OCR MEI M1 (Mechanics 1)

1 A ring is moving up and down a vertical pole. The displacement, \(s \mathrm {~m}\), of the ring above a mark on the pole is modelled by the displacement-time graph shown in Fig. 1. The three sections of the graph are straight lines. \begin{figure}[h] \end{figure}

1. Calculate the velocity of the ring in the interval \(0 < t < 2\) and in the interval \(2 < t < 3.5\).
2. Sketch a velocity-time graph for the motion of the ring during the 4 seconds.
3. State the direction of motion of the ring when
   (A) \(t = 1\),
   (B) \(t = 2.75\),
   (C) \(t = 3.25\).

2 Fig. 2 shows an acceleration-time graph modelling the motion of a particle. \begin{figure}[h] \end{figure} At \(t = 0\) the particle has a velocity of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive direction.

1. Find the velocity of the particle when \(t = 2\).
2. At what time is the particle travelling in the negative direction with a speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) ?

3 A cyclist starts from rest and takes 10 seconds to accelerate at a constant rate up to a speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). After travelling at this speed for 20 seconds, the cyclist then decelerates to rest at a constant rate over the next 5 seconds.

1. Sketch a velocity-time graph for the motion.
2. Calculate the distance travelled by the cyclist.

4 Fig. 1 is the velocity-time graph for the motion of a body. The velocity of the body is \(v \mathrm {~m} \mathrm {~s} { } ^ { 1 }\) at time \(t\) seconds. \begin{figure}[h] \end{figure} The displacement of the body from \(t = 0\) to \(t = 100\) is 1400 m . Find the value of \(V\).