslant 800\).
   (a) Find the values of \(t\) for which the actual acceleration of the tractor is given correctly by the approximate velocity-time graph in part (i). For the interval \(0 \leqslant t \leqslant 400\), the approximate velocity of the tractor in part (i) is denoted by \(v _ { 1 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   (b) Express \(v _ { 1 }\) in terms of \(t\) and hence show that \(v _ { 1 } - v = 0.00005 ( t - 200 ) ^ { 2 } - 1\).
   (c) Deduce that \(- 1 \leqslant v _ { 1 } - v \leqslant 1\).