OCR M1 (Mechanics 1) 2009 January

1
\includegraphics[max width=\textwidth, alt={}, center]{470e70de-66ba-4dcc-a205-0c92f29471b1-2_227_878_269_635} A particle \(P\) of mass 0.5 kg is travelling with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a smooth horizontal plane towards a stationary particle \(Q\) of mass \(m \mathrm {~kg}\) (see diagram). The particles collide, and immediately after the collision \(P\) has speed \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(Q\) has speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Given that both particles are moving in the same direction after the collision, calculate \(m\).
2. Given instead that the particles are moving in opposite directions after the collision, calculate \(m\).

2 A trailer of mass 500 kg is attached to a car of mass 1250 kg by a light rigid horizontal tow-bar. The car and trailer are travelling along a horizontal straight road. The resistance to motion of the trailer is 400 N and the resistance to motion of the car is 900 N . Find both the tension in the tow-bar and the driving force of the car in each of the following cases.

1. The car and trailer are travelling at constant speed.
2. The car and trailer have acceleration \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

3
\includegraphics[max width=\textwidth, alt={}, center]{470e70de-66ba-4dcc-a205-0c92f29471b1-2_570_679_1512_731} Three horizontal forces act at the point \(O\). One force has magnitude 7 N and acts along the positive \(x\)-axis. The second force has magnitude 9 N and acts along the positive \(y\)-axis. The third force has magnitude 5 N and acts at an angle of \(30 ^ { \circ }\) below the negative \(x\)-axis (see diagram).

1. Find the magnitudes of the components of the 5 N force along the two axes.
2. Calculate the magnitude of the resultant of the three forces. Calculate also the angle the resultant makes with the positive \(x\)-axis.

4
\includegraphics[max width=\textwidth, alt={}, center]{470e70de-66ba-4dcc-a205-0c92f29471b1-3_200_897_269_625} A block of mass 3 kg is placed on a horizontal surface. A force of magnitude 20 N acts downwards on the block at an angle of \(30 ^ { \circ }\) to the horizontal (see diagram).

1. Given that the surface is smooth, calculate the acceleration of the block.
2. Given instead that the block is in limiting equilibrium, calculate the coefficient of friction between the block and the surface.

5 A car is travelling at \(13 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along a straight road when it passes a point \(A\) at time \(t = 0\), where \(t\) is in seconds. For \(0 \leqslant t \leqslant 6\), the car accelerates at \(0.8 t \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Calculate the speed of the car when \(t = 6\).
2. Calculate the displacement of the car from \(A\) when \(t = 6\).
3. Three \(( t , x )\) graphs are shown below, for \(0 \leqslant t \leqslant 6\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{470e70de-66ba-4dcc-a205-0c92f29471b1-3_382_458_1366_340} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{470e70de-66ba-4dcc-a205-0c92f29471b1-3_382_460_1366_881} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{470e70de-66ba-4dcc-a205-0c92f29471b1-3_384_461_1366_1420} \captionsetup{labelformat=empty} \caption{Fig. 3}
   \end{figure} (a) State which of these three graphs is most appropriate to represent the motion of the car.
   (b) For each of the two other graphs give a reason why it is not appropriate to represent the motion of the car.

6 Small parcels are being loaded onto a trolley. Initially the parcels are 2.5 m above the trolley.

1. A parcel is released from rest and falls vertically onto the trolley. Calculate
   (a) the time taken for a parcel to fall onto the trolley,
   (b) the speed of a parcel when it strikes the trolley.
2. 
   \includegraphics[max width=\textwidth, alt={}, center]{470e70de-66ba-4dcc-a205-0c92f29471b1-4_327_723_603_751} Parcels are often damaged when loaded in the way described, so a ramp is constructed down which parcels can slide onto the trolley. The ramp makes an angle of \(60 ^ { \circ }\) to the vertical, and the coefficient of friction between the ramp and a parcel is 0.2 . A parcel of mass 2 kg is released from rest at the top of the ramp (see diagram). Calculate the speed of the parcel after sliding down the ramp.

7
\includegraphics[max width=\textwidth, alt={}, center]{470e70de-66ba-4dcc-a205-0c92f29471b1-4_227_901_1352_623} Two particles \(P\) and \(Q\) have masses 0.7 kg and 0.3 kg respectively. \(P\) and \(Q\) are simultaneously projected towards each other in the same straight line on a horizontal surface with initial speeds of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively (see diagram). Before \(P\) and \(Q\) collide the only horizontal force acting on each particle is friction and each particle decelerates at \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The particles coalesce when they collide.

1. Given that \(P\) and \(Q\) collide 2 s after projection, calculate the speed of each particle immediately before the collision, and the speed of the combined particle immediately after the collision.
2. Given instead that \(P\) and \(Q\) collide 3 s after projection,
   (a) sketch on a single diagram the \(( t , v )\) graphs for the two particles in the interval \(0 \leqslant t < 3\),
   (b) calculate the distance between the two particles at the instant when they are projected.