6
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5b10afa1-1c45-4370-a0e6-ad8fd626df9a-4_664_969_264_589}
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\caption{Fig. 1}
\end{figure}
A cyclist \(P\) travels along a straight road starting from rest at \(A\) and accelerating at \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) up to a speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). He continues at a constant speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), passing through the point \(B 20 \mathrm {~s}\) after leaving \(A\). Fig. 1 shows the ( \(t , v\) ) graph of \(P\) 's journey for \(0 \leqslant t \leqslant 20\). Find
- the time for which \(P\) is accelerating,
- the distance \(A B\).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5b10afa1-1c45-4370-a0e6-ad8fd626df9a-4_607_937_1420_605}
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\caption{Fig. 2}
\end{figure}
Another cyclist \(Q\) travels along the same straight road in the opposite direction. She starts at rest from \(B\) at the same instant that \(P\) leaves \(A\). Cyclist \(Q\) accelerates at \(2 \mathrm {~ms} ^ { - 2 }\) up to a speed of \(8 \mathrm {~ms} ^ { - 1 }\) and continues at a constant speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), passing through the point \(A 20 \mathrm {~s}\) after leaving \(B\). Fig. 2 shows the \(( t , x )\) graph of \(Q\) 's journey for \(0 \leqslant t \leqslant 20\), where \(x\) is the displacement of \(Q\) from \(A\) towards \(B\). - Sketch a copy of Fig. 1 and add to your copy a sketch of the ( \(t , v\) ) graph of \(Q\) 's journey for \(0 \leqslant t \leqslant 20\).
- Sketch a copy of Fig. 2 and add to your copy a sketch of the \(( t , x )\) graph of \(P\) 's journey for \(0 \leqslant t \leqslant 20\).
- Find the value \(t\) at the instant that \(P\) and \(Q\) pass each other.
\includegraphics[max width=\textwidth, alt={}, center]{5b10afa1-1c45-4370-a0e6-ad8fd626df9a-5_447_739_269_703}
The upper edge of a smooth plane inclined at \(70 ^ { \circ }\) to the horizontal is joined to an edge of a rough horizontal table. Particles \(A\) and \(B\), of masses 0.3 kg and 0.2 kg respectively, are attached to the ends of a light inextensible string. The string passes over a smooth pulley which is fixed at the top of the smooth inclined plane. Particle \(A\) is held in contact with the rough horizontal table and particle \(B\) is in contact with the smooth inclined plane with the string taut (see diagram). The coefficient of friction between \(A\) and the horizontal table is 0.4 . Particle \(A\) is released from rest and the system starts to move.
- Find the acceleration of \(A\) and the tension in the string.
The string breaks when the speed of the particles is \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
- Assuming \(A\) does not reach the pulley, find the distance travelled by \(A\) after the string breaks.
- Assuming \(B\) does not reach the ground before \(A\) stops, find the distance travelled by \(B\) from the time the string breaks to the time that \(A\) stops.