OCR MEI M1 (Mechanics 1) 2011 January

1 An object C is moving along a vertical straight line. Fig. 1 shows the velocity-time graph for part of its motion. Initially C is moving upwards at \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and after 10 s it is moving downwards at \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). \begin{figure}[h] \end{figure} C then moves as follows.

• In the interval \(10 \leqslant t \leqslant 15\), the velocity of C is constant at \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) downwards.
• In the interval \(15 \leqslant t \leqslant 20\), the velocity of C increases uniformly so that C has zero velocity at \(t = 20\).
  1. Complete the velocity-time graph for the motion of C in the time interval \(0 \leqslant t \leqslant 20\).
  2. Calculate the acceleration of C in the time interval \(0 < t < 10\).
  3. Calculate the displacement of C from \(t = 0\) to \(t = 20\).

2 Fig. 2 shows two forces acting at A. The figure also shows the perpendicular unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) which are respectively horizontal and vertically upwards. The resultant of the two forces is \(\mathbf { F } \mathbf { N }\). \begin{figure}[h] \end{figure}

1. Find \(\mathbf { F }\) in terms of \(\mathbf { i }\) and \(\mathbf { j }\), giving your answer correct to three significant figures.
2. Calculate the magnitude of \(\mathbf { F }\) and the angle that \(\mathbf { F }\) makes with the upward vertical.

3 Two cars, P and Q, are being crashed as part of a film 'stunt'.
At the start

• P is travelling directly towards Q with a speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
• Q is instantaneously at rest and has an acceleration of \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) directly towards P .
  \(P\) continues with the same velocity and \(Q\) continues with the same acceleration. The cars collide \(T\) seconds after the start.
  1. Find expressions in terms of \(T\) for how far each of the cars has travelled since the start.

At the start, \(P\) is 90 m from \(Q\).

Show that \(T ^ { 2 } + 4 T - 45 = 0\) and hence find \(T\).

4 At time \(t\) seconds, a particle has position with respect to an origin O given by the vector $$\mathbf { r } = \binom { 8 t } { 10 t ^ { 2 } - 2 t ^ { 3 } } ,$$ where \(\binom { 1 } { 0 }\) and \(\binom { 0 } { 1 }\) are perpendicular unit vectors east and north respectively and distances are in metres.

1. When \(t = 1\), the particle is at P . Find the bearing of P from O .
2. Find the velocity of the particle at time \(t\) and show that it is never zero.
3. Determine the time(s), if any, when the acceleration of the particle is zero.

5 Fig. 5 shows two boxes, A of mass 12 kg and B of mass 6 kg , sliding in a straight line on a rough horizontal plane. The boxes are connected by a light rigid rod which is parallel to the line of motion. The only forces acting on the boxes in the line of motion are those due to the rod and a constant force of \(F \mathrm {~N}\) on each box. \begin{figure}[h] \end{figure} The boxes have an initial speed of \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and come to rest after sliding a distance of 0.375 m .

1. Calculate the deceleration of the boxes and the value of \(F\).
2. Calculate the magnitude of the force in the rod and state, with a reason, whether it is a tension or a thrust (compression).